# IGraph/M

the igraph interface for Mathematica
version 0.6.5 (December 21, 2022)

This notebook can be opened using the command IGDocumentation[] or through the Documentation Centre. It cannot be saved, so feel free to edit and evaluate input cells, and experiment!

The documentation is currently incomplete. Contributions are very welcome!

# Introduction

IGraph/M provides a Mathematica interface to the popular igraph network analysis package, as well as many other functions for working with graphs in Mathematica. The purpose of IGraph/M is not to replace Mathematica’s built-in graph theory functionality, but to complement it. Thus the IGraph/M interface is designed to interoperate seamlessly with built-in functions and datatypes, while also being familiar to users of other igraph interfaces (R, Python or C).

The full igraph functionality is not yet exposed. Priority is given to functionality that is not currently built into Mathematica. While many of the functions that IGraph/M provides overlap with built-in ones, like IGBetweenness and BetweeneessCentrality, there are usually some relevant differences. For example, IGBetweenness uses edge weights, while the built-in function BetweennessCentrality does not.

# Basic usage

The package can be loaded using

Needs["IGraphM"]

The list of included functions can be queried with the command below. Notice that their names always have the IG prefix. Click on the name of a function to see its usage message.

?IGraphM*

Or just type a question mark followed by the symbol’s name:

?IGVersion

IGVersion[]
"IGraph/M 0.6.5 (December 21, 2022)
igraph 0.9.10-23-g5635203bd (Dec 21 2022)
Mac OS X x86 (64-bit)"

IGraph/M functions work directly with Mathematica’s built-in Graph datatype. No new special graph datatype is introduced.

Let’s take a look at a few examples. Let us first generate a graph using the built-in functions of Mathematica.

SeedRandom[42];
g = RandomGraph[BarabasiAlbertGraphDistribution[100, 2]]

We can compute the betweenness centrality of each vertex either using IGraph/M, …

IGBetweenness[g]
{1118.26, 1058.15, 540.601, 127.365, 1175.53, 678.175, 206.929, \
128.576, 204.019, 535.316, 487.858, 391.669, 0., 135.039, 0., \
52.5324, 104.28, 12.2286, 75.8798, 110.155, 68.8282, 13.9095, \
46.4209, 99.3299, 0., 168.196, 213.871, 358.855, 0., 64.9572, \
5.12619, 102.369, 17.978, 15.569, 95.7266, 8.45843, 25.4984, 13.0274, \
0., 71.2012, 47.2895, 32.4444, 8.20833, 0., 27.0286, 10.9357, \
4.60238, 0., 14.7095, 24.7944, 79.125, 7.38301, 22.0817, 43.9635, \
11.7135, 10.9952, 40.8782, 11.2429, 0., 60.0431, 9.36667, 32.4529, \
85.4487, 100.431, 15.205, 93.2876, 60.0548, 9.2, 0., 0., 10.512, \
9.37438, 8.42222, 45.7937, 3.61667, 9.23333, 53.3897, 11.4012, \
22.0959, 5.24091, 10.2647, 8.66017, 9.97438, 11.0429, 15.8765, \
12.7798, 0., 30.1744, 0., 0., 4.0373, 9.7, 1., 10.4883, 0., 0., \
13.7861, 13.8594, 1.7, 2.80952}

… or using Mathematica’s built-ins, and obtain the same result.

BetweennessCentrality[g]
{1118.26, 1058.15, 540.601, 127.365, 1175.53, 678.175, 206.929, \
128.576, 204.019, 535.316, 487.858, 391.669, 0., 135.039, 0., \
52.5324, 104.28, 12.2286, 75.8798, 110.155, 68.8282, 13.9095, \
46.4209, 99.3299, 0., 168.196, 213.871, 358.855, 0., 64.9572, \
5.12619, 102.369, 17.978, 15.569, 95.7266, 8.45843, 25.4984, 13.0274, \
0., 71.2012, 47.2895, 32.4444, 8.20833, 0., 27.0286, 10.9357, \
4.60238, 0., 14.7095, 24.7944, 79.125, 7.38301, 22.0817, 43.9635, \
11.7135, 10.9952, 40.8782, 11.2429, 0., 60.0431, 9.36667, 32.4529, \
85.4487, 100.431, 15.205, 93.2876, 60.0548, 9.2, 0., 0., 10.512, \
9.37438, 8.42222, 45.7937, 3.61667, 9.23333, 53.3897, 11.4012, \
22.0959, 5.24091, 10.2647, 8.66017, 9.97438, 11.0429, 15.8765, \
12.7798, 0., 30.1744, 0., 0., 4.0373, 9.7, 1., 10.4883, 0., 0., \
13.7861, 13.8594, 1.7, 2.80952}

Let us now assign weights to the edges. Many IGraph/M functions, including IGBetweenness, support edge weights.

wg = SetProperty[g, EdgeWeight -> RandomReal[1, EdgeCount[g]]];
IGBetweenness[wg]
{1569., 1509., 697., 506., 1510., 948., 173., 0., 106., 827., 663., \
379., 0., 318., 0., 360., 0., 0., 83., 129., 1., 0., 227., 582., 0., \
91., 236., 213., 0., 60., 0., 334., 1., 53., 549., 0., 0., 0., 0., \
10., 0., 0., 0., 0., 68., 68., 17., 357., 27., 16., 80., 0., 0., 0., \
437., 0., 0., 0., 52., 22., 0., 0., 62., 139., 93., 187., 1., 7., 0., \
0., 0., 16., 0., 69., 10., 98., 0., 1., 4., 21., 0., 0., 0., 0., 0., \
0., 0., 43., 0., 0., 98., 0., 0., 0., 0., 0., 0., 63., 25., 4.}

Notice that Mathematica 13.0 does not include functionality to compute weighted vertex betweenness. The built-in function BetweennessCentrality[] ignores the weights.

BetweennessCentrality[wg]
{1118.26, 1058.15, 540.601, 127.365, 1175.53, 678.175, 206.929, \
128.576, 204.019, 535.316, 487.858, 391.669, 0., 135.039, 0., \
52.5324, 104.28, 12.2286, 75.8798, 110.155, 68.8282, 13.9095, \
46.4209, 99.3299, 0., 168.196, 213.871, 358.855, 0., 64.9572, \
5.12619, 102.369, 17.978, 15.569, 95.7266, 8.45843, 25.4984, 13.0274, \
0., 71.2012, 47.2895, 32.4444, 8.20833, 0., 27.0286, 10.9357, \
4.60238, 0., 14.7095, 24.7944, 79.125, 7.38301, 22.0817, 43.9635, \
11.7135, 10.9952, 40.8782, 11.2429, 0., 60.0431, 9.36667, 32.4529, \
85.4487, 100.431, 15.205, 93.2876, 60.0548, 9.2, 0., 0., 10.512, \
9.37438, 8.42222, 45.7937, 3.61667, 9.23333, 53.3897, 11.4012, \
22.0959, 5.24091, 10.2647, 8.66017, 9.97438, 11.0429, 15.8765, \
12.7798, 0., 30.1744, 0., 0., 4.0373, 9.7, 1., 10.4883, 0., 0., \
13.7861, 13.8594, 1.7, 2.80952}

Let us delete the minimum feedback edge set to obtain an acyclic graph:

acg = EdgeDelete[g, IGFeedbackArcSet[g]]

And try out a few of igraph’s layout algorithms.

{IGLayoutGraphOpt[acg], IGLayoutKamadaKawai[acg],
IGLayoutFruchtermanReingold[acg]}

Layout functions typically have many options to tune:

Options[IGLayoutGraphOpt]
{"MaxIterations" -> 500, "NodeCharge" -> 0.001, "NodeMass" -> 30,
"SpringLength" -> 0, "SpringConstant" -> 1, "MaxStepMovement" -> 5,
"Continue" -> False, "Align" -> True}

Increasing the number of iterations will usually improve the result.

IGLayoutGraphOpt[acg, "MaxIterations" -> 5000]

A final note

Please refer to the usage messages for information on how to use each function. For more information on the meaning of various function options, the algorithms used by the functions, references, etc. please refer to the C/igraph documentation. The igraph documentation provides article references for most nontrivial algorithms.

The following sections provide general information on each functionality area, and show common usage patterns.

# Graph creation

All the graph creation functions in IGraph/M take any standard Mathematica Graph option such as VertexLabels, EdgeLabels, VertexStyle, GraphStyle, PlotTheme, etc.

IGLCF[{5, -5}, 7, GraphStyle -> "SmallNetwork"]

## Deterministic graph generators▴

### IGShorthand▴

IGShorthand provides an easy way to create small graphs from a simple and quick-to-type notation.

?IGShorthand

The available options are:

• SelfLoops -> True keeps self-loops in the graph.

• MultiEdges -> True keeps parallel edges in the graph.

Construct a cycle graph.

IGShorthand["1-2-3-4-1"]

Vertex labels are shown by default. They can be turned off using VertexLabels -> None.

IGShorthand["1-2-3-1", VertexLabels -> None]

The interpretation of - as directed or undirected is controlled by the DirectedEdges option.

IGShorthand["1-2-3-1", DirectedEdges -> True]

Directed edges can be input using ->, <- or <->.

IGShorthand["Jim -> Suzy <- Joe"]

<-> is interpreted as a pair of directed edges.

IGShorthand["1<->2->3"]

Mixed graphs, containing both directed and undirected edges, are supported. Note that mixed graphs are not allowed as input to most IGraph/M functions.

IGShorthand["1-2<-3"]

Disconnected components are separated by commas.

IGShorthand["1, 2-3, 4-5-6"]

Groups of vertices can be given using the colon separator. Edges will be connected to each vertex in the group. This makes it easy to specify a complete graph …

IGShorthand["A:B:C:D:E -- A:B:C:D:E"]

… or a complete bipartite graph.

IGLayoutBipartite@IGShorthand["a:b:c - 1:2:3:4"]

Vertex names are taken as strings, except when they can be interpreted as an integer.

IGShorthand["xyz - 137"] // VertexList // InputForm
{"xyz", 137}

Spaces are allowed in vertex names, and edges can be specified using any number of - characters.

IGShorthand["Sophus Lie --- Camille Jordan"]

Self-loops and parallel edges are removed by default because these are often created as an undesired by-product of vertex groups. They can be re-enabled using the SelfLoops or MultiEdges options when desired.

IGShorthand["1:2:3 - 1:2:3", SelfLoops -> True]

IGShorthand["1:2:3 - 1:2:3", SelfLoops -> True, MultiEdges -> True]

IGShorthand["1-2-1-3", MultiEdges -> True]

The vertex order will follow the order of appearance of vertices in the input string. To control the order, simply list vertices at the beginning of the shorthand specification.

IGShorthand["4-3-1-2-4"] // VertexList
{4, 3, 1, 2}
IGShorthand["1,2,3,4, 4-3-1-2-4"] // VertexList
{1, 2, 3, 4}

Create an interactive graph editor that dynamically visualizes betweenness.

Manipulate[
IGShorthand[spec, VertexSize -> {"Scaled", 0.06},
EdgeStyle -> Gray] //
IGVertexMap[ColorData["NeonColors"],
VertexStyle -> IGBetweenness/*Rescale],
{{spec, "1-2-3-1-4"},
InputField[#, String, ContinuousAction -> True] &},
Initialization :> Needs["IGraphM"]
]

### IGEmptyGraph▴

?IGEmptyGraph

IGEmptyGraph is a convenience function for creating graphs with no edges.

Create a null graph.

IGEmptyGraph[] // VertexCount
0

Create an empty graph on 15 vertices.

IGEmptyGraph[15]

The built-in EmptyGraphQ returns True for these graphs.

EmptyGraphQ[%]
True

### IGLCF▴

?IGLCF

creates a graph based on the LCF notation.

The Möbius–Kantor graph is [5, -5]^8.

IGLCF[{5, -5}, 8, GraphStyle -> "DiagramGreen"]

The Pappus graph is [5, 7, -7, 7, -7, -5]^3.

IGLCF[{5, 7, -7, 7, -7, -5}, 3, GraphStyle -> "ThickEdge"]

The cuboctahedral graph is [4, 2]^6.

IGLayoutKamadaKawai3D@IGLCF[{4, 2}, 6]

### IGChordalRing▴

?IGChordalRing

IGChordalRing[n, w] constructs an extended chordal ring based on the offset specification vector or matrix $$w$$ as follows:

1. It creates a cycle graph (i.e. ring) on $$n$$ vertices.

2. For each vertex $$i$$ on the ring, it adds a chord to a vertex $$w[[(i \bmod p)]]$$ steps ahead counter-clockwise on the ring.

3. If $$w$$ is a matrix, the procedure is carried out for each row.

The number of vertices $$n$$ must be an integer multiple of the number of columns in the matrix $$w$$.

The available options are:

• DirectedEdges -> True creates a graph with directed edges.

• SelfLoops -> False prevents the creation of self-loops.

• MultiEgdes -> False prevents the creation of multi-edges.

Create an extended chordal graph.

IGChordalRing[15, {3, 4, 8}, GraphStyle -> "Business"]

Negative offsets are allowed.

IGChordalRing[15, {{3, 4, 8}, {-3, -4, -8}}]

IGChordalGraph may create self-loops or multi-edges. This can be prevented by setting the SelfLoops or MultiEdges options to False.

IGChordalRing[15, {{3, 4, 8}, {-3, -4, -8}}, MultiEdges -> False]

Create a chordal graph with directed edges.

IGChordalRing[8, {2, 3}, DirectedEdges -> True,
GraphStyle -> "DiagramGold"]

Colour the chords of the ring based on which entry of the $$w$$ vector they correspond to.

w = {2, 3, 4};
IGChordalRing[12, w, GraphStyle -> "ThickEdge",
EdgeStyle -> Opacity[1/2]] // IGEdgeMap[
ColorData[97],
EdgeStyle -> Function[g,
Table[
If[i <= VertexCount[g], 0, Mod[i, Length[w], 1]], {i,
EdgeCount[g]}]
]
]

### IGSquareLattice▴

?IGSquareLattice

creates a square lattice graph of the given dimensions. The available options are:

• "Radius" controls the size of the neighbourhood within which vertices will be connected.

• "Periodic" -> True creates a periodic lattice.

• "Mutual" -> True inserts directed edges in both directions when DirectedEdges -> True is used.

In previous versions, IGSquareLattice was called IGMakeLattice. This name can still be used as a synonym for the sake of backwards compatibility, however, it will be removed in a future version.

To create other types of lattices, see IGTriangleLattice and IGLatticeMesh.

IGSquareLattice[{3, 4}, GraphStyle -> "VintageDiagram"]

IGSquareLattice[{10, 10}, "Periodic" -> True]

Graph3D@IGSquareLattice[{5, 4, 3}, GraphStyle -> "Prototype"]

Graph3D@IGSquareLattice[{2, 5}, DirectedEdges -> True,
"Periodic" -> True, PlotTheme -> "NeonColor"]

### IGTriangularLattice▴

?IGTriangularLattice

IGTriangularLattice can create a triangular grid graph in the shape of a triangle or a rectangle. To generate other types of lattices, see IGSquareLattice and IGLatticeMesh.

The available options are:

• DirectedEdges -> True creates a directed graph.

• "Periodic" -> True creates a periodic lattice.

Generate a triangular lattice on an equilateral triangle with 6 vertices along each of its edges.

IGTriangularLattice[6, GraphStyle -> "SmallNetwork"]

Create a directed triangle lattice on a rectangle. Notice the vertex labelling and that the arrows are oriented from smaller index vertices to larger index ones, making this an acyclic graph.

IGTriangularLattice[{4, 4}, DirectedEdges -> True,
VertexShapeFunction -> "Name", PerformanceGoal -> "Quality"]

Create a triangle lattice and colour its vertices.

IGTriangularLattice[{8, 6}, VertexSize -> Large, EdgeStyle -> Gray] //
IGVertexMap[ColorData[98], VertexStyle -> IGMinimumVertexColoring]

Take a hexagonal subgraph of a triangle lattice.

g = IGTriangularLattice[13];
center = First@GraphCenter[g];
VertexDelete[g,
]

Create a periodic (i.e. toroidal topology) triangle lattice.

Graph3D@IGTriangularLattice[{24, 8}, "Periodic" -> True]

### IGKaryTree▴

?IGKaryTree

The available options are:

• DirectedEdges -> True creates a directed tree.
IGKaryTree[15]

IGKaryTree[10, 3, DirectedEdges -> True]

### IGSymmetricTree▴

?IGSymmetricTree

IGSymmetricTree creates a tree where successive layers (i.e. vertices at the same distance from the root) have the specified number of children.

Create a tree where the root has 4 children, its children have 3 children, and so on.

IGSymmetricTree[{4, 3, 2, 1}]

Create a directed tree.

IGSymmetricTree[{4, 2}, DirectedEdges -> True]

IGSymmetricTree is guaranteed to label vertices in breadth-first order. Deeper layers have higher integer labels.

IGSymmetricTree[{3, 3}, GraphStyle -> "DiagramBlue"]

### IGBetheLattice▴

?IGBetheLattice

A Bethe lattice, also called a regular tree, is an infinite tree in which all vertices have the same degree. IGBetheLattice[n, k] computes the first n layers of such a tree. Each non-leaf vertex will have degree k. The default degree is 3.

IGBetheLattice differs from CompleteKaryTree in that the degree of the root will be the same as the degree of other non-lead nodes.

IGBetheLattice[5, GraphStyle -> "Prototype", VertexSize -> Large]

Generate a tree where non-leaf nodes have total degree 5, and use directed edges.

IGBetheLattice[5, 4, DirectedEdges -> True]

Colour vertices based on their distance from the root (i.e. the “layer” they are part of).

IGVertexMap[
ColorData[68],
VertexStyle -> (First@IGDistanceMatrix[#, {1}] &),
IGBetheLattice[5, GraphStyle -> "BasicBlack", VertexSize -> 0.4]
]

Visualize the nested line graph of a degree-4 regular tree.

Graph3D@Nest[LineGraph, IGBetheLattice[5, 4], 2]

### IGFromPrufer▴

?IGFromPrufer

A Prüfer sequence is a unique representation of an $$n$$-vertex labelled tree as $$n-2$$ integers between $$1$$ and $$n$$.

IGFromPrufer[{1, 1, 2, 2}, VertexLabels -> "Name"]

Use IGToPrufer to convert a tree back to its Prüfer sequence.

IGToPrufer[%]
{1, 1, 2, 2}

Generate all labelled trees on 4 nodes:

IGFromPrufer[#, VertexLabels -> "Name"] & /@ Tuples[Range[4], {2}]

Of these, only two are non-isomorphic.

DeleteDuplicates[CanonicalGraph /@ %]

### IGExpressionTree▴

?IGExpressionTree

IGExpressionTree constructs the tree graph corresponding to an arbitrary Mathematica expression. The vertices of the tree will be the positions of the corresponding subexpressions.

IGExpressionTree takes all standard Graph options. The VertexLabels option takes the following special values:

• VertexLabels -> "Head" labels branch vertices with the Head of the corresponding subexpression and leaf vertices with the corresponding atomic expression.

• VertexLabels -> "Subexpression" labels vertices with the corresponding subexpression.

• VertexLabels -> "Name" labels vertices with their name, i.e. the position of the corresponding subexpression.

• VertexLabels -> None uses no labels.

IGExpressionTree constructs a graph corresponding to the structure of a Mathematica expression.

tree = IGExpressionTree[expr = 1 + x^2]

The expression tree is similar to what TreeForm displays, but unlike TreeForm’s output, it is a Graph object that works with all graph functions.

TreeForm[expr]

The vertex names are the position specifications of the corresponding subexpressions.

VertexList[tree]
{{1}, {2, 1}, {2, 2}, {2}, {}}
Extract[expr, %]
{1, x, 2, x^2, 1 + x^2}

Place the vertex labels in the centre and construct an undirected graph.

IGExpressionTree[x^2 + y^2,
GraphStyle -> "DiagramGold",
VertexLabels -> Placed["Head", Center], VertexSize -> Large
]

Create an undirected graph, labelled with subexpressions.

IGExpressionTree[Normal[Sin[x] + O[x]^5], DirectedEdges -> False,
VertexLabels -> "Subexpression"]

Certain trees are easier to construct through their corresponding nested expression.

IGExpressionTree[#, VertexLabels -> "Index"] & /@ Groupings[5, 3]

An equivalent of IGSymmetricTree can be easily implemented using IGExpressionTree.

IGExpressionTree[ConstantArray[1, {3, 4, 5}], VertexLabels -> None,
GraphLayout -> "RadialEmbedding"]

Define a tree through a substitution system.

IGExpressionTree[
Nest[ReplaceAll[{0 -> {0, 1}, 1 -> {0}}], {0, 1}, 3],
VertexLabels -> None, GraphStyle -> "VibrantColor"
]

To format each node so that it fits a label, it is necessary to set an explicit VertexShapeFunction.

IGExpressionTree[First@Roots[x^2 + a x + 1 == 0, x],
VertexLabels -> "Subexpression",
PerformanceGoal -> "Quality",
ImageSize -> 280
] //
IGVertexMap[
Function[e, Inset[Panel[e], #1] &],
VertexShapeFunction -> IGVertexProp[VertexLabels]
] // RemoveProperty[#, VertexLabels] &

### IGCompleteGraph▴

?IGCompleteGraph

The available options are:

• DirectedEdges -> True creates a directed graph.

• SelfLoops -> True includes self-loops.

Create an undirected complete graph with loops.

IGCompleteGraph[5, SelfLoops -> True]

Create a directed complete graph with loops.

IGCompleteGraph[6, SelfLoops -> True, DirectedEdges -> True]

Create a list of complete graphs starting with the null graph.

IGCompleteGraph /@ Range[0, 3]

Create a complete graph on the given vertices.

IGCompleteGraph[{"a", "b", "c", "d"}, GraphStyle -> "DiagramBlue"]

### IGCompleteAcyclicGraph▴

?IGCompleteAcyclicGraph

Create a complete acyclic directed graph on 5 vertices.

IGCompleteAcyclicGraph[5]

Create a complete acyclic graph on the given vertices. The directed edges always run from vertices that appear earlier in the list to those that appear later.

IGCompleteAcyclicGraph[CharacterRange["a", "f"],
GraphStyle -> "DiagramGold"]

### IGKautzGraph▴

?IGKautzGraph

The vertices of the Kautz graph $$K_m^n$$ are strings of length $$n+1$$, composed of $$m+1$$ distinct symbols, with the restriction that two adjacent symbols in the string may not be the same. A vertex $$s_1s_2\text{\ldots s}_ns_{n+1}$$ connects to all other vertices of the form $$s_2\text{\ldots s}_{n+1}x$$, where $$x$$ can be any symbol distinct from $$s_{n+1}$$.

The Kautz graph $$K_m^n$$ has $$(m+1) m^n$$ vertices, with each vertex having in-degree and out-degree $$m$$. Therefore, it has $$(m+1) m^{n+1}$$ edges.

VertexCount@IGKautzGraph[3, 2] == (3 + 1)*3^2
True
VertexOutDegree@IGKautzGraph[3, 2]
{3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, \
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

The line graph of Kautz graph $$K_m^n$$ is $$K_m^{n+1}$$.

IGIsomorphicQ[
LineGraph@IGKautzGraph[2, 2],
IGKautzGraph[2, 3]
]
True

Visualize the Kautz graph $$K_2^3$$ on 3 characters with string length 4 in three dimensions.

Graph3D@IGKautzGraph[2, 3]

Label the vertices of the Kautz graph on 3 characters with string length 2.

labels =
StringJoin /@ DeleteCases[Tuples[{"A", "B", "C"}, {2}], {c_, c_}];
IGKautzGraph[2, 1,
VertexLabels -> Thread[Range[6] -> (Placed[#, Center] &) /@ labels],
VertexSize -> Large, VertexShapeFunction -> "Capsule",
PerformanceGoal -> "Quality",
PlotTheme -> "CoolColor", VertexLabelStyle -> White
]

### IGDeBruijnGraph▴

?IGDeBruijnGraph

IGDeBruijnGraph[3, 2, GraphStyle -> "BackgroundBlue",
EdgeStyle -> Thick]

### IGRealizeDegreeSequence▴

?IGRealizeDegreeSequence

This function constructs an undirected graph with the given degree sequence, or a directed graph with the given in- and out-degree sequences. For constructing simple graphs, it uses the Havel–Hakimi algorithm (undirected case) or Kleitman–Wang algorithm (directed case). These algorithms work by selecting a “hub” vertex, and connecting up all its free (out-)degrees to other vertices with the largest degrees. In the directed case, the “largest” degrees are determined by lexicographic ordering of (in, out)-degree pairs. For constructing a loop-free multigraph, a similar algorithm is used, but the hub is connected to only one other vertex in each step, instead of to as many as its degree. If self-loops are allowed as well, the same algorithm is used, and if a loop-free result cannot be created, an appropriate number of self-loops will be added to the very last hub. The order in which hub vertices are selected is controlled by the Method option.

To randomly sample multiple realizations of a degree sequence, use IGDegreeSequenceGame, or first create a single graph with IGRealizeDegreeSequence, then randomly rewire it using IGRewire.

The available options are:

• SelfLoops -> True allows creating self-loops (disallowed by default).

• MultiEdges -> True allows creating multi-edges (disallowed by default).

• The Method option controls the order in which hub vertices are chosen.

Available Method option values:

• "SmallestFirst" will choose a smallest-degree vertex in each step of the algorithm (this is the default). This results in a disassortative network. In the undirected case, this method is guaranteed to construct a connected graph, if one exists, both when constructing simple graphs or multigraphs. See Horvát and Modes (2020), as well as http://szhorvat.net/pelican/hh-connected-graphs.html for the proof. In the directed case, it tends to construct weakly-connected graphs, but this is not guaranteed.

• "LargestFirst" will choose a largest-degree vertex. This results in an assortative network. This method tends to construct disconnected graphs. This is the most common variant of the Havel–Hakimi algorithm implemented in other packages.

• "Index" will choose vertices in the order of their indices.

degseq = VertexDegree@IGGiantComponent@RandomGraph[{50, 50}]
{3, 4, 4, 4, 2, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 5, 2, 4, 3, 1, 2, 5, 4, \
3, 1, 3, 2, 1, 2, 2, 3, 1, 3, 1, 1, 1, 1, 1}

The default Method -> "SmallestFirst" tends to create highly disassortative graphs. The result is guaranteed to be connected if the input degree sequence was potentially connected.

IGRealizeDegreeSequence[degseq]

N@GraphAssortativity[%]
-0.524347

Method -> "LargestFirst" tends to create highly assortative disconnected graphs.

IGRealizeDegreeSequence[degseq, Method -> "LargestFirst"]

N@GraphAssortativity[%]
0.904728

Allow parallel edges.

IGRealizeDegreeSequence[degseq, MultiEdges -> True,
Method -> #] & /@ {"SmallestFirst", "LargestFirst", "Index"}

Create a directed graph.

g = IGBarabasiAlbertGame[50, 1]

indegseq = VertexInDegree[g];
outdegseq = VertexOutDegree[g];
IGRealizeDegreeSequence[outdegseq, indegseq]

Verify that the degrees sequences of the result match the input to the function.

{VertexOutDegree[%] == outdegseq, VertexInDegree[%] == indegseq}
{False, False}

#### References

• S. L. Hakimi, On Realizability of a Set of Integers as Degrees of the Vertices of a Linear Graph, Journal of the Society for Industrial and Applied Mathematics 10, 3 (1962). http://eudml.org/doc/19050

• V. Havel, Poznámka O Existenci Konečných Grafů (A Remark on the Existence of Finite Graphs), Časopis Pro Pěstování Matematiky 80, 4 (1955). https://www.jstor.org/stable/2098746

• D. J. Kleitman and D. L. Wang, Algorithms for Constructing Graphs and Digraphs with Given Valences and Factors, Discrete Mathematics 6, 1 (1973). https://doi.org/10.1016/0012-365X(73)90037-X

• Sz. Horvát and C. D. Modes, Connectivity matters: Construction and exact random sampling of connected graphs (2020). https://arxiv.org/abs/2009.03747

### IGGraphAtlas▴

?IGGraphAtlas

This function is provided for convenience for those who have the book An Atlas of Graphs by Ronald C. Read and Robin J. Wilson, and for those who wish to replicate results obtained with other packages that include this database. For all other purposes, use Mathematica’s built-in GraphData function.

Retrieve graph number 789:

IGGraphAtlas[789]

### IGFamousGraph▴

?IGFamousGraph

This function returns various “named” graphs from the igraph C library’s built-in database. It is included in IGraph/M for compatibility with other igraph interfaces. It is recommended to use Mathematica’s built-in GraphData function instead. See the documentation of the igraph C library for the list of supported graph names.

Create Krackhardt’s kite graph:

g = IGFamousGraph["Krackhardt_Kite"]

Krackhardt’s kite was devised to illustrate the difference between various centrality measures:

#[g] & /@ {
IGVertexMap[0.1 # &, VertexSize -> IGBetweenness],
IGVertexMap[# &, VertexSize -> IGCloseness],
IGVertexMap[# &, VertexSize -> IGHarmonicCentrality],
IGVertexMap[0.1 # &, VertexSize -> VertexDegree]
}

### IGFromNauty▴

?IGFromNauty

IGFromNauty converts a Graph6, Digraph6 or Sparse6 string to a graph. These formats originate with the nauty suite of programs and are supported by many other graph theory software.

Interpret a Graph6 string.

IGFromNauty["GrHOk"]

Interpret a Sparse6 string. These start with a : character.

IGFromNauty[":IESgTlVF"]

Interpret a Digraph6 string. These start with a & character.

IGFromNauty["&FKB?oMB_W?"]

IGFromNauty does not support headers or whitespace in the string. To handle these, or to interpret a multiline string, use IGImportString[…, "Nauty"].

IGFromNauty[">>graph6<<DYw"]

$Failed IGImportString[ ">>graph6<<DYw Dhs Dxo DVW ", "Nauty"] ## Random graph generators▴ These graph creation functions use igraph’s random graph generator, which can be seeded using IGSeedRandom. ?IG*Game* ### Basic random graphs▴ ?IGErdosRenyiGameGNM ?IGErdosRenyiGameGNP IGErdosRenyiGameGNM uniformly samples graphs with $$n$$ vertices and $$m$$ edges. This random graph model is known as the Erdős–Rényi $$G(n,m)$$ model. In IGErdosRenyiGameGNP, each edge is present with the same and independent probability. This model is known as the Erdős–Rényi $$G(n,p)$$ model or Gilbert model. The available options are: • DirectedEdges -> True produces a directed graph. • SelfLoops -> True allows self-loops. Create a random graph with 10 vertices and 20 edges. IGErdosRenyiGameGNM[10, 20, GraphStyle -> "VintageDiagram"] Create a directed graph and allow self-loops. IGErdosRenyiGameGNM[10, 35, DirectedEdges -> True, SelfLoops -> True] Insert each edge with a probability of 20%. IGErdosRenyiGameGNP[20, 0.2, GraphStyle -> "RoyalColor"] The $$G(n,p)$$ model produces connected graphs with high probability for $$p>\ln (n)/n$$. n = 300; ListPlot[ Table[ {p, Mean@ Boole@Table[ConnectedGraphQ@IGErdosRenyiGameGNP[n, p], {50}]}, {p, 0, 0.05, 0.0005} ], GridLines -> {{Log[n]/n}, None} ] ### Random bipartite graphs▴ ?IGBipartiteGameGNM ?IGBipartiteGameGNP IGBipartiteGameGNM and IGBipartiteGameGNP are equivalent to IGErdosRenyiGNM and IGErdosRenyiGNP, but they generate bipartite graphs. The available options are: • DirectedEdges -> True creates a directed graph. • "Bidirectional" -> True allows directed edges to run in either direction between the two partitions. The default is False, which means that edges will run only from the first partition to the second. This option is ignored for undirected graphs. IGBipartiteGameGNP[5, 5, 0.5, VertexLabels -> "Name"] Create a bipartite directed graph with edges running either uni-directionally or bidirectionally between the two partitions. IGLayoutBipartite@ IGBipartiteGameGNM[10, 10, 30, DirectedEdges -> True, "Bidirectional" -> #] & /@ {False, True} ### IGTreeGame▴ ?IGTreeGame IGTreeGame samples uniformly from the set of labelled trees. Available options: • DirectedEdges -> True will create a directed tree, with edges oriented away from the root. • Method can be used to choose the tree generation algorithm. All methods sample labelled trees uniformly. Available Method options: • "PruferCode", works by generating a random Prüfer sequence, then constructing a tree from it. It does not currently support directed trees. • "LoopErasedRandomWalk", uses a loop-erased random walk to uniformly sample the spanning trees of the complete graph. IGTreeGame[250, GraphLayout -> "LayeredEmbedding", PlotTheme -> "PastelColor"] There are several distinct labellings of isomorphic trees. All of these are generated with equal probability. Table[ IGTreeGame[3, VertexLabels -> Automatic], {100} ] // DeleteDuplicatesBy[AdjacencyMatrix] Generate directed trees. Table[IGTreeGame[6, DirectedEdges -> True, GraphLayout -> "LayeredDigraphEmbedding"], {5}] Generate a random sparse connected graph by first creating a tree, then adding cycle edges. Note that this method does not sample connected graphs uniformly. randomConnected[nodeCount_, edgeCount_] := Module[{tree}, tree = IGTreeGame[nodeCount]; EdgeAdd[tree, RandomSample[EdgeList@GraphComplement[tree], edgeCount - nodeCount + 1]] ] randomConnected[100, 120] Colour the nodes of a random tree by their inverse average distance to other nodes. IGVertexMap[ ColorData["SolarColors"], VertexStyle -> Rescale@*IGCloseness, IGTreeGame[1000, Background -> Black, ImageSize -> Large, EdgeStyle -> LightGray] ] ### IGDegreeSequenceGame▴ ?IGDegreeSequenceGame IGDegreeSequenceGame implements various random sampling methods for graphs with a given degree sequence. To quickly construct a single realization of a degree sequence, use IGRealizeDegreeSequence. IGDegreeSequenceGame takes the following values for its Method option: • "ConfigurationModel" implements the configuration model: it connects up vertex stubs randomly. It may generate both self-loops and multi-edges. Undirected graphs are generated with probability proportional to $$\left(\prod _{i<j}A_{ij}!\prod _iA_{ii}\text{!!}\right){}^{-1}$$, where $$A$$ is the adjacency matrix, having twice the number of loops for each vertex on the diagonal. Directed ones are generated with probability proportional to $$\left(\prod _{i,j}A_{ij}!\right){}^{-1}$$. All simple graphs are generated with the same probability, but the probability of multigraphs and graphs with self-loops differs from that of simple graphs and depends on their specific structure. • "ConfigurationModelSimple" also implements the configuration model, but it rejects non-simple graphs. It samples uniformly from the set of all simple graphs with the given degree sequence. This method can be very slow for dense graphs. • "FastSimple" is a fast generation algorithm that avoids self-loops and multi-edges. This method can generate any simple graph with the given degree sequence, but it does not sample them uniformly. • "VigerLatapy" can sample undirected, connected simple graphs uniformly and uses Monte-Carlo methods to randomize the graphs. This generator should be favoured if undirected and connected graphs are to be generated and execution time is not a concern. igraph uses the original implementation of Fabien Viger; see https://www-complexnetworks.lip6.fr/~latapy/FV/generation.html and the corresponding paper at https://arxiv.org/abs/cs/0502085. The default method is "FastSimple". Note that it does not sample uniformly. degseq = VertexDegree@RandomGraph[{50, 100}]; IGDegreeSequenceGame[degseq, Method -> "ConfigurationModel"] SimpleGraphQ[%] False IGDegreeSequenceGame[degseq, Method -> "ConfigurationModelSimple"] SimpleGraphQ[%] True The configuration model algorithm is too slow to construct even small dense graphs. ds = VertexDegree@RandomGraph[{10, Binomial[10, 2] - 5}] {9, 7, 7, 9, 9, 8, 8, 7, 8, 8} TimeConstrained[ IGDegreeSequenceGame[ds, Method -> "ConfigurationModelSimple"], 1] $Aborted

Graphs that are almost complete can be sampled by generating the complement first.

GraphComplement@
IGDegreeSequenceGame[9 - ds, Method -> "ConfigurationModelSimple"]

ds == VertexDegree[%]
True

### IGKRegularGame▴

?IGKRegularGame

In a $$k$$-regular graph all vertices have degree $$k$$. The current implementation is able to generate any $$k$$-regular graph, but it does not sample them with precisely the same probability.

The available options are:

• DirectedEdges -> True creates a directed graph.

• MultiEdges -> True allows the creation of parallel edges.

IGKRegularGame[10, 3]

Not all parameters are valid:

IGKRegularGame[5, 3]

$Failed There are no graphs with 5 vertices each having degree 3. IGGraphicalQ[{3, 3, 3, 3, 3}] False ### IGGrowingGame▴ ?IGGrowingGame IGGrowingGame[n, k] creates a random graph by successively adding vertices to the graph until the vertex count n is reached. At each step, k new edges are added as well. The available options are: • DirectedEdges -> True creates a directed graph. • "Citation" -> True connects newly added edges to the newly added vertex. IGGrowingGame[50, 2] With "Citation" -> True, the newly added edges are connected to the newly added vertices. IGGrowingGame[50, 1, "Citation" -> True] Note that while this model can be used to generate random trees, it will not sample them uniformly. If uniform sampling is desired, use IGTreeGame instead. Create a directed citation graph. IGGrowingGame[20, 2, DirectedEdges -> True, "Citation" -> True, GraphStyle -> "Web"] ### IGBarabasiAlbertGame▴ ?IGBarabasiAlbertGame IGBarabasiAlbertGame implements a preferential attachment model. It generates a graph by sequentially adding new vertices with the specified number of edges ($$k$$). The edges will connect to existing vertices with probability $$d^{\beta }+A$$, where $$d$$ is the in-degree of the existing vertex. The default parameters are $$\beta =1$$ and $$A=1$$. The available options are: • DirectedEdges -> False creates an undirected graph. • "TotalDegreeAttraction" -> True computes the attachment probability based on the the total degree of existing vertices (i.e. the sum of in- and out-degrees), not their in-degree. Always assumed to be True when using DirectedEdges -> True. • "StartingGraph" -> g will use graph g as the starting point for building the preferential attachment graph. The vertex names of g are ignored; the result always uses positive integers as vertex names. Available Method option values: • "Bag" works by putting the IDs of the vertices into a bag exactly as many times as their (in-)degree, plus once more. Then the required number of cited vertices are drawn from the bag, with replacement. This method might generate multi-edges. It only works if $$\beta =1$$ and $$A=1$$. • "PSumTree" uses a partial prefix-sum tree to generate the graph. It does not generate multi-edges and works for any $$\beta$$ and $$A$$ values. • "PSumTreeMultiple" works like "PSumTree" but allows multi-edges. The built-in BarabasiAlbertGraphDistribution is equivalent to using $$A=0$$ and DirectedEdges -> False in IGBarabasiAlbertGame, while the built-in PriceGraphDistribution is equivalent DirectedEdges -> True. IGBarabasiAlbertGame[100, 1] Use attachment probability proportional to degree^1.5 + 1. IGBarabasiAlbertGame[100, 2, {1.5, 1}] The "Bag" method may generate parallel edges: IGBarabasiAlbertGame[100, 2, Method -> "Bag"] MultigraphQ[%] True Create a graph with the given out-degree sequence. The $$k^{\text{th}}$$ entry in the degree sequence list must be no greater than $$k$$. IGBarabasiAlbertGame[12, {1, 2, 3, 2, 1, 3, 4, 5, 1, 5, 2}, PlotTheme -> "Minimal"] VertexOutDegree[%] {0, 1, 2, 3, 2, 1, 3, 4, 5, 1, 5, 2} Create a preferential attachment graph using a 4-node complete graph as the starting point. IGBarabasiAlbertGame[10, 1, "StartingGraph" -> CompleteGraph[4]] ### IGWattsStrogatzGame▴ ?IGWattsStrogatzGame The two-argument form produces results equivalent to that of the built-in WattsStrogatzGraphDistribution. IGWattsStrogatzGame[30, 0.05, PlotTheme -> "Web"] The extended form allows for multi-dimensional lattices. Create a graph by randomly rewiring a two-dimensional toroidal lattice of $$10\times 10$$ nodes: Graph3D@IGWattsStrogatzGame[10, 0.01, {2, 1}] ### IGStaticFitnessGame▴ ?IGStaticFitnessGame IGStaticFitnessGame generates a random graph by connecting vertices based on their fitness score. The algorithm starts with $$n$$ vertices and no edges. Two vertices are selected with probabilities proportional to their fitness scores (for directed graphs, a starting vertex is selected based on its out-fitness and an end vertex based on its in-fitness). If they are not yet connected, an edge is inserted between them. The procedure is repeated until the number of edges reaches $$m$$. The expected degree of each vertex is proportional to its fitness score. This is exactly true when self-loops and multi-edges are allowed, and approximately true otherwise. IGStaticFitnessGame approximates the Chung–Lu model in which each edge i <-> j is present independently, with probability $p_{ij}= \begin{array}{ll} \{ & \begin{array}{ll} \frac{f_if_j}{2m} & \text{if} i\neq j \\ \frac{f_if_j}{4m} & \text{if} i=j \\ \end{array} \\ \end{array} ,$ where $$m=\frac{1}{2}\sum _kf_k$$. Unlike the Chung–Lu algorithm, which would require $$O\left(m^2\right)$$ computation steps, IGStaticFitnessGame runs in $$O(m)$$ time. The available options are: • SelfLoops -> True allows the creation of self-loops. • MultiEdges -> True allows the creation of parallel edges. Create an undirected graph with four high-degree nodes and 40 low-degree ones. weights = Join[{10, 10, 10, 10}, ConstantArray[1, 40]]; IGStaticFitnessGame[Total[weights]/2, weights] VertexDegree[%] {5, 5, 12, 8, 2, 2, 1, 2, 2, 1, 1, 2, 2, 0, 2, 1, 2, 0, 0, 3, 1, 1, \ 0, 0, 1, 2, 0, 3, 0, 2, 1, 2, 1, 1, 1, 2, 0, 1, 2, 0, 0, 3, 2, 1} Create a directed graph. IGStaticFitnessGame[30, Range[10], Range[10, 1, -1]] When self-loops and multi-edges are allowed, the expected degree of each vertex is proportional to its fitness score. degrees = {3, 3, 2, 2, 2, 1, 1}; Table[ VertexDegree@IGStaticFitnessGame[ Total[degrees]/2, degrees, SelfLoops -> True, MultiEdges -> True ], {1000} ] // N // Mean {3.03, 2.97, 2.023, 1.957, 2.056, 0.977, 0.987} When generating simple graphs, this holds only approximately. degrees = {3, 3, 2, 2, 2, 1, 1}; Table[ VertexDegree@IGStaticFitnessGame[ Total[degrees]/2, degrees ], {1000} ] // N // Mean {2.703, 2.625, 2.04, 2.071, 2.086, 1.23, 1.245} ### IGStaticPowerLawGame▴ ?IGStaticPowerLawGame IGStaticPowerLawGame generates a directed or undirected random graph where the degrees of vertices follow power-law distributions with prescribed exponents. For directed graphs, the exponents of the in- and out-degree distributions may be specified separately. This function is equivalent to IGStaticFitnessGame with a fitness vector $$f$$ where $$f_i=i^{-\alpha }$$ and $$\alpha =\frac{1}{\text{exponent}-1}$$. Note that significant finite size effects may be observed for exponents smaller than 3 in the original formulation of the game. This function removes the finite size effects by default by assuming that the fitness of vertex $$i$$ is $$\left(i+i_0\right){}^{-\alpha }$$, where $$i_0$$ is a constant chosen appropriately to ensure that the maximum degree is less than the square root of the number of edges times the average degree; see the paper of Chung and Lu, and Cho et al. for more details. The available options are: • SelfLoops -> True allows the creation of self-loops. • MultiEdges -> True allows the creation of parallel edges. • "FiniteSizeCorrection" -> False disables finite size correction, which is used by default. Create a graph with a power-law degree distribution of exponent 2.5. g = IGStaticPowerLawGame[100000, 200000, 2.5]; Histogram[VertexDegree[g], "Log", {"Log", "PDF"}] Create a directed graph with power-law in- and out-degree distributions. IGStaticPowerLawGame[50, 150, 3, 3] #### References • Goh K-I, Kahng B, Kim D: Universal behaviour of load distribution in scale-free networks. Phys Rev Lett 87(27):278701, 2001. • Chung F and Lu L: Connected components in a random graph with given degree sequences. Annals of Combinatorics 6, 125-145, 2002. • Cho YS, Kim JS, Park J, Kahng B, Kim D: Percolation transitions in scale-free networks under the Achlioptas process. Phys. Rev. Lett. 103:135702, 2009. ### IGStochasticBlockModelGame▴ ?IGStochasticBlockModelGame The ratesMatrix argument gives the connection probability between and within blocks (groups of vertices). The blockSizes argument gives the size of each block (vertex group). The available options are: • DirectedEdges -> True creates a directed graph. • SelfLoops -> True allows the creation of self-loops. IGAdjacencyMatrixPlot[%] ### IGPreferenceGame▴ ?IGPreferenceGame Experimental: This is experimental functionality that may change without notice. IGPreferenceGame[n, w, p] first samples n vertices of different types, each having type $$i$$ with probability proportional to $$w_i$$. Then it connects vertices with types $$i$$ and $$j$$ with probability $$p_{ij}$$. This is similar to a stochastic block model, but the vertex types are chosen randomly. The available options are: • DirectedEdges -> True creates a directed graph. • SelfLoops -> True allows self-loops. Generate a graph with three groups of vertices of different sizes, with intra-group connections being much more frequent than inter-group ones. IGPreferenceGame[60, {3, 6, 10}, 0.05 + 0.5 IdentityMatrix[3]] Generate a directed graph with low-probability intra-group connections and high probability unidirectional inter-group connections. ### IGAsymmetricPreferenceGame▴ ?IGAsymmetricPreferenceGame Experimental: This is experimental functionality that may change without notice. IGAsymmetricPreferenceGame[n, w, p] is similar to IGPreferenceGame[n, w, p], but it assigns a separate out-type and in-types to each vertex. The probability of a vertex having out-type $$i$$ and in-type $$j$$ is proportional to $$w_{ij}$$. The probability of connecting a vertex with out-type $$i$$ to another one with in-type $$j$$ is $$p_{ij}$$. The available options are: • SelfLoops -> True allows self-loops. ### IGForestFireGame▴ ?IGForestFireGame The forest fire model is a growing graph model. In every time step, a new vertex is added to the graph. The new vertex chooses the specified number of ambassadors (default: 1) and starts a simulated forest fire at their locations. The fire spreads through the directed edges. The spreading probability along an edge is given by pForward. The fire may also spread backwards on an edge with probability pForward * rBackward. When the fire has ended, the newly added vertex connects to all the vertices that were burned in the fire. The forest fire model intends to reproduce the following network characteristics, observed in real networks: • Heavy-tailed in-degree and out-degree distributions. • Community structure. • Densification power-law. The network is densifying in time, according to a power-law rule. • Shrinking diameter. The diameter of the network decreases in time. The available options are: • DirectedEdges -> False generates an undirected graph. Generate a graph with only forward burning. IGForestFireGame[30, 0.2, 0, GraphLayout -> "SpringEmbedding"] Generate a graph from the forest fire model, and visualize its community structure. IGForestFireGame[100, 0.2, 1, 2, DirectedEdges -> False, GraphLayout -> {"EdgeLayout" -> "HierarchicalEdgeBundling"}] Plot the cumulative in-degree distribution for different backward to forward burning probability ratios. Table[ Histogram[ VertexInDegree@ IGForestFireGame[2000, 0.4, r, 2, DirectedEdges -> True], "Log", {"Log", "SurvivalCount"}, PlotLabel -> Row[{"r=", r}] ], {r, 0, 0.8, 0.2} ] #### References • Jure Leskovec, Jon Kleinberg and Christos Faloutsos. Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery from Data (TKDD), 2007. https://doi.org/10.1145/1217299.1217301 • Jure Leskovec, Jon Kleinberg and Christos Faloutsos. Graphs over time: densification laws, shrinking diameters and possible explanations. KDD ’05: Proceeding of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining, 177–187, 2005. ### IGCallawayTraitsGame▴ ?IGCallawayTraitsGame This function simulates a growing random graph according to the following algorithm: At each time step, a new vertex is added. Its type is randomly selected according to the type weights. Then k existing pairs of vertices are selected randomly, and each pair attempts to connect. The probability of success for given types of vertices is given by the preference matrix. This algorithm may create self-loops and multi-edges. The available options are: • DirectedEdges -> True creates a directed graph. ### IGEstablishmentGame▴ ?IGEstablishmentGame This function simulates a growing random graph according to the following algorithm: At each time step, a new vertex is added. Its type is randomly selected according to the type weights. It attempts to connect to k distinct existing vertices. The probability of success for given types of vertices is given by the preference matrix. The available options are: • DirectedEdges -> True creates a directed graph. ### IGGeometricGame▴ ?IGGeometricGame Available options: • "Periodic" -> True assumes a toroidal topology IGGeometricGame[50, 0.2] Use a toroidal topology and draw “wraparound” edges with dashed lines. IGGeometricGame[50, 0.2, "Periodic" -> True] // IGEdgeMap[ If[EuclideanDistance @@ # > 0.2, Dashed, None] &, EdgeStyle -> IGEdgeVertexProp[VertexCoordinates] ] # Graph modification ### IGRewire▴ ?IGRewire IGRewire will try to rewire the edges of the graph the given number of times by switching random pairs of edges as below, thus preserving the graph’s degree sequence. or The switches succeed only if they would not create multi-edges. The parameter $$n$$ specifies the number of switch attempts, not the number of successful switches. For directed graphs, the switches are such that they preserve both the in- and out-degree sequence. The vertex ordering of the graph is retained. Warning: Most graph properties, such as edge weights, will be lost. The available options are: • SelfLoops -> True allows the creation of self-loops. Generate a random network with scale-free degree distribution: IGRewire[IGBarabasiAlbertGame[200, 2, DirectedEdges -> False], 10000] Use SelfLoops -> True to allow creating loops. Table[IGRewire[PathGraph@Range[4], 100, SelfLoops -> True], {5}] IGRewrire never creates any multi-edges. Multigraphs are allowed as input, but a warning is given. Uniformly sample simple labelled graphs with a given degree sequence by first creating a single realization, then rewiring it a sufficient amount of times. degseq = {3, 3, 2, 2, 1, 1}; Table[ IGRewire[IGRealizeDegreeSequence[degseq], 100], {1000} ] // CountsBy[AdjacencyMatrix] // KeySort // KeyMap[AdjacencyGraph[#, VertexShapeFunction -> "Name"] &] ### IGRewireEdges▴ ?IGRewireEdges IGRewireEdges randomly rewires each edge of the graph with the given probability. The vertex ordering is retained. For directed graphs, it can optionally rewire only the starting point or endpoint of directed edges, thus preserving the out- or in-degree sequence. In this case, the MultiEdges option is ignored and multi-edges may be created. Warning: Most graph properties, such as edge weights, will be lost. The available options are: • SelfLoops -> True allows the creation of self-loops. • MultiEdges -> True allows the creation of multi-edges. Create a random graph with 10 vertices and 20 edges, while allowing for multi-edges: IGRewireEdges[RandomGraph[{10, 20}], 1, MultiEdges -> True] EdgeCount[%] 20 Rewire the endpoint of each edge, preserving the out-degree sequence. g = RandomGraph[{10, 30}, DirectedEdges -> True]; {VertexInDegree[g], VertexOutDegree[g]} {{5, 4, 1, 2, 2, 2, 5, 3, 3, 3}, {2, 6, 4, 2, 2, 4, 3, 2, 2, 3}} rg = IGRewireEdges[g, 1, "Out"]; {VertexInDegree[rg], VertexOutDegree[rg]} {{2, 0, 2, 7, 3, 3, 3, 3, 2, 5}, {2, 6, 4, 2, 2, 4, 3, 2, 2, 3}} Note that multi-edges were created. MultigraphQ[rg] True ### IGVertexContract▴ ?IGVertexContract IGVertexContract[g, {set1, set2, …}] will simultaneously contract multiple vertex sets into single vertices. The name of a contracted vertex will be the same as the first element of the corresponding set. Vertex ordering is not retained. Edge ordering is retained only when using both SelfLoops -> True and MultiEdges -> True. Warning: Most graph properties, such as edge weights, will be lost. The available options are: • SelfLoops -> True keeps any self-loops created during contraction. • MultiEdges -> True keeps any parallel edges created during contraction. IGVertexContract[g, {{1, 2, 3}, {4, 5}}, VertexLabels -> "Name"] IGVertexContract[g, {{1, 2, 3}, {4, 5}}, SelfLoops -> True] IGVertexContract[g, {{1, 2, 3}, {4, 5}}, SelfLoops -> True, MultiEdges -> True] IGVertexContract[g, {{1, 2, 3}, {4, 5}}, MultiEdges -> True] When using both SelfLoops -> True and MultiEdges -> True, the edge ordering is maintained relative to the input graph. This allows easily transferring edge weights, and combining them if necessary. g = IGShorthand["a-b-c-d-a,a-c", EdgeWeight -> {1, 2, 3, 4, 5}, EdgeLabels -> "EdgeWeight"] IGWeightedSimpleGraph[ IGVertexContract[g, {{"a", "b"}}, SelfLoops -> True, MultiEdges -> True, EdgeWeight -> IGEdgeProp[EdgeWeight][g] ], EdgeLabels -> "EdgeWeight", VertexLabels -> "Name" ] ### IGConnectNeighborhood▴ ?IGConnectNeighborhood IGConnectNeighborhood[g, k] connects each vertex in g to its order k neighbourhood. This operation is also known as the $$k^{\text{th}}$$ power of the graph. IGConnectNeighborhood differs from the built-in GraphPower in that it preserves parallel edges and self-loops. Warning: Most graph properties, such as edge weights, will be lost. Connect each vertex to its second order neighbourhood: IGConnectNeighborhood[CycleGraph[15]] Connect each vertex to its third order neighbourhood: IGConnectNeighborhood[GridGraph[{10, 10}], 3] ### IGMycielskian▴ ?IGMycielskian IGMycielskian applies the Mycielski construction to an undirected graph on $$n\geq 2$$ vertices to obtain a larger graph (the Mycielskian) on $$2 n+1$$ vertices. If the graph has less than 2 vertices, then instead of applying the standard Mycielski construction, IGMycielskian simply adds one vertex and one edge. If the original graph has chromatic number $$k$$, its Mycielskian has chromatic number $$k+1$$. The Mycielski construction preserves the triangle-free property of the graph. g = CycleGraph[4] {IGChromaticNumber[g], IGTriangleFreeQ[g]} {2, True} mg = IGMycielskian[g] {IGChromaticNumber[mg], IGTriangleFreeQ[mg]} {3, True} Construct triangle-free graphs with successively larger chromatic numbers. NestList[IGMycielskian, IGEmptyGraph[], 5] IGChromaticNumber /@ % {0, 1, 2, 3, 4, 5} ### IGSmoothen▴ ?IGSmoothen IGSmoothen suppresses all degree-2 vertices, thus obtaining the smallest topologically equivalent (i.e. homeomorphic) graph. See also IGHomeomorphicQ. The vertex names are preserved, and the weights of merged edges are summed up. All other graph properties are discarded. In directed graphs, only those vertices are smoothened which have one incoming and one outgoing edge. Available options: • DirectedEdges -> False ignores edge directions in the input graph. The smallest topological equivalent of a path graph consists of two connected vertices. The result may contain self-loops. The smallest topological equivalent of a cycle graph is a single vertex with a self-loop. IGSmoothen[CycleGraph[10]] The result may also contain multi-edges. If the input is directed, only those vertices are smoothed which have one incoming and one outgoing edge. Use DirectedEdges -> False to treat the input graph as undirected. The result is always a weighted graph. When contracting edges, their weights are added up. If the input graph was not weighted, all of its edge weights are considered to be 1. Thus, the graph distance of any two vertices in the result is always the same as it was in the input graph. g = IGGiantComponent@RandomGraph[{100, 100}] tm = IGSmoothen[g] IGEdgeWeightedQ[tm] True IGDistanceMatrix[g, VertexList[tm], VertexList[tm]] == IGDistanceMatrix[tm] True The result does not contain any degree-2 vertices, except possibly isolated vertices with self-loops. Union@VertexDegree[tm] {1, 3, 4, 5, 6} The vertex coordinates, as well as any other graph properties are discarded. g = IGMeshGraph@IGLatticeMesh["Hexagonal", {3, 3}] IGSmoothen[g] Vertex coordinates can be transferred to the new graph as follows: IGSmoothen[g, VertexCoordinates -> {v_ :> PropertyValue[{g, v}, VertexCoordinates]}] An alternative and faster method uses IGVertexMap and IGVertexAssociate: IGSmoothen[g] // IGVertexMap[IGVertexAssociate[GraphEmbedding][g], VertexCoordinates -> VertexList] Create a tree in which every non-leaf node has a degree of at least 3. IGSmoothen[IGTreeGame[100], GraphLayout -> "RadialEmbedding"] Let us compute the effective resistance of a resistor network by repeated smoothing (merger of resistors in series) and simplification (merger of resistors in parallel). Resistances are stored as edge weights. A zero-resistance input and output terminal is added to prevent the premature smoothing of these points. Merge resistors in series … reducedGrid = IGSmoothen[resistorGrid] … then merge resistors in parallel and check the resulting edge weights. reducedGrid = IGWeightedSimpleGraph[reducedGrid, 1/Total[1/{##}] &, EdgeLabels -> "EdgeWeight"] Repeat until a single resistor remains. reducedGrid = IGSmoothen[reducedGrid] reducedGrid = IGWeightedSimpleGraph[reducedGrid, 1/Total[1/{##}] &, EdgeLabels -> "EdgeWeight"] reducedGrid = IGSmoothen[reducedGrid, EdgeLabels -> "EdgeWeight"] IGEdgeProp[EdgeWeight][reducedGrid] {3.} # Structural properties ## Centrality measures▴ Centralities are various measures that quantify the “importance” of vertices or edges in graphs. ### Betweenness▴ ?IGBetweenness ?IGBetweennessCutoff ?IGEdgeBetweenness ?IGEdgeBetweennessCutoff The betweenness of a vertex or edge is, roughly speaking, the number of shortest paths passing through it. More formally, the betweenness of vertex $$i$$ is $$b_i=\sum _{i\neq s\neq t} \frac{g_{st}^{(i)}}{g_{st}}$$, where $$g_{st}$$ is the total number of shortest paths (geodesics) between vertices $$s$$ and $$t$$, and $$g_{st}^{(i)}$$ is the number of shortest paths between vertices $$s$$ and $$t$$ that pass through $$i$$. Weighted graphs and multigraphs are supported by all betweenness functions in IGraph/M. Note that as of Mathematica 13.0, the built-in BetweennessCentrality function ignores edge weights and multi-edges, which causes it to yield different results from IGBetweenness. Available options: • Normalized -> True will compute the normalized betweenness by dividing the result by the number of (ordered or unordered) vertex pairs used in the shortest path calculation. Thus the normalization factor is $$(V-1) (V-2)$$ for directed graphs and $$\frac{1}{2} (V-1) (V-2)$$ for undirected graphs. The normalized value lies between 0 and 1. Visualize the vertex and edge betweenness of a weighted geometrical graph, where weights represent Euclidean distances. pts = RandomPoint[Disk[], 100]; IGMeshGraph[ DelaunayMesh[pts], EdgeStyle -> Thick, VertexStyle -> EdgeForm[None] ] // IGVertexMap[ ColorData["SolarColors"], VertexStyle -> Rescale@*IGBetweenness ]/* IGEdgeMap[ ColorData["SolarColors"], EdgeStyle -> Rescale@*IGEdgeBetweenness ] Compute the betweenness of a subset of vertices. g = ExampleData[{"NetworkGraph", "DolphinSocialNetwork"}]; Take[VertexList[g], 5] {"Beak", "Beescratch", "Bumper", "CCL", "Cross"} IGBetweenness[g, %] {34.9212, 390.384, 16.6032, 4.34405, 0.} Visualize the betweenness of a periodic grid with slightly randomized edge weights. n = 40; IGSquareLattice[{n, n}, "Periodic" -> True, VertexCoordinates -> Tuples[Range[n], {2}], EdgeWeight -> {_ :> RandomReal[{.99, 1.01}]}, GraphStyle -> "BasicBlack", EdgeShapeFunction -> None, VertexSize -> 1 ] // IGVertexMap[ ColorData["BlueGreenYellow"], VertexStyle -> Rescale@*IGBetweenness ] #### Possible issues Betweenness computation involves comparing the lengths of paths, and deciding which specific path is the shortest, and which paths have equal lengths. When non-integer edge weights are used, the path length computation is subject to roundoff errors, which may cause the path length comparison to fail. igraph mitigates this by comparing the lengths using tolerances, however, there is still a small risk that roundoff errors may affect the result. To avoid this potential problem entirely, use integer weights. For example, if the weights are rational, multiply them by the least common multiple of their denominators. ### Closeness▴ ?IGCloseness ?IGClosenessCutoff ?IGNeighborhoodCloseness The normalized closeness centrality of a vertex is the inverse average shortest path length to other vertices. Weighted graphs are supported. Available options: • Normalized -> False will compute the non-normalized closeness, i.e. the inverse of the sum of shortest path lengths to all other vertices. There is no standard definition of closeness centrality for disconnected graphs. When the graph is disconnected, IGraph/M will only consider the distances to reachable vertices. In the undirected case, this effectively computes the closeness separately for each connected component. Use IGNeighborhoodCloseness to obtain both the closeness values, as well as how many vertices were reachable from each vertex. This information allows for computing various generalizations of closeness centrality for disconnected graphs. Visualize the closeness of nodes in a weighted geometrical graph where weights correspond to Euclidean distances. pts = RandomPoint[Polygon@CirclePoints[3], 75]; IGVertexMap[ ColorData["Rainbow"], VertexStyle -> Rescale@*IGCloseness, IGMeshGraph[DelaunayMesh[pts], GraphStyle -> "BasicBlack"] ] For isolated vertices, Indeterminate is returned. IGCloseness@IGShorthand["1,2-3"] {Indeterminate, 1., 1.} ### Harmonic centrality▴ ?IGHarmonicCentrality ?IGHarmonicCentralityCutoff The harmonic centrality of a vertex is the average inverse shortest path length to all other vertices. The inverse shortest path length to unreachable vertices is considered to be zero. Available options: • Normalized -> False computes the non-normalized harmonic centrality, i.e. the sum of inverse shortest path length to all other vertices. RandomGraph[{30, 40}, VertexSize -> Large] // IGVertexMap[ColorData["Rainbow"], VertexStyle -> IGHarmonicCentrality/*Rescale] ### PageRank▴ ?IGPageRank ?IGPersonalizedPageRank The PageRank centrality of a vertex is the fraction of time a random walker would spend on that vertex. The walker jumps from vertex to vertex randomly, following outward edges with probabilities proportional to their weights. Additionally, after each step, with a probability $$1-d$$ the walk is restarted from a random vertex. $$d$$ is called the damping factor. If the walker is stuck in a sink vertex (i.e. a vertex with no outgoing edges), the walk is also restarted. In the standard version of PageRank, when the walk is restarted, the starting vertex is chosen uniformly. In the personalized version, it is chosen with probabilities proportional to the values in the reset parameter. Weighted graphs and multigraphs are supported, and self-loops are taken into consideration. Note that as of Mathematica 13.0, the built-in PageRankCentrality function ignores self-loops. The default damping factor is 0.85. The following Method options are available: • "Arnoldi" uses ARPACK, and solves PageRank as an eigenvalue problem. • "PRPACK" uses PRPACK and uses the algebraic method. It is the default method, and usually much faster than "Arnoldi". Plot the logarithmic histogram of PageRank scores of the network of webpage in the nd.edu domain. ndWeb = ExampleData[{"NetworkGraph", "WorldWideWeb"}]; Histogram[IGPageRank[ndWeb], "Log", {"Log", "PDF"}, Frame -> True, FrameLabel -> {"PageRank", "PDF"}] The personalization weights may be given as a vector of the same length as the vertex list … g = RandomGraph[{10, 20}, DirectedEdges -> True]; IGPersonalizedPageRank[g, RandomReal[1, VertexCount[g]]] {0.0433579, 0.129471, 0.23356, 0.113496, 0.000101892, 0.12872, \ 0.260584, 0.024101, 0.0324582, 0.0341504} … or as an association from vertex names to weights, in which case the weight of non-included vertices is taken to be zero. IGPersonalizedPageRank[g, <|1 -> 1.5, 3 -> 0.5|>] {0.297935, 0.0717967, 0.168933, 0.290878, 0., 0.0376334, 0.132824, \ 0., 0., 0.} Personalize PageRank by always restarting the walk from one of two vertices (29 or 74) on a grid graph: g = IGSquareLattice[{10, 10}, VertexSize -> Large]; g // IGVertexMap[ColorData["Rainbow"], VertexStyle -> (IGPersonalizedPageRank[#, <|29 -> 1, 74 -> 1|>, 0.99] &/*Rescale)] ### LinkRank▴ ?IGLinkRank ?IGPersonalizedLinkRank LinkRank is the equivalent of PageRank for edges. The LinkRank of an edge is the relative frequency of traversing that edge by a random walker. For a detailed description of the random walk process, see the PageRank section. The LinkRank of edges can be computed from the PageRank by simply dividing the PageRank of each vertex between its outgoing edges, proportionally with their edge weights. The LinkRank scores of the out-edges of a vertex add up to the PageRank of that vertex. The LinkRank scores of all edges in the graph add up to 1. Weighted graphs and multigraphs are supported, and self-loops are taken into consideration. The available Method options are the same as for IGPageRank. Visualize both the LinkRank and PageRank of a random directed graph. maxNorm = #/Max[#] &; g = RandomGraph[{15, 30}, DirectedEdges -> True, EdgeStyle -> Thick, VertexSize -> Large, GraphStyle -> "BasicBlack"]; g // IGEdgeMap[ColorData["Rainbow"], EdgeStyle -> IGLinkRank/*maxNorm] // IGVertexMap[ColorData["Rainbow"], VertexStyle -> IGPageRank/*maxNorm] Visualize the personalized version of LinkRank and PageRank, always starting the random walk from vertex 1. pers = <|1 -> 1|>; Graph[g, VertexLabels -> "Name"] // IGEdgeMap[ColorData["Rainbow"], EdgeStyle -> (IGPersonalizedLinkRank[#, pers] &)/*maxNorm] // IGVertexMap[ColorData["Rainbow"], VertexStyle -> (IGPersonalizedPageRank[#, pers] &)/*maxNorm] ### Eigenvector centrality▴ ?IGEigenvectorCentrality Eigenvector centrality is based on the idea that the importance (centrality) of a vertex should be affected not only by how many other vertices point to it, but also by the importance of its neighbours. The eigenvector centrality of a vertex is proportional to the sum of centralities of its neighbours. Mathematically, the eigenvector centrality is the leading eigenvector of the adjacency matrix. Eigenvector centrality is meaningful for connected graphs only. Disconnected graphs should be decomposed into their components, and the eigenvector centrality computed separately for each. The vertex centrality scores will be comparable only within components, not between separate components. In undirected graphs, the diagonal of the adjacency matrix is assumed to contain twice the number of self-loops on each vertex. This makes the undirected result consistent with the directed one when each undirected edge is replaced by reciprocal directed ones. For directed graphs, the left eigenvector of the adjacency matrix is calculated. In other words, the centrality of a vertex is proportional to the sum of centralities of vertices pointing to it. Weighted and directed graphs are supported. The available options are: • Normaized -> True will scale the result so that the maximum centrality is 1. The default is True. • DirectedEdges -> False ignores edge directions. ### Kleinberg’s hub and authority scores▴ ?IGHubScore ?IGAuthorityScore Weighted graphs are supported. The available options are: • Normalized -> True scales the result so that the maximum centrality is 1. The default is True. ### Burt’s constraint score▴ ?IGConstraintScore Weighted graphs are supported. ## Centralization▴ ?IG*Centralization Centralization is computed from centrality values in a way equivalent to Total[Max[centralities] - centralities]. With the (default) option Normalized -> True, the result is normalized by dividing by the highest possible centralization score of any graph of the same directedness on the same number of vertices. g = IGBarabasiAlbertGame[100, 2, DirectedEdges -> False]; IGBetweennessCentralization[g] 0.194343 IGClosenessCentralization[g] 0.275726 IGDegreeCentralization[g, SelfLoops -> False] 0.144919 IGEigenvectorCentralization[g] 0.820631 For most centrality types, the highest centralization is achieved by the StarGraph. IGBetweennessCentralization@StarGraph[5] 1. In the case of the degree centralization, the highest possible centralization score depends on whether self-loops are allowed. This is controlled by the SelfLoops option of IGDegreeCentralization. The default is SelfLoops -> True. {0.666667, 1., 1.} ## Topological sorting and acyclic graphs▴ ### IGDirectedAcyclicGraphQ▴ ?IGDirectedAcyclicGraphQ IGDirectedAcyclicGraphQ tests if a graph is directed and has no directed cycles. IGDirectedAcyclicGraphQ /@ {IGShorthand["1->2->3->1"], IGShorthand["1->2->3<-1"]} {False, True} IGDirectedAcyclicGraphQ returns True for graphs with no edges. IGDirectedAcyclicGraphQ[IGEmptyGraph[3]] True ### IGTopologicalOrdering▴ ?IGTopologicalOrdering IGTopologicalOrdering is to the built-in TopologicalSort as Ordering is to Sort: it returns the permutation which sorts vertices in topological order. When vertices are ordered topologically, all directed edges point from earlier vertices to later ones. Graphs must be acyclic for topological sorting to be possible. IGDirectedAcyclicGraphQ[g] True p = IGTopologicalOrdering[g] {5, 8, 9, 4, 6, 1, 2, 3, 10, 7} VertexList[g][[p]] {"E", "H", "I", "D", "F", "A", "B", "C", "J", "G"} If the vertices are laid out from left to right in topological order, all edges will point from left to right. curvedEdge[offset_][{a_, ___, b_}, ___] := Arrow@BezierCurve[{a, (a + b)/2 + offset Reverse[b - a], b}] Graph[g, EdgeShapeFunction -> curvedEdge[2/3] (* in M12.0 or later simply use {{"CurvedEdge", "Curvature"->1.5}} *) ] // IGVertexMap[{#, 0} &, VertexCoordinates -> IGTopologicalOrdering/*Ordering] When the graph contains cycles, $Failed is returned.

IGTopologicalOrdering[IGShorthand["1->2->3->4->5, 5->3, 5->6"]]

$Failed ### IGFeedbackArcSet▴ ?IGFeedbackArcSet IGFeedbackArcSet[] returns a set of directed edges (also called arcs) the removal of which makes the graph acyclic. With Method -> "IntegerProgramming", it finds an exact minimal feedback arc set through integer programming using the triangle inequality formulation. With Method -> "EadesLinSmyth", it finds a feedback arc set (not necessarily minimal) using the fast “GR” heuristic of Eades, Lin and Smyth (1993). The following directed graph is not acyclic. g = RandomGraph[{10, 20}, DirectedEdges -> True, VertexLabels -> "Name"] {AcyclicGraphQ[%], IGDirectedAcyclicGraphQ[%]} {False, False} Find a set of edges whose removal breaks all cycles. IGFeedbackArcSet[g] {1 $DirectedEdge] 9, 3 \[DirectedEdge] 8, 4 \[DirectedEdge] 8} ag = EdgeDelete[g, %] IGDirectedAcyclicGraphQ[ag] True Vertices of a directed acyclic graph can be sorted topologically. IGTopologicalOrdering returns a permutation that sorts them this way, and thus makes the graph’s adjacency matrix upper triangular. perm = IGTopologicalOrdering[ag] {9, 8, 4, 5, 6, 7, 1, 10, 2, 3} With[{am = AdjacencyMatrix[ag]}, ArrayPlot /@ {am, am[[perm, perm]]} ] #### References ## Chordal graphs▴ Chordal graphs are graphs that do not contain induced cycles with more than three vertices. ### IGChordalQ▴ ?IGChordalQ A graph is chordal if each of its cycles of four or more nodes has a chord, i.e. an edge joining two non-adjacent vertices in the cycle. Equivalently, all chordless cycles in a chordal graph have at most 3 vertices. Chordal graphs are also called rigid circuit graphs or triangulated graphs. Grid graphs are not chordal because they have chordless 4 cycles. g = GridGraph[{2, 3}, VertexLabels -> "Name"] IGChordalQ[g] False Adding chords to the 4 cycles makes them chordal. EdgeAdd[g, {1 <-> 4, 4 <-> 5}] IGChordalQ[%] True ### IGChordalCompletion▴ ?IGChordalCompletion IGChordalCompletion computes a set of edges that, when added to a graph, make it chordal. The edge set returned is not usually minimal, i.e. some of the edges may not be necessary to create a chordal graph. g = CycleGraph[5] completion = IGChordalCompletion[g]; HighlightGraph[EdgeAdd[g, completion], completion] ### IGMaximumCardinalitySearch▴ ?IGMaximumCardinalitySearch The maximum cardinality search algorithm visits the vertices of the graph in such an order so that every time the vertex with the most already visited neighbours is visited next. Ties are broken arbitrarily. Then vertices are assigned ranks $$\alpha$$ in decreasing order from the vertex count of the graph to 1. IGMaximumCardinalitySearch returns these ranks. The visiting order is animated below: ranks = AssociationThread[VertexList[g], IGMaximumCardinalitySearch[g]] <|1 -> 10, 2 -> 3, 3 -> 8, 4 -> 6, 5 -> 7, 6 -> 2, 7 -> 1, 8 -> 5, 9 -> 4, 10 -> 9|> verts = Keys@Reverse@Sort[ranks] Table[ HighlightGraph[ Graph[g, VertexLabels -> "Name"], Take[verts, i] ], {i, VertexCount[g]} ] // ListAnimate {1, 10, 3, 5, 4, 8, 9, 2, 6, 7} The rank $$\alpha$$ is useful for deciding the chordality of a graph. A graph is chordal if and only if any two neighbors of a vertex which are higher in rank than it are connected to each other. Label the vertices of a graph with their ranks. g = IGShorthand["a-b-c-d-a-e-f-g-h-e-g"]; IGVertexMap[Row[{#1, ": ", #2}] &, VertexLabels -> {VertexList, IGMaximumCardinalitySearch}, g] Notice that vertex b has two higher-rank neighbours that are not connected to each other. This graph is not chordal. Use IGChordalCompletion to determine which edges to add to it to make it chordal. IGChordalCompletion[g] {"c" <-> "a"} #### References • R. E. Tarjan, M. Yannakakis: Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs, SIAM J. Comput., 13(3), 566–579 (1984). https://doi.org/10.1137/0213035 ## Clustering coefficient▴ ?IG*ClusteringCoefficient Clustering coefficients are measures of the degree to which vertices in a graph tend to cluster together. They are also referred to as transitivity, as they measure how often two vertices that are connected through a third one are also directly connected. All clustering coefficient calculations in IGraph/M ignore edge directions. ### IGGlobalClusteringCoefficient▴ ?IGGlobalClusteringCoefficient The clustering coefficient of an undirected graph is defined as \[C=\frac{\text{number of closed ordered triplets}}{\text{number of connected ordered triplets}}$ The available options are: • "ExcludeIsolates" -> True will cause Indeterminate to be returned if the graph has no connected triplets. With the default "ExcludeIsolates" -> False, 0 is returned. The following graph has 10 connected ordered triplets, namely {3, 1, 2}, {2, 1, 3}, {1, 2, 3}, {3, 2, 1}, {2, 3, 1}, {2, 3, 4}, {1, 3, 4}, {1, 3, 2}, {4, 3, 2}, {4, 3, 1}. Out of these, only 6 are closed: {1, 3, 2}, {1, 2, 3}, {2, 1, 3}, {2, 3, 1}, {3, 2, 1}, {3, 1, 2}. Thus the clustering coefficient is 6/10 = 0.6. 0.6 ### IGLocalClusteringCoefficient▴ ?IGLocalClusteringCoefficient The local clustering coefficient of a vertex is defined as $C=\frac{\text{number of connected pairs of neighbours}}{\text{total number of pairs of neighbours}}$ The available options are: • "ExcludeIsolates" -> True will cause Indeterminate to be returned for degree 0 and degree 1 vertices. With the default "ExcludeIsolates" -> False, 0 is returned. The the following graph, vertex 4 has two neighbours which are disconnected, making its local clustering zero. However, vertex 5 has only one neighbour, thus computing the local clustering for it arguably does not make sense. Setting "ExcludeIsolates" -> True serves to distinguish these two cases by returning Indeterminate for vertex 5. {1., 1., 0.333333, 0., 0.} {1., 1., 0.333333, 0., Indeterminate} ### IGAverageLocalClusteringCoefficient▴ ?IGAverageLocalClusteringCoefficient The available options are: • "ExcludeIsolates" -> True will cause degree 0 and degree 1 vertices to be excluded from the calculation. With "ExcludeIsolates" -> True, the local clustering coefficient of vertex 4 will be excluded from the calculation of the average. {0.583333, 0.777778} When the graph has no vertices with degree of at least 2, and "ExcludeIsolates" -> True is set, the result will be Indeterminate. Indeterminate ### IGWeightedClusteringCoefficient▴ ?IGWeightedClusteringCoefficient IGWeightedClusteringCoefficient computes the weighted local clustering coefficient. This function expects a weighted graph as input. The available options are: • "ExcludeIsolates" -> True will cause Indeterminate to be returned for degree 0 and degree 1 vertices. With the default "ExcludeIsolates" -> False, 0 is returned. #### References ## Neighbour degrees▴ ### IGAverageNeighborDegree▴ ?IGAverageNeighborDegree IGAverageNeighborDegree computes the average of the degrees of each vertex’s neighbours. In weighted graphs, a weighted average is used: $k_{\text{nn},u}=\frac{1}{s_u}\sum _v w_{uv}k_v$ $$k_{\text{nn},u}$$ denotes the average neighbour degree of vertex $$u$$, $$k_v$$ is the degree of vertex $$v$$, $$w_{uv}$$ is the weighted adjacency matrix, and $$s_u=\sum _vw_{uv}$$ is the strength of vertex $$u$$. IGAverageNeighborDegree is similar to MeanNeighborDegree, with a few differences: it can compute the measure for only a subset of vertices, the interpretation of degrees and neighbours can be controlled independently in directed graphs, and for vertices which have no neighbours it returns Indeterminate instead of 0. Average neighbour degree in a star graph: IGAverageNeighborDegree[StarGraph[4]] {1., 3., 3., 3.} Compute the result only for vertices 1 and 3: IGAverageNeighborDegree[StarGraph[4], {1, 3}] {1., 3.} All computes the result for all vertices (the default): IGAverageNeighborDegree[StarGraph[4], All] {1., 3., 3., 3.} When a vertex has no neighbours, Indeterminate is returned: IGAverageNeighborDegree[IGShorthand["1,2-3"]] {Indeterminate, 1., 1.} In directed graphs, the out-degrees of out-neighbours are considered by default. IGAverageNeighborDegree[g] {1., 1., 1.} Use in-degrees of in-neighbours instead: IGAverageNeighborDegree[g, All, "In"] {Indeterminate, 1., 1.} Use the in-degrees of all neighbours: IGAverageNeighborDegree[g, All, "In", "All"] {2., 1.33333, 1.33333} Compute a weighted neighbour degree average. The weights used in averaging are taken from the edge weights: {2.2, 1.85714, 3., 3., 3., 3.} #### References ### IGAverageDegreeConnectivity▴ ?IGAverageDegreeConnectivity IGAverageDegreeConnectivity computes the average neighbour degree as a function of the vertex degree. The $$i$$th element of the result is the average of the IGAverageNeighborDegree result for all vertices of degree $$i$$. g = RandomGraph[{30, 50}]; IGAverageDegreeConnectivity[g] {3., 3.92857, 3.44444, 4.21429, 4.46667, 4.33333, Indeterminate, \ 4.125} An equivalent implementation of IGAverageDegreeConnectivity is: Transpose[{VertexDegree[g], IGAverageNeighborDegree[g]}] // GroupBy[#, First -> Last, Mean] & // Lookup[#, Range@Max@VertexDegree[g], Indeterminate] & {3., 3.92857, 3.44444, 4.21429, 4.46667, 4.33333, Indeterminate, \ 4.125} Compute the average degree connectivity curve for a scale free network: ListPlot[ IGAverageDegreeConnectivity@IGStaticPowerLawGame[1000, 2000, 2], FrameLabel -> {"degree", "average neighbour degree"}, PlotTheme -> "Detailed" ] #### References ## Shortest paths▴ The length of a path between two vertices is the number of edges the path consists of. Functions that use edge weights consider the path length to be the sum of edge weights along the path. ### IGDistanceMatrix▴ ?IGDistanceMatrix IGDistanceMatrix takes the following Method options: • Automatic selects a method automatically. As of IGraph/M 0.5, "Unweighted" is selected for unweighted graphs, "Dijkstra" for weighted graphs with only positive weights, and "Johnson" otherwise. • "Unweighted" ignores weights • "Dijkstra" uses Dijkstra’s algorithm. All weights must be non-negative. • "BellmanFord" uses the Bellman–Ford algorithm. Negative weights are supported but all cycles must have a non-negative total weight. • "Johnson" uses the Johnson algorithm. Negative weights are supported but all cycles must have a non-negative total weight. The igraph C core may override explicit method settings when appropriate. For example, if the graph is not weighted, it always uses "Unweighted". ### IGDistanceCounts▴ ?IGDistanceCounts IGDistanceCounts[graph] counts all-pair unweighted shortest path lengths in the graph. counts unweighted shortest path lengths for paths starting at the given vertices. For weighted path lengths, or to restrict the computation to both certain start and end vertex sets, use IGDistanceHistogram[]. Compute how the shortest path length distribution changes as we rewire a grid graph k times. Table[ ListPlot[ Normalize[IGDistanceCounts@IGRewire[GridGraph[{50, 50}], k], Total], Joined -> True, Filling -> Bottom, PlotLabel -> StringTemplate["rewiring steps: "][k] ], {k, {0, 5, 10, 20, 50, 100}} ] ### IGNeighborhoodSize▴ ?IGNeighborhoodSize IGNeighborhoodSize returns the number of vertices reachable within a certain distance range from a given vertex, or from multiple given vertices. Scale vertices proportionally to the number of their second order neighbours: g = IGBarabasiAlbertGame[50, 2, DirectedEdges -> False]; IGVertexMap[# &, VertexSize -> (Rescale@IGNeighborhoodSize[#, All, {2}] &), g] ### IGDistanceHistogram▴ ?IGDistanceHistogram IGDistanceHistogram[] computes the weighted shortest path length histogram between the specified start and end vertex sets. The start and end vertex sets do not need to be the same. Note that if the graph is undirected, path lengths between s and t will be double counted (from s -> t and t -> s) if s and t appear both in the starting and ending vertex sets. IGDistanceHistogram[] is useful when the result of IGDistanceMatrix[] (or GraphDistanceMatrix[]) does not fit in memory. ### IGAveragePathLength▴ ?IGAveragePathLength IGAveragePathLength computes the average pairwise distances between vertices. Available options: • Method can take the values "Unweighted", "Dijkstra", "BellmanFord", "Johnson" and Automatic. Automatic uses "Unweighted" if no edge weights are present, "Dijkstra" if all weights are non-negative and "Johnson" otherwise. • "ByComponents" controls how unconnected graphs are handled. If False, Infinity is returned. If True, vertex pairs between which there is no path are excluded from the calculation. ### IGGirth▴ ?IGGirth IGGirth computes the girth of a graph, i.e. the length of its shortest cycle. IGGirth ignores multi-edges and self-loops. Edge directions and edge weights are also ignored. 3 If the graph has no cycles, ∞ is returned. IGGirth@IGShorthand["1-2"] ∞ ### IGDiameter and IGFindDiameter▴ ?IGDiameter The diameter of a graph is the length of the longest shortest path between any two vertices. The available options are: • Method can take the values "Unweighted", "Dijkstra" or Automatic. "Dijkstra" takes edge weights into account. Automatic chooses based on whether the graph is weighted. • "ByComponents" controls how unconnected graphs are handled. If False, Infinity is returned. If True, the longest shortest path is returned. In the undirected case, this is the largest diameter of any connected component. 2 For the null graph, Indeterminate is returned. IGDiameter[IGEmptyGraph[]] Indeterminate ?IGFindDiameter {1, 2, 4} HighlightGraph[g, PathGraph@IGFindDiameter[g], GraphHighlightStyle -> "DehighlightFade", PlotTheme -> "RoyalColor" ] ### IGEccentricity▴ ?IGEccentricity The eccentricity of a vertex is the longest shortest path to any other vertex. IGEccentricity computes the unweighted eccentricity of each vertex within the connected component where it is contained. IGEccentricity@CycleGraph[8] {4, 4, 4, 4, 4, 4, 4, 4} Connected components are considered separately. IGEccentricity[IGDisjointUnion[{CycleGraph[3], CycleGraph[8]}]] {1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4} ### IGRadius▴ ?IGRadius The radius of a graph is the smallest eccentricity of any of its vertices, i.e. the eccentricity of the graph center. ### IGVoronoiCells▴ ?IGVoronoiCells IGVoronoiCells[graph, centers] partitions a graph’s vertices into groups based on which given centre vertex they are the closest to. Edge weights are considered for the distance calculations. Available options: • "Tiebreaker" sets the function used to decide which cell a vertex should belong to if its distance to several different centres is equal. The default is to use the first qualifying cell. Possible useful settings are First, Last, RandomChoice. g = PathGraph[Range[5], VertexLabels -> "Name", VertexSize -> Medium] IGVoronoiCells[g, {2, 4}] <|2 -> {1, 2, 3}, 4 -> {4, 5}|> HighlightGraph[g, Values[%]] In the event of a tie, a vertex is added to the first qualifying cell. The tiebreaker function can be changed as below. IGVoronoiCells[g, {2, 4}, "Tiebreaker" -> Last] <|2 -> {1, 2}, 4 -> {3, 4, 5}|> Table[IGVoronoiCells[g, {2, 4}, "Tiebreaker" -> RandomChoice], {5}] {<|2 -> {1, 2, 3}, 4 -> {4, 5}|>, <|2 -> {1, 2}, 4 -> {3, 4, 5}|>, <|2 -> {1, 2, 3}, 4 -> {4, 5}|>, <|2 -> {1, 2, 3}, 4 -> {4, 5}|>, <|2 -> {1, 2}, 4 -> {3, 4, 5}|>} Find Voronoi cells on a grid. g = GridGraph[{10, 10}, VertexSize -> Medium, GraphStyle -> "BasicBlack"]; centers = RandomSample[VertexList[g], 3]; HighlightGraph[g, Append[ Subgraph[g, #] & /@ Values@IGVoronoiCells[g, centers], Style[centers, Black] ], GraphHighlightStyle -> "DehighlightHide" ] Edge weights are interpreted as distances. g = IGMeshGraph@DelaunayMesh@RandomPoint[Disk[], 200]; centers = RandomSample[VertexList[g], 3]; HighlightGraph[g, Append[ Subgraph[g, #] & /@ Values@IGVoronoiCells[g, centers], Style[centers, Black] ], GraphHighlightStyle -> "DehighlightGray" ] ### IGShortestPathTree▴ ?IGShortestPathTree Experimental: This is experimental functionality that may change in the future. g = IGTriangularLattice[4]; HighlightGraph[g, IGShortestPathTree[g, 1], GraphHighlightStyle -> "Thick"] ## Efficiency measures▴ ### IGGlobalEfficiency▴ ?IGGlobalEfficiency IGGlobalEfficiency[graph] computes the global efficiency of a graph. The global efficiency is defined as the average inverse shortest path length between all pairs of vertices, $E_{\text{global}}=\frac{1}{V(V-1)}\sum _{u,v} \frac{1}{d_{uv}},$ where $$d_{uv}$$ is the graph distance from vertex $$u$$ to vertex $$v$$ and $$V$$ is the number of vertices. When $$v$$ is not reachable from $$u$$, $$1\left/d_{uv}\right.$$ is taken to be 0. Available options: • DirectedEdges -> False ignores edge directions when computing shortest path lengths. Compute the global efficiency of a network … g = ExampleData[{"NetworkGraph", "ProteinInteraction"}]; IGGlobalEfficiency[g] 0.0997348 … and that of its spanning tree. IGGlobalEfficiency@IGSpanningTree[g] 0.00106384 #### References ### IGLocalEfficiency▴ ?IGLocalEfficiency IGLocalEfficiency[graph] computes the local efficiency around each vertex of a graph. The local efficiency around a vertex $$u$$ is defined as the average pairwise inverse shortest path length between the neighbours of $$u$$ after excluding $$u$$ itself from the graph, $E_{\text{local}}(u)=\frac{1}{k_u\left(k_u-1\right)}\sum _{v,w\in N(u)} \frac{1}{d_{vw}},$ where $$k_u$$ is the degree of vertex $$u$$, $$N(u)$$ denotes its neighbourhood and $$d_{vw}$$ is the graph distance from vertex $$v$$ to vertex $$w$$. If $$u$$ has less than two neighbours, $$E_{\text{local}}(u)$$ is taken to be 0. Available options: • DirectedEdges -> False ignores edge directions when computing shortest path lengths. Size the vertices of a graph according to the corresponding local efficiency g = ExampleData[{"NetworkGraph", "ZacharyKarateClub"}]; IGVertexMap[1.5 # &, VertexSize -> IGLocalEfficiency, g] Plot the local efficiency versus the local clustering coefficient. ListPlot[ Transpose[{IGLocalClusteringCoefficient[g], IGLocalEfficiency[g]}], PlotTheme -> "Detailed" ] Compute the local efficiency of a subset of vertices only. g = RandomGraph[{10, 20}, DirectedEdges -> True]; IGLocalEfficiency[g, {1, 2, 3}] {0.506944, 0.408333, 0.5} By default, both in- and out-neighbours are considered when determining the neighbourhoods of vertices. We can also consider only in-neighbours or only out-neighbours. {IGLocalEfficiency[g, All, "All"], IGLocalEfficiency[g, All, "In"], IGLocalEfficiency[g, All, "Out"]} {{0.506944, 0.408333, 0.5, 0.388889, 0.275, 0.305556, 0.375, 0.336111, 0.493333, 0.35}, {1., 0.625, 0.5, 0.5, 0.25, 0.25, 0.375, 0.1, 0.416667, 0.}, {0.125, 0., 0., 0.25, 0.388889, 0.25, 0., 0.5, 0.520833, 0.35}} Ignore edge directions when computing shortest paths. IGLocalEfficiency[g, DirectedEdges -> False] {0.722222, 0.833333, 1., 0.75, 0.561667, 0.5, 0.583333, 0.583333, \ 0.7, 0.5} #### References ### IGAverageLocalEfficiency▴ ?IGAverageLocalEfficiency IGAverageLocalEfficiency[graph] computes the average local efficiency of a network. See IGLocalEfficiency for a definition of this graph measure. Plot the decrease in average local efficiency during sequential edge removals. g = RandomGraph[{30, 60}, DirectedEdges -> True]; ListPlot[ Table[ {k, IGAverageLocalEfficiency@ Graph[VertexList[g], Take[EdgeList[g], k]]}, {k, EdgeCount[g]} ], AxesLabel -> {"edge count", "local efficiency"} ] IGAverageLocalEfficiency simply gives the average of the values returned by IGLocalEfficiency. {IGAverageLocalEfficiency[g], Mean@IGLocalEfficiency[g]} {0.210095, 0.210095} Use only the out-neighbourhood while computing the local efficiency. IGAverageLocalEfficiency[g, "Out"] 0.142476 ## Bipartite graphs▴ The vertices of a bipartite graph can be divided into two groups (partitions) such that connections run only between the two partitions, but never within a single partition. ?IGBipartite* ### IGBipartiteQ▴ ?IGBipartiteQ Generate a graph and verify that it is bipartite. g = IGBipartiteGameGNM[5, 5, 10, VertexLabels -> "Name"] IGBipartiteQ[g] True Verify that no edges run between two disjoint vertex subsets of the graph. IGBipartiteQ[g, {{1, 2, 3}, {6, 7, 8}}] True ### IGBipartitePartitions▴ ?IGBipartitePartitions Find a bipartite partitioning of a graph. IGBipartitePartitions[g] {{1, 2, 3, 4}, {5, 6, 7, 8}} Ensure that the partitions are returned in such an order that the first one contains vertex 5. IGBipartitePartitions[g, 5] {{5, 6, 7, 8}, {1, 2, 3, 4}} $Failed is returned for non-bipartite graphs.

IGBipartitePartitions[CompleteGraph[4]]

\$Failed

We can use IGPartitionsToMembership or IGKVertexColoring[…, 2] to obtain a partition index for each vertex.

IGPartitionsToMembership[g]@IGBipartitePartitions[g]
{1, 1, 1, 1, 2, 2, 2, 2}
IGKVertexColoring[g, 2]
{{1, 1, 1, 1, 2, 2, 2, 2}}

### IGBipartiteProjections▴

?IGBipartiteProjections

The following bipartite graph described the relationship between diseases and genes.

g = ExampleData[{"NetworkGraph", "BipartiteDiseasomeNetwork"}]

parts = Values@GroupBy[
First -> Last
];

Construct a disease-disease and gene-gene network from it.

IGBipartiteProjections[g, parts]

### IGBipartiteIncidenceMatrix and IGBipartiteIncidenceGraph▴

?IGBipartiteIncidenceGraph

?IGBipartiteIncidenceMatrix

Compute an incidence matrix. The default partitioning used by IGBipartiteIncidenceMatrix is the one returned by IGBipartitePartitions.

g = IGBipartiteGameGNM[5, 5, 10, VertexLabels -> "Name"]

bm = IGBipartiteIncidenceMatrix[g];
MatrixForm[bm, TableHeadings -> IGBipartitePartitions[g]]

Reconstruct a graph from an incidence matrix.

IGBipartiteIncidenceGraph[bm, VertexLabels -> "Name",
GraphLayout -> "BipartiteEmbedding"]

Compute an incidence matrix using a given partitioning / vertex ordering. It is allowed to pass only a subset of vertices.

IGBipartiteIncidenceMatrix[g, {{1, 2, 3}, {6, 7, 8}}]

Reconstruct the bipartite graph while specifying vertex names.

IGBipartiteIncidenceGraph[{{a, b, c}, {d, e, f}}, %,
VertexLabels -> "Name"]

## Similarity measures▴

The functions in this section characterize the similarity of vertex pairs within a graph.

### IGBibliographicCoupling▴

?IGBibliographicCoupling

The bibliographic coupling of two vertices in a directed graph is the number of other vertices they both connect to. The bibliographic coupling matrix can also be obtained using am . am$Transpose] - DiagonalMatrix@VertexInDegree[g], where am is the adjacency matrix of the graph g. ?IGStaticPowerLawGame Create a random graph and compute its bibliographic coupling matrix. g = IGStaticPowerLawGame[10, 25, 2, 4, GraphLayout -> "CircularEmbedding", GraphStyle -> "BasicBlack"] cc = IGBibliographicCoupling[g]; MatrixForm[cc, TableHeadings -> {VertexList[g], VertexList[g]}] Construct the weighted graph corresponding to the bibliographic coupling of vertices and visualize it. ccg = IGWeightedAdjacencyGraph[cc, VertexCoordinates -> GraphEmbedding[g], GraphStyle -> "ThickEdge"] // IGEdgeMap[Thickness[0.02 #] &, EdgeStyle -> IGEdgeProp[EdgeWeight]] Overlay the bibliographic coupling graph with the original directed graph. Show[ccg, g] ### IGCocitationCoupling▴ ?IGCocitationCoupling The co-citation coupling of two vertices in a directed graph is the number of other vertices that connect to both of them. The co-citation coupling matrix can also be obtained using am\[Transpose] . am - DiagonalMatrix@VertexOutDegree[g], where am is the adjacency matrix of the graph g. ### IGDiceSimilarity▴ ?IGDiceSimilarity The Dice similarity coefficient of two vertices is twice the number of common neighbours divided by the sum of the degrees of the vertices. For directed graphs, out-neighbours are considered. Edge multiplicities are not taken into account. The available options are: • SelfLoops -> True will include self-loops in the calculation of the similarity score. ### IGJaccardSimilarity▴ ?IGJaccardSimilarity The Jaccard similarity coefficient of two vertices is the number of common neighbours divided by the number of vertices that are neighbours of at least one of the two vertices being considered. For directed graphs, out-neighbours are considered. Edge multiplicities are not taken into account. The available options are: • SelfLoops -> True will include self-loops in the calculation of the similarity score. Construct and visualize a weighted graph of Jaccard similarities between vertices of an animal social network: g = ExampleData[{"NetworkGraph", "DolphinSocialNetwork"}] IGWeightedAdjacencyGraph[IGZeroDiagonal@IGJaccardSimilarity[g], VertexCoordinates -> GraphEmbedding[g]] // IGEdgeMap[GrayLevel[0, #] &, EdgeStyle -> IGEdgeProp[EdgeWeight]] Compare it to the inverse log-weighted similarity: IGWeightedAdjacencyGraph[Rescale@IGInverseLogWeightedSimilarity[g], VertexCoordinates -> GraphEmbedding[g]] // IGEdgeMap[GrayLevel[0, #] &, EdgeStyle -> IGEdgeProp[EdgeWeight]] ### IGInverseLogWeightedSimilarity▴ ?IGInverseLogWeightedSimilarity The inverse log-weighted similarity of two vertices is the number of their common neighbours, weighted by the inverse natural logarithm of the neighbours’ degrees. It is also known as the Adamic–Adar index. It is based on the assumption that two vertices should be considered more similar if they share a low-degree common neighbour, since high-degree common neighbours are more likely to appear even by pure chance. Formally, the similarity of vertices $$u$$ and $$v$$ is \[A(u,v)=\sum _{w\in \mathcal{N}(u)\cap \mathcal{N}(v)} \frac{1}{\ln d_w},$

where $$\mathcal{N}(u)$$ denotes the neighbourhood of vertex $$u$$ and $$d_w$$ denotes the degree of vertex $$w$$.

Isolated vertices will have zero similarity to any other vertex. Self-similarities are not calculated.

In directed graphs, the out-neighbours of each vertex are considered, weighted by the inverse logarithm of their in-degrees.

## Connectivity and graph components▴

### IGConnectedQ and IGWeaklyConnectedQ▴

?IGConnectedQ

?IGWeaklyConnectedQ

IGConnectedQ checks if the graph is (strongly) connected. It is equivalent to ConnectedGraphQ. IGWeaklyConnectedQ check if a directed graph is weakly connected. It is equivalent to WeaklyConnectedGraphQ. Both of these functions use the implementation from the core igraph library, and will always be consistent with it for edge cases such as the null graph.

This graph is connected.

True

This directed graph is only weakly connected.

False

True

The null graph is considered disconnected by convention.

IGConnectedQ@IGEmptyGraph[0]
False

### IGConnectedComponentSizes and IGWeaklyConnectedComponentSizes▴

?IGConnectedComponentSizes

?IGWeaklyConnectedComponentSizes

IGWeaklyConnectedComponentsSizes and IGConnectedComponentSizes return the sizes of the graph’s weakly or strongly connected components in decreasing order.

In large graphs, these functions will be faster than the equivalent Length /@ ConnectedComponents[g].

The emergence of a giant component as the number of edges in a random graph increases.

Table[
{m, First@IGConnectedComponentSizes@RandomGraph[{1000, m}]},
{m, 5, 2000, 5}
] // ListPlot

The number of weakly and strongly connected components versus the number of edges in a random directed graph.

Table[
With[{g = RandomGraph[{1000, m}, DirectedEdges -> True]},
{{m, Length@IGWeaklyConnectedComponentSizes[g]}, {m,
Length@IGConnectedComponentSizes[g]}}
],
{m, 5, 3000, 5}
] // Transpose // ListPlot

### IGFindMinimumCuts▴

?IGFindMinimumCuts

IGFindMinimalCuts[g, s, t] finds all **smallest-weight (i.e. minimum) edge cuts that disconnect vertex t from vertex s.

g = ExampleData[{"NetworkGraph", "MetabolicNetworkAeropyrumPernix"}];
IGFindMinimumCuts[g, 30, 160]
{{30 DirectedEdge] 1000177, 30 \[DirectedEdge] 1000178}, {30 \[DirectedEdge] 1000178, 1000177 \[DirectedEdge] 10}, {1000080 \[DirectedEdge] 160, 1000092 \[DirectedEdge] 160}} Visualize all minimum cuts between two vertices in a random cubic graph. g = IGKRegularGame[20, 3]; HighlightGraph[g, Join[#, {1, 20}], GraphHighlightStyle -> "Dotted", VertexSize -> Large] & /@ IGFindMinimumCuts[g, 1, 20] Warning: IGFindMinimumCuts takes edge weights into account, but it is only safe to use with integer weights. If the weights are not integers, then numerical roundoff errors may prevent the function from detecting that two cuts have the same total weight. Create an integer-weighted graph with more than one minimum cut between vertices 1 and 10: g = IGTryUntil[Length@IGFindMinimumCuts[#, 1, 10] > 2 &][ RandomGraph[{10, 30}, DirectedEdges -> True, EdgeWeight -> RandomInteger[{1, 10}, 30]]] IGFindMinimumCuts[g, 1, 10] {{1 \[DirectedEdge] 3, 1 \[DirectedEdge] 6, 1 \[DirectedEdge] 7, 1 \[DirectedEdge] 8}, {1 \[DirectedEdge] 3, 2 \[DirectedEdge] 3, 2 \[DirectedEdge] 10, 4 \[DirectedEdge] 10}, {2 \[DirectedEdge] 10, 3 \[DirectedEdge] 10, 4 \[DirectedEdge] 10}} Multiplying the weights by 0.1 causes IGFindMinimumCuts to return fewer results because some of the weights are no longer exactly representable in binary: IGFindMinimumCuts[IGEdgeMap[0.1 # &, EdgeWeight, g], 1, 10] {{1 \[DirectedEdge] 3, 1 \[DirectedEdge] 6, 1 \[DirectedEdge] 7, 1 \[DirectedEdge] 8}} If only a single minimum cut is needed, use IGMinimumCut: IGMinimumCut[g, 1, 10] {2 \[DirectedEdge] 10, 3 \[DirectedEdge] 10, 4 \[DirectedEdge] 10} The size (total weight) of the cut can be obtained with IGMinimumCutValue: IGMinimumCutValue[g, 1, 10] 14. #### References ### IGFindMinimalCuts▴ ?IGFindMinimalCuts IGFindMinimalCuts[g, s, t] finds all unweighted minimal edge cuts that disconnect vertex t from vertex s. IGFindMinimalCuts[g, 1, 10] {{1 \[DirectedEdge] 2, 1 \[DirectedEdge] 5, 1 \[DirectedEdge] 10}, {1 \[DirectedEdge] 2, 1 \[DirectedEdge] 10, 5 \[DirectedEdge] 10}, {1 \[DirectedEdge] 5, 1 \[DirectedEdge] 10, 2 \[DirectedEdge] 3, 2 \[DirectedEdge] 5}, {1 \[DirectedEdge] 10, 2 \[DirectedEdge] 3, 5 \[DirectedEdge] 10}, {1 \[DirectedEdge] 5, 1 \[DirectedEdge] 10, 2 \[DirectedEdge] 5, 3 \[DirectedEdge] 4, 3 \[DirectedEdge] 5}, {1 \[DirectedEdge] 10, 3 \[DirectedEdge] 4, 5 \[DirectedEdge] 10}, {1 \[DirectedEdge] 5, 1 \[DirectedEdge] 10, 2 \[DirectedEdge] 5, 3 \[DirectedEdge] 5, 4 \[DirectedEdge] 10}, {1 \[DirectedEdge] 10, 4 \[DirectedEdge] 10, 5 \[DirectedEdge] 10}} The set of all minimum cuts is a subset of the minimal ones. IGFindMinimumCuts[g, 1, 10] {{1 \[DirectedEdge] 2, 1 \[DirectedEdge] 5, 1 \[DirectedEdge] 10}, {1 \[DirectedEdge] 2, 1 \[DirectedEdge] 10, 5 \[DirectedEdge] 10}, {1 \[DirectedEdge] 10, 2 \[DirectedEdge] 3, 5 \[DirectedEdge] 10}, {1 \[DirectedEdge] 10, 3 \[DirectedEdge] 4, 5 \[DirectedEdge] 10}, {1 \[DirectedEdge] 10, 4 \[DirectedEdge] 10, 5 \[DirectedEdge] 10}} SubsetQ[%%, %] True Visualize all minimal cuts between two vertices, from smallest to largest, in an undirected graph. g = IGGiantComponent@RandomGraph[{8, 12}]; HighlightGraph[g, Join[#, {1, 8}], GraphHighlightStyle -> "Dashed", VertexSize -> Medium] & /@ SortBy[Length]@IGFindMinimalCuts[g, 1, 8] #### References ### Vertex separators▴ A vertex separator is a set of vertices whose removal disconnects the graph. ?IGMinimalSeparators ?IGMinimumSeparators ?IGVertexSeparatorQ ?IGMinimalVertexSeparatorQ g = ExampleData[{"NetworkGraph", "Friendship"}] separators = IGMinimumSeparators[g] {{"Anna", "Rose"}, {"Larry", "Rudy"}, {"Larry", "James"}, {"Rudy", "Linda"}, {"Anna", "Nora"}, {"Anna", "Ben"}, {"Anna", "Larry"}, {"Anna", "Linda"}, {"Anna", "James"}} Removing any of these vertex sets will disconnect the graph: VertexDelete[g, #] & /@ separators IGVertexSeparatorQ[g, #] & /@ separators {True, True, True, True, True, True, True, True, True} IGMinimalVertexSeparatorQ[g, #] & /@ separators {True, True, True, True, True, True, True, True, True} Removing Anna, Nora and Larry also disconnects the graph, thus this vertex set is a separator: IGVertexSeparatorQ[g, {"Anna", "Nora", "Larry"}] True But it is not minimal: IGMinimalVertexSeparatorQ[g, {"Anna", "Nora", "Larry"}] False IGMinimumSeparators returns only those vertex separators which are of the smallest possible size in the graph. IGMinimalSeparators returns all separators which cannot be made smaller by removing a vertex from them. The former is a subset of the latter. IGMinimalSeparators[g] {{1, 6}, {0, 2}, {1, 3, 4}, {2, 5}, {3, 4, 6}, {0, 5}, {2, 6}, {1, 5}, {0, 3, 4}} IGMinimumSeparators[g] {{2, 5}, {1, 5}, {1, 6}, {0, 5}, {2, 6}, {0, 2}} SubsetQ[%%, %] True ### IGEdgeConnectivity▴ ?IGEdgeConnectivity IGEdgeConnectivity ignores edge weights. To take edge weights into account, use IGMinimumCutValue instead. Compute the edge connectivity of the dodecahedral graph. IGEdgeConnectivity[GraphData["DodecahedralGraph"]] 3 The edge connectivity of the singleton graph is returned as 0. IGEdgeConnectivity[IGEmptyGraph[1]] 0 ### IGVertexConnectivity▴ ?IGVertexConnectivity According to Steinitz’s theorem, the skeleton of every convex polyhedron is a 3-vertex-connected planar graph. g = GraphData["DodecahedralGraph"] IGVertexConnectivity[g] 3 To find the specific vertex sets that disconnect the graph, use IGMinimumSeparators or IGMinimalSeparators. IGMinimumSeparators[g] {{14, 15, 16}, {5, 6, 13}, {7, 14, 19}, {8, 15, 20}, {2, 11, 19}, {2, 12, 20}, {3, 11, 16}, {4, 12, 16}, {10, 14, 17}, {9, 15, 18}, {5, 7, 12}, {6, 8, 11}, {2, 17, 18}, {9, 13, 19}, {10, 13, 20}, {3, 5, 17}, {4, 6, 18}, {1, 3, 9}, {1, 4, 10}, {1, 7, 8}} The vertex connectivity of the singleton graph is returned as 0. IGVertexConnectivity[IGEmptyGraph[1]] 0 ### IGBiconnectedQ▴ ?IGBiconnectedQ IGBiconnectedQ checks if a graph is biconnected. Edge directions are ignored. False Since IGBiconnectedComponents does not return any isolated vertices, Length@IGBiconnectedComponents[g] == 1 cannot be used to check if a graph is biconnected. Use IGBiconnectedQ instead. {{4, 3, 2, 1}} The singleton graph is not considered to be biconnected, but the two-vertex complete graph is. Table[IGBiconnectedQ@CompleteGraph[k], {k, 1, 2}] {False, True} ### IGBiconnectedComponents and IGBiconnectedEdgeComponents▴ ?IGBiconnectedComponents ?IGBiconnectedEdgeComponents IGBiconnectedCompoments returns the vertices of the maximal biconnected components of the graph. IGBiconnectedEdgeComponents returns the edges of the components. Edge directions are ignored and isolated vertices are excluded. IGBiconnectedComponents is equivalent to KVertexConnectedComponents[…, 2], except that isolated vertices are not returned as individual components. The articulation vertices will be part of more than a single component, thus the biconnected components are not disjoint subsets of the vertex set. {{3, 2, 1}, {4, 1}} However, each edge is part of precisely one biconnected components. {{1 <-> 3, 2 <-> 3, 1 <-> 2}, {1 <-> 4}} Thus, visualizing biconnected components is best done by colouring the edges, not the vertices. HighlightGraph[g, IGBiconnectedEdgeComponents[g], GraphStyle -> "ThickEdge"] ### IGArticulationPoints▴ ?IGArticulationPoints IGArticulationPoints finds vertices whose removal increases the number of (weakly) connected components in the graph. Edge directions are ignored. g = Graph[{1 -> 2, 2 -> 3, 3 -> 1, 3 -> 4, 4 -> 5, 5 -> 6, 6 -> 4}, DirectedEdges -> False, VertexLabels -> Automatic] IGArticulationPoints[g] {4, 3} VertexDelete[g, #] & /@ % Articulation points are also size-1 separators. IGMinimumSeparators[g] {{4}, {3}} Highlight the articulation points of a cactus graph. HighlightGraph[g, IGArticulationPoints[g]] Compute the block-cut tree of a connected graph. The blocks are the biconnected components. Together with the articulation vertices they form a bipartite graph, specifically a tree. RelationGraph[ MemberQ, Join[IGBiconnectedComponents[g], IGArticulationPoints[g]], DirectedEdges -> False, GraphStyle -> "ClassicDiagram", VertexSize -> {3, 1}/7, VertexLabelStyle -> 8 ] ### IGBridges▴ ?IGBridges A bridge is an edge whose removal disconnects the graph (or increases the number of connected components if the graph was already disconnected). Edge directions are ignored. IGShorthand["1-2-3-1-4-5-6-4"] IGBridges[%] {1 <-> 4} Highlight bridges in a network. g = ExampleData[{"NetworkGraph", "FlorentineFamilies"}]; HighlightGraph[g, IGBridges[g]] ### IGSourceVertexList and IGSinkVertexList▴ ?IGSourceVertexList ?IGSinkVertexList Find and highlight the source and sink vertices of a random acyclic graph. g = DirectedGraph[RandomGraph[{10, 20}], "Acyclic", VertexLabels -> "Name", VertexSize -> Large, EdgeStyle -> Gray] IGSourceVertexList[g] {1, 2} IGSinkVertexList[g] {9, 10} HighlightGraph[g, {IGSourceVertexList[g], IGSinkVertexList[g]}] Undirected graphs have neither source nor sink vertices because undirected edges are counted as bidirectional. {} The exception is isolated vertices, which are counted both as sources and sinks. Through[{IGSourceVertexList, IGSinkVertexList}@Graph[{1, 2}, {}]] {{1, 2}, {1, 2}} These are merely convenience functions that can be implemented straightforwardly as Pick[VertexList[g], VertexOutDegree[g], 0] {9, 10} ### IGIsolatedVertexList▴ ?IGIsolatedVertexList IGIsolatedVertexList returns the vertices which form their own weakly connected components. This includes vertices with no connections, as well as vertices with only self-loops. {1, 5} ### IGGiantComponent▴ ?IGGiantComponent IGGiantComponent is a convenience function that returns the largest weakly connected component of graph. If there are multiple components of largest size, there is no guarantee about which one would be returned. If this is a concern, use WeaklyConnectedComponents or WeaklyConnectedGraphComponents instead. g = RandomGraph[{200, 200}]; HighlightGraph[ g, IGGiantComponent[g] ] // IGLayoutFruchtermanReingold IGGiantComponent takes all standard graph options. IGGiantComponent[g, GraphStyle -> "BasicGreen", GraphLayout -> "SpringEmbedding"] Size of the giant component of a random subgraph of a grid graph. g = IGSquareLattice[{30, 30}, "Periodic" -> True]; Table[ {k, VertexCount@ IGGiantComponent@Subgraph[g, RandomSample[VertexList[g], k]]}, {k, 1, VertexCount[g], 1} ] // ListPlot ### IGPercolationCurve▴ ?IGPercolationCurve Experimental: This is experimental functionality that may change in the future. IGPercolationCurve computes the percolation curve for a sequence of edge additions (interpretable as edge removals in reverse order). The ith element of the result is the mean degree and the fraction of vertices within the largest component before adding the ith edge. IGPercolationCurve[graph] is equivalent to IGPercolationCurve[RandomSample@EdgeList[graph], VertexCount[graph]]. Plot the averaged percolation curve of a grid graph over many random edge removals. ListLinePlot@Mean@Table[IGPercolationCurve@GridGraph[{50, 50}], {100}] Percolation curve for a random geometric graph when edges are removed in order of decreasing or increasing betweenness. g = IGGeometricGame[500, 0.1]; edgeOrder = EdgeList[g][[Ordering@IGEdgeBetweenness[g]]]; ListLinePlot[IGPercolationCurve /@ {edgeOrder, Reverse[edgeOrder]}, PlotLegends -> {"decreasing betweenness", "increasing betweenness"}, FrameLabel -> {"mean degree", "largest component fraction"}, Frame -> True, PlotRange -> All] IGPercolationCurve also accepts a list of pairs in addition to a list of edge expressions. IGPercolationCurve@RandomInteger[{1, 10}, {20, 2}] {{0., 0.1}, {0., 0.2}, {0.2, 0.2}, {0.4, 0.2}, {0.6, 0.2}, {0.8, 0.3}, {1., 0.4}, {1.2, 0.4}, {1.4, 0.6}, {1.6, 0.7}, {1.8, 0.7}, {2., 0.9}, {2.2, 0.9}, {2.4, 1.}, {2.6, 1.}, {2.8, 1.}, {3., 1.}, {3.2, 1.}, {3.4, 1.}, {3.6, 1.}, {3.8, 1.}} IGPercolationCurve works efficiently on large networks. g = ExampleData[{"NetworkGraph", "WorldWideWeb"}]; EdgeCount[g] 1497134 ListLinePlot[IGPercolationCurve[g], MaxPlotPoints -> 1000] ## Trees▴ A tree is a connected graph that contains no undirected cycles. ### IGTreeQ▴ ?IGTreeQ IGTreeQ checks if a graph is a tree. An undirected tree is a connected graph with no cycles. A directed tree is similar, with its edges oriented either away from a root vertex (out-tree or arborescence) or towards a root vertex (in-tree or anti-arborescence). True False By convention, the null graph is not a tree. IGTreeQ[IGEmptyGraph[0]] False This is an out-tree. True It is not also an in-tree. False It becomes an in-tree if we reverse its edges. True This graph is neither an out-tree nor an in-tree. False However, it becomes a tree if we ignore edge directions. True ### IGForestQ▴ ?IGForestQ IGForestQ is a convenience function that tests if all connected components of a graph are trees. This graph is not a tree, but it is a forest. {False, True} By convention, the null graph is not a tree, but it is a forest. {IGTreeQ[IGEmptyGraph[0]], IGForestQ[IGEmptyGraph[0]]} {False, True} Use the second argument to test for forests of out-trees or in-trees. By default, directed graphs are checked to be out-forests. g = IGShorthand["1->2<-3, 4<-5->6"] {IGForestQ[g], IGForestQ[g, "Out"], IGForestQ[g, "In"], IGForestQ[g, "All"]} {False, False, False, True} ### IGStrahlerNumber▴ ?IGStrahlerNumber IGStrahlerNumber computes the Horton–Strahler index of each vertex in a rooted tree. The tree must be directed—this is how the root is encoded. The Horton–Strahler index of the tree itself is the index of the root, i.e. the largest returned index. This measure is also called stream order, as it was originally used to characterize river networks. tree = IGTreeGame[30, DirectedEdges -> True, GraphLayout -> "LayeredDigraphEmbedding"] IGVertexMap[# &, VertexLabels -> IGStrahlerNumber, tree] To get the Horton–Strahler number of the tree, find the maximal element. Max@IGStrahlerNumber[tree] 12 IGStrahlerNumber requires a directed (i.e. rooted) tree as input. Failed Orient undirected trees, effectively specifying a root vertex, before passing them to IGStrahlerNumber. {5, 4, 3, 1, 2, 2, 1, 1, 1, 1} ### IGTreelikeComponents▴ ?IGTreelikeComponents IGTreelikeComponents finds the tree-like components of an undirected graph by repeatedly identifying and removing degree-1 vertices. Vertices in the tree-like components are not part of any undirected cycle, nor are they on a path connecting vertices that belong to a cycle. g = RandomGraph[{100, 100}]; HighlightGraph[ g, IGTreelikeComponents[g] ] // IGLayoutFruchtermanReingold Highlight both the edges and vertices of tree-like components. g = IGGiantComponent@RandomGraph[{50, 50}]; HighlightGraph[ g, Join[ Union @@ (IncidenceList[g, #] &) /@ IGTreelikeComponents[g], IGTreelikeComponents[g] ] ] IGLayoutFruchtermanReingold[%] Remove tree-like components. VertexDelete[g, IGTreelikeComponents[g]] Vertices incident to multi-edges or loop-edges are not part of tree-like components. {1} ### IGFromPrufer▴ ?IGFromPrufer ### IGToPrufer▴ ?IGToPrufer ## Spanning trees▴ A spanning tree of a graph is a subgraph that is a tree and contains all the graph’s vertices. ### IGSpanningTree▴ ?IGSpanningTree IGSpanningTree[RandomGraph[{8, 20}], GraphStyle -> "DiagramGold"] Find the shortest set of paths connecting a set of points in the plane: pts = RandomReal[1, {10, 2}]; g = IGMeshGraph@DelaunayMesh[pts]; tree = IGSpanningTree[g, VertexCoordinates -> pts] The edge weights are preserved in the result. IGEdgeWeightedQ[tree] True Compute the total path length. Total@IGEdgeProp[EdgeWeight][tree] 2.0021 Find a maximum spanning tree by negating the weights before running the algorithm. IGSpanningTree[IGEdgeMap[Minus, EdgeWeight, g], VertexCoordinates -> pts] Find the minimum and maximum spanning trees of a network, using its edge betweenness as edge weights. g = ExampleData[{"NetworkGraph", "ZacharyKarateClub"}]; HighlightGraph[ g, EdgeList@ IGSpanningTree@IGEdgeMap[#, EdgeWeight -> IGEdgeBetweenness, g], GraphHighlightStyle -> "Thick", ImageSize -> Small ] & /@ { Identity, (* minimum spanning tree *) Minus (* maximum spanning tree *) } ### IGRandomSpanningTree▴ ?IGRandomSpanningTree IGRandomSpanningTree samples the spanning trees (or forests) of a graph uniformly by performing a loop-erased random walk. Edge directions are ignored. If a spanning forest of the entire graph is requested using IGRandomSpanningTree[g], then the vertex names and ordering are preserved. If a spanning tree of only a single component is requested using IGRandomSpanningTree[{g, v}], then this is not the case. Highlight a few random spanning trees of the Petersen graph. g = PetersenGraph[]; HighlightGraph[g, #, GraphHighlightStyle -> "Thick"] & /@ IGRandomSpanningTree[g, 9] If the input is a multi-graph, each edge will be considered separately for the purpose of spanning tree calculations. Thus the following graph has not 3, but 5 different spanning trees. Two pairs of these are indistinguishable based on their adjacency matrix due to the indistinguishability of the two parallel 1 <-> 2 edges. However, since all 5 spanning trees are generated with equal probability, two of the 3 adjacency matrices will appear twice as frequently as the third one. g = IGShorthand["1-2-3-1,1-2", MultiEdges -> True, ImageSize -> Small] IGRandomSpanningTree[g, 10000] // CountsBy[AdjacencyMatrix] // KeySort // KeyMap[MatrixForm] Edge directions are ignored for the purpose of spanning tree calculation. Thus the result may not be an out-tree. IGRandomSpanningTree@WheelGraph[11, DirectedEdges -> True] Create mazes by taking random spanning trees of grid graphs. g = GridGraph[{10, 10}, GraphStyle -> "Web"]; HighlightGraph[g, IGRandomSpanningTree[g], GraphHighlightStyle -> "DehighlightHide"] g = GridGraph[{6, 6, 6}, VertexCoordinates -> Tuples[Range[6], {3}]]; HighlightGraph[g, IGRandomSpanningTree[g], GraphHighlightStyle -> "DehighlightHide"] Generate a random spanning tree of the component containing vertex 8. ### IGSpanningTreeCount▴ ?IGSpanningTreeCount IGSpanningTreeCount computes the number of spanning trees of a graph using Kirchhoff’s theorem. Multigraphs and directed graphs are supported. 3 The number of spanning trees of a directed graph, rooted in any vertex. 3 The number of spanning trees rooted in vertex 1. 1 IGSpanningTreeCount[PetersenGraph[]] 2000 IGSpanningTreeCount works on large graphs. g = HypercubeGraph[6] IGSpanningTreeCount[g] 1657509127047778993870601546036901052416000000 Edge multiplicities are taken into account. Thus the following graph has not 3, but 5 different spanning trees. 5 ## Dominance▴ In a directed graph, a vertex $$d$$ is said to dominate a vertex $$v$$ if every path from the root to $$v$$ passes through $$d$$. We say that $$d$$ is an immediate dominator of $$v$$ if it does not dominate any other dominator of $$v$$. A dominator tree of a graph consists of the same vertices as the graph, and the children of a vertex are those other vertices which it immediately dominates. ### IGDominatorTree▴ ?IGDominatorTree Find the dominator tree of a directed graph. IGDominatorTree[g, "a", GraphStyle -> "VintageDiagram"] Vertices that cannot be reached from the specified root are left isolated in the returned graph. IGDominatorTree[g, "b", GraphStyle -> "VintageDiagram"] IGDominatorTree accepts all standard Graph options. ### IGImmediateDominators▴ ?IGImmediateDominators Directly find the immediate dominators of vertices in a graph. <|"b" -> "a", "c" -> "b", "d" -> "a", "e" -> "a", "f" -> "e"|> The immediate dominator of a vertex is its parent in the dominator tree. tree = IGDominatorTree[g, "a", VertexLabels -> Automatic] IGAdjacencyList[tree, "In"] <|"a" -> {}, "b" -> {"a"}, "c" -> {"b"}, "d" -> {"a"}, "e" -> {"a"}, "f" -> {"e"}|> Neither the root, nor vertices unreachable from the root are included in the keys of the returned association. IGImmediateDominators[g, "b"] <|"c" -> "b", "d" -> "c"|> ## $$k$$-cores▴ ?IGCoreness A $$k$$-core of a graph is a maximal subgraph in which each vertex has degree at least $$k$$. The coreness of a vertex is the highest order of $$k$$-cores that contain it. IGCoreness[g] {3, 3, 3, 3, 2, 2, 2, 2} {KCoreComponents[g, 2], KCoreComponents[g, 3]} {{{1, 2, 3, 4, 6, 7, 8, 5}}, {{1, 2, 3, 4}}} By default, edge directions are ignored, and multi-edges are considered. IGCoreness[g] {4, 4, 4, 3} Use the second argument to consider only in- or out-degrees. IGCoreness[g, "In"] {2, 2, 2, 1} IGCoreness[g, "Out"] {2, 2, 2, 2} ## Matchings▴ A matching of a graph is a subset of its edges that share no vertices between them. ### IGMaximumMatching▴ ?IGMaximumMatching A matching of a graph is also known as an independent edge set. IGMaximumMatching ignores edge directions and edge weights. g = RandomGraph[{10, 20}]; IGMaximumMatching[g] {7 <-> 10, 6 <-> 8, 3 <-> 9, 2 <-> 5, 1 <-> 4} HighlightGraph[g, IGMaximumMatching[g], GraphHighlightStyle -> "Thick"] ### IGMatchingNumber▴ ?IGMatchingNumber The matching number of a graph is the size of its maximum matchings. ## Graph traversal▴ ### IGUnfoldTree▴ ?IGUnfoldTree IGUnfoldTree creates a tree based on the breadth-first traversal of a graph. Each time a graph vertex is found, a new tree vertex is created. Available options: • DirectedEdges -> False will ignore edge directions in directed graphs. Otherwise, the search is done only along edge directions. The original vertex that generates a tree node is stored in the "OriginalVertex" property. IGVertexProp["OriginalVertex"][tree] {1, 2, 3, 4, 6, 7, 8, 5, 3, 4, 4, 8} We can label the tree nodes with the name of the original vertex either using pattern matching in VertexLabels along with PropertyValue IGLayoutReingoldTilford[ tree, "RootVertices" -> {1}, VertexLabels -> (v_ :> PropertyValue[{tree, v}, "OriginalVertex"]) ] … or using IGVertexMap. IGLayoutReingoldTilford[tree, "RootVertices" -> {1}] // IGVertexMap[# &, VertexLabels -> IGVertexProp["OriginalVertex"]] In directed graphs, the search is done along edge directions. It may be necessary to give multiple starting roots to fully unfold a weakly connected (or unconnected) graph. IGUnfoldTree[Graph[{1 -> 2, 2 -> 3}], {2, 1}] // IGVertexMap[# &, VertexLabels -> IGVertexProp["OriginalVertex"]] Use DirectedEdges -> False to ignore edge directions during the search. Edge directions are still preserved in the result. IGUnfoldTree[Graph[{1 -> 2, 2 -> 3}], {2}, DirectedEdges -> False] // IGVertexMap[# &, VertexLabels -> IGVertexProp["OriginalVertex"]] ## Other structural properties▴ ### IGNullGraphQ▴ ?IGNullGraphQ IGNullGraphQ returns True only for the null graph, i.e. the graph that has no vertices. IGNullGraphQ[IGEmptyGraph[]] True For graphs that have vertices, but no edges, it returns False. IGNullGraphQ[IGEmptyGraph[5]] False In contrast, the built-in EmptyGraphQ tests if there are no edges: EmptyGraphQ[IGEmptyGraph[5]] True ### IGCompleteQ▴ ?IGCompleteQ IGCompleteQ tests if a graph is complete, i.e. if all pairs of vertices are connected. IGCompleteQ@IGCompleteGraph[10] True IGCompleteQ@IGCompleteGraph[5, DirectedEdges -> True] True IGCompleteQ ignores self-loops and multi-edges. True Check if each connected component of a graph is a clique. g = GraphData[{8, 911}] AllTrue[ConnectedGraphComponents[g], IGCompleteQ] True The null graph is considered complete. IGCompleteQ@IGEmptyGraph[] True ### IGCactusQ▴ ?IGCactusQ IGCactusQ tests if a graph is a cactus. A cactus graph is a connected undirected graph in which any two simple cycles share at most one vertex. Equivalently, a cactus is a connected graph in which every edge belongs to at most one simple cycle. True IGCactusQ[GridGraph[{2, 3}]] False IGCactusQ supports multigraphs and ignores self-loops. {True, False} The null graph is not considered to be a cactus, but the singleton graph is. IGCactusQ /@ {IGEmptyGraph[0], IGEmptyGraph[1]} {False, True} Currently, IGCactusQ does not support directed graphs. IGCactusQ[Graph[{1 -> 2}]] Failed # Motifs and subgraphs ## Motifs▴ IGraph/M’s motif-related functions count the number of times each possible connectivity pattern of $$k$$ vertices (i.e. induced subgraph of size $$k$$) occurs in a graph. The patterns are called motifs. As of IGraph/M 0.6, size 3 and 4 motifs are supported in directed graphs and size 3 to 6 in undirected graphs. Only (weakly) connected subgraphs are considered. To count larger induced subgraphs, see IGLADSubisomorphismCount. To identify where a subgraph occurs, see IGLADFindSubisomorphisms. To count non-connected size-3 subgraphs, use IGTriadCensus. igraph’s motif functions use the RAND-ESU algorithm, which is able to uniformly sample a random subset of motifs (connected subgraphs), and can thus estimate motif counts even in very large graphs. See the description of IGMotifs for an example. #### References • S. Wernicke, Efficient Detection of Network Motifs, IEEE/ACM Trans. Comput. Biol. Bioinforma. 3, 347 (2006). ### IGMotifs▴ ?IGMotifs IGMotifs counts how many times each motif (i.e. induced subgraph) of the given size occurs in the graph. For subgraphs that are not weakly connected, Indeterminate is returned. Available options are: • DirectedEdges -> False treats the graph as undirected and DirectedEdges -> True treats the graph as directed. The default is DirectedEdges -> Automatic, which respects the directedness of the graph. Motifs are returned by their IGIsoclass, i.e. the same order as listed in IGData. mot3 = Graph[#, ImageSize -> 36, VertexSize -> 0.1] & /@ IGData[{"AllDirectedGraphs", 3}] Let us count size-3 motifs in the following graph, and summarize them a table. For non-weakly-connected subgraphs, Indeterminate is returned. g = RandomGraph[{10, 40}, DirectedEdges -> True] Grid[{mot3, IGMotifs[g, 3]}\[Transpose], Frame -> All] Empty graphs are treated as undirected by default. To treat them as directed, use DirectedEdges -> True. The result will be different as the number of non-isomorphic graphs on $$k$$ vertices is not the same in the directed and undirected cases. IGMotifs[IGEmptyGraph[5], 3, DirectedEdges -> #] & /@ {Automatic, True, False} {{Indeterminate, Indeterminate, 0, 0}, {Indeterminate, Indeterminate, 0, Indeterminate, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {Indeterminate, Indeterminate, 0, 0}} #### Example: metabolic network Let us find the size-4 motifs that stand out in the E. coli metabolic network by comparing the motif counts to that of a rewired graph: g = ExampleData[{"NetworkGraph", "MetabolicNetworkEscherichiaColi"}]; rg = IGRewire[g, 50000]; {Indeterminate, Indeterminate, Indeterminate, 1.30949, Indeterminate, \ Indeterminate, Indeterminate, 1.31896, 0.353667, Indeterminate, \ Indeterminate, Indeterminate, 0.578637, 0.58617, 0., Indeterminate, \ 0.016559, 0., 0., 4.94656, 0., 0., Indeterminate, Indeterminate, \ 1.29702, 0.346928, 0.165433, Indeterminate, Indeterminate, 0.674934, \ 0., 0.0190229, 0., Indeterminate, Indeterminate, 0., 0., 0., 0., \ Indeterminate, 0.28928, 0.708281, 0., 0.300212, 0., 0.188359, \ 0.0603697, 0., 0., 0., 0., 0., 1.37343, 0., 0.0753138, 0., 0., 0., \ 0., 0., 0., 0., Indeterminate, 0., 0., 0., 31.1911, 0., 0.21875, 0., \ 0., 0., 0., 0.186047, 0., 0., 1.25741, 0., 0., 0., 0., 0., 0., 0., \ 0., 0., 0., 0., 0., 0., 0., Indeterminate, 0.341903, 0.129316, \ 1.38161, 0., 0.0207404, 0., 0.245275, 0., 0.046376, 0., 0., 0., 0., \ 0., 0., 0., 0., 0., 0.589124, 0., 0.275862, 0., 0., 0., 0., 0., 0., \ 0., Indeterminate, 0.429719, 0., 0., 0., 2.11212, 0., 0., 44.6957, \ 0., 0.557143, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \ 0., 2.22222, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \ 0.0769231, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., Indeterminate, 0., \ 0., 0., 0., Indeterminate, 0., 0., 0., 0., 0., 0., Indeterminate, 0., \ Indeterminate} largeRatios = Select[ratios, # > 5 &] {31.1911, 44.6957} There are two motifs that are more than 30 times more common in the metabolic network than in the rewired graph. Extract[IGData[{"AllDirectedGraphs", 4}], FirstPosition[ratios, #]] & /@ largeRatios The Davidson–Harel algorithm attempts to reduce edge crossings and can draw these subgraphs in a clearer way: IGLayoutDavidsonHarel /@ % #### Estimating motif counts in large graphs IGMotifs uses the RAND-ESU algorithm which can uniformly sample a random subset of motifs, and thus estimate motif counts even in very large graphs. To enable random sampling, set a cutoff probability for stopping the search at each level of the ESU tree. The length of the cutoff probability vector, n, must be the same as the motif size. The number of sampled motifs is, on average, a fraction of the total number. bigG = ExampleData[{"NetworkGraph", "WorldWideWeb"}]; {VertexCount[bigG], EdgeCount[bigG]} {325729, 1497134} Sample a fraction $$0.1^3=0.001$$ of all motifs. IGMotifs[bigG, 3, 1 - 0.1 {1, 1, 1}] // AbsoluteTiming {0.502014, {Indeterminate, Indeterminate, 34953, Indeterminate, 1549, 1646, 27314, 378, 291, 681, 3917, 2, 49, 171, 71, 7010}} Sample 12.5% of motifs, i.e. a fraction of $$0.5^3$$. IGMotifs[bigG, 3, 1 - 0.5 {1, 1, 1}] // AbsoluteTiming {9.29356, {Indeterminate, Indeterminate, 36350526, Indeterminate, 299314, 530929, 5424564, 67262, 34048, 166963, 516326, 1730, 4461, 204276, 12329, 800823}} ### IGMotifsVertexParticipation▴ ?IGMotifsVertexParticipation IGMotifsVertexParticipation counts how many times each vertex participates in each motif. For each vertex, the result is returned in the same format as with IGMotifs. Available options are: • DirectedEdges -> False treats the graph as undirected and DirectedEdges -> True treats the graph as directed. The default is DirectedEdges -> Automatic, which respects the directedness of the graph. Count how many times each vertex appears in each 3-motif in a directed graph. mot = IGMotifsVertexParticipation[g, 3] <|"A" -> {Indeterminate, Indeterminate, 0, Indeterminate, 2, 0, 0, 2, 1, 1, 0, 2, 0, 0, 0, 0}, "B" -> {Indeterminate, Indeterminate, 0, Indeterminate, 1, 1, 0, 3, 0, 2, 0, 1, 1, 1, 0, 0}, "C" -> {Indeterminate, Indeterminate, 1, Indeterminate, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0}, "D" -> {Indeterminate, Indeterminate, 0, Indeterminate, 2, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0}, "E" -> {Indeterminate, Indeterminate, 1, Indeterminate, 0, 0, 0, 2, 1, 2, 0, 1, 1, 0, 0, 0}, "F" -> {Indeterminate, Indeterminate, 1, Indeterminate, 3, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0}|> The sum of the participation counts in 3-motifs is 3 times the total motif counts of the graph. Total[mot] === 3 IGMotifs[g, 3] True ### IGMotifsTotalCount and IGMotifsTotalCountEstimate▴ ?IGMotifsTotalCount ?IGMotifsTotalCountEstimate IGMotifsTotalCount[graph, motifSize] counts the number of weakly connected subgraphs of the given size in a graph. All subgraph sizes greater than 2 are supported. IGMotifsTotalCountEstimate[graph, motifSize, sampleSize] estimates the total number of motifs by taking a random subset of vertices of the specified size, and counting motifs in which these vertices participate. The total number is estimated as motifCount*vertexCount/sampleSize. IGMotifsTotalCountEstimate[graph, motifSize, vertices] uses the specified vertices as the sample. Let us create a graph. g = RandomGraph[{20, 50}]; The number of size-4 subgraphs it has is: Binomial[VertexCount[g], 4] 4845 However, only a small fraction of these is connected: IGMotifsTotalCount[g, 4] 779 IGMotifsTotalCount is effectively equivalent to (but much faster than) the following: Count[Subsets[VertexList[g], {4}], subset_ /; WeaklyConnectedGraphQ@Subgraph[g, subset]] 779 Estimate the count of connected subgraphs by subsampling: at each level of the ESU tree, continue only with probability 0.9. IGMotifsTotalCount[g, 4, 1 - 0.9 {1, 1, 1, 1}]/0.9^4 833.714 Estimate the count of connected subgraphs by considering a random subset of 15 vertices (out of a total of 20). IGMotifsTotalCountEstimate[g, 4, 15] 994 Use the first 15 vertices tot estimate the count. IGMotifsTotalCountEstimate[g, 4, Range[15]] 1038 ## Triad and dyad census▴ ?IGTriadCensus ?IGDyadCensus See IGData["MANTriadLabels"] for the mapping between MAN labels and graphs. IGTriadCensus[g] does not return triad counts in the same order as IGMotifs[g, 3], i.e. ordered according to the triads’ IGIsoclass[]. To get the result ordered by isoclass, use Lookup[IGTriadCensus[g], Keys@IGData["MANTriadLabels"]] IGData["MANTriadLabels"] are ordered according to isoclass. net = ExampleData[{"NetworkGraph", "MetabolicNetworkActinobacillusActinomycetemcomitans"}]; IGDyadCensus[net] <|"Mutual" -> 32, "Asymmetric" -> 2304, "Null" -> 490192|> IGTriadCensus[net] <|"003" -> 160429739, "012" -> 2191799, "102" -> 30579, "021D" -> 11774, "021U" -> 10566, "021C" -> 22853, "111D" -> 496, "111U" -> 583, "030T" -> 0, "030C" -> 0, "201" -> 27, "120D" -> 0, "120U" -> 0, "120C" -> 0, "210" -> 0, "300" -> 0|> ## Finding triangles▴ ### IGTriangles▴ ?IGTriangles Highlight all triangles in a graph. g = RandomGraph[{8, 16}, VertexSize -> Large]; HighlightGraph[g, Subgraph[g, #], ImageSize -> Tiny, GraphHighlightStyle -> "Thick"] & /@ IGTriangles[g] ### IGAdjacenctTriangleCount▴ ?IGAdjacentTriangleCount Label a graph’s vertices based on the number of adjacent triangles. RandomGraph[{8, 16}, VertexSize -> Large] // IGVertexMap[Placed[#, Center] &, VertexLabels -> IGAdjacentTriangleCount] ### IGTriangleFreeQ▴ Triangle-free graphs do not have any fully connected subgraphs of size 3. Equivalently, they do not have any cliques (other than 2-cliques, which are edges). ?IGTriangleFreeQ Mycielski graphs are triangle-free. IGTriangleFreeQ@GraphData[{"Mycielski", 10}] True # Isomorphism and the automorphism group igraph implements three isomorphism testing algorithms: BLISS, VF2 and LAD. These support slightly different functionality. Naming: Most of IGraph/M’s isomorphism related functions include the name of the algorithm as a prefix, e.g. IGBlissIsomorphicQ. Functions named as …GetIsomorphism will find a single isomorphism. Functions named as …FindIsomorphisms can find multiple isomorphisms. Both return a result in a format compatible with the built-in FindGraphIsomorphism. Additionally, IGIsomorphicQ[] and IGSubisomorphicQ[] try to select the best algorithm for the given graphs. For graphs without multi-edges, they use igraph’s default algorithm selection. For multigraphs, they use VF2 after internally transforming the multigraphs to edge- and vertex-coloured simple graphs, in a manner similar to IGColoredSimpleGraph. ## Basic functions▴ ### IGIsomorphicQ▴ ?IGIsomorphicQ ?IGGetIsomorphism IGIsomorphicQ decides if two graphs are isomorphic. IGIsomorphicQ[IGShorthand["a-b-c-a-d"], IGShorthand["1-2,3-4-2-3"]] True IGIsomorphicQ supports multigraphs. True False Get a specific mapping between the vertices of the graphs. {<|1 -> 4, 2 -> 3, 3 -> 1, 4 -> 2|>} When the graphs are not isomorphic, an empty list is returned. IGGetIsomorphism[CycleGraph[4], IGCompleteGraph[4]] {} ### IGSubisomorphicQ▴ ?IGSubisomorphicQ ?IGGetSubisomorphism IGSubisomorphicQ decides if a subgraph is part of a larger graph. A dodecahedral graph does not contain a [1, 2, 3] symmetric tree. target = GraphData["DodecahedralGraph"]; pattern = IGSymmetricTree[{1, 2, 3}]; IGSubisomorphicQ[pattern, target] False It does contain a [3, 2, 1] tree. pattern = IGSymmetricTree[{3, 2, 1}]; IGSubisomorphicQ[pattern, target] True Let us retrieve a specific mapping … {iso} = IGGetSubisomorphism[pattern, target] {<|1 -> 1, 2 -> 14, 3 -> 15, 4 -> 16, 5 -> 3, 6 -> 9, 7 -> 4, 8 -> 10, 9 -> 7, 10 -> 8, 11 -> 19, 12 -> 17, 13 -> 20, 14 -> 18, 15 -> 11, 16 -> 12|>} … and highlight it. HighlightGraph[target, VertexReplace[pattern, Normal[iso]], GraphHighlightStyle -> "Thick" ] IGSubisomorphicQ supports multigraphs. True {<|"a" -> 1, "b" -> 2|>} True False ## Bliss▴ The Bliss library was developed by Tommi Junttila and Petteri Kaski. It is capable of canonical labelling of directed or undirected vertex coloured graphs. Bliss generally outperforms Mathematica’s built-in isomorphisms functions (including finding and counting automorphisms) as of Mathematica 12.1. However, this advantage will only be apparent for large and difficult graphs. For small ones the overhead of having to copy the graph and convert it to igraph’s internal format is much larger than the actual computation time. ?IGBliss* All Bliss functions take a "SplittingHeuristics" option, which can influence the performance of the method. Possible values are: • "First" – First non-unit cell. Very fast but may result in large search spaces on difficult graphs. Use for large but easy graphs. • "FirstSmallest" – First smallest non-unit cell. Fast, should usually produce smaller search spaces than "First". • "FirstLargest" – First largest non-unit cell. Fast, should usually produce smaller search spaces than "First". • "FirstMaximallyConnected" – First maximally non-trivially connected non-unit cell. Not so fast, should usually produce smaller search spaces than "First", "FirstSmallest" and "FirstLargest". • "FirstSmallestMaximallyConnected" – First smallest maximally non-trivially connected non-unit cell. Not so fast, should usually produce smaller search spaces than "First", "FirstSmallest" and "FirstLargest". • "FirstLargestMaximallyConnected" – First largest maximally non-trivially connected non-unit cell. Not so fast, should usually produce smaller search spaces than "First", "FirstSmallest" and "FirstLargest". The default setting is "FirstLargest", which performs well on average on sparse graphs. Note: The result of the IGBlissCanonicalLabeling, IGBlissCanonicalPermutation and IGBlissanonicalGraph functions depend on the choice of "SplittingHeuristics". See the Bliss documentation for more information. #### Basic examples Let us take the cuboctahedral graph from GraphData … g1 = GraphData["CuboctahedralGraph"] … and also generate it based on its LCF notation. g2 = IGLCF[{4, 2}, 6] The two graphs are isomorphic: IGBlissIsomorphicQ[g1, g2] True One particular mapping between them is the following: IGBlissGetIsomorphism[g1, g2] {<|1 -> 1, 2 -> 2, 3 -> 5, 4 -> 12, 5 -> 9, 6 -> 4, 7 -> 10, 8 -> 3, 9 -> 6, 10 -> 11, 11 -> 8, 12 -> 7|>} How many mappings are there in total? The same number as the number of automorphisms of either graph. IGBlissAutomorphismCount[g1] 48 Bliss cannot generate all 48 of these mappings directly. We can either use VF2 for this … IGVF2FindIsomorphisms[g1, g2] // Length 48 … or we can use the automorphism group computed by the IGBlissAutomorphismGroup function. group = IGBlissAutomorphismGroup[g1] PermutationGroup[{Cycles[{{2, 3}, {4, 5}, {8, 9}, {10, 11}}], Cycles[{{2, 4}, {3, 5}, {6, 7}, {8, 10}, {9, 11}}], Cycles[{{1, 2}, {3, 6}, {5, 8}, {7, 10}, {11, 12}}]}] GroupOrder[group] 48 Ask for all 48 vertex permutations that create isomorphic graphs: PermutationReplace[VertexList[g1], group] {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {1, 3, 2, 5, 4, 6, 7, 9, 8, 11, 10, 12}, {1, 4, 5, 2, 3, 7, 6, 10, 11, 8, 9, 12}, {1, 5, 4, 3, 2, 7, 6, 11, 10, 9, 8, 12}, {2, 1, 6, 4, 8, 3, 10, 5, 9, 7, 12, 11}, {2, 4, 8, 1, 6, 10, 3, 7, 12, 5, 9, 11}, {2, 6, 1, 8, 4, 3, 10, 9, 5, 12, 7, 11}, {2, 8, 4, 6, 1, 10, 3, 12, 7, 9, 5, 11}, {3, 1, 6, 5, 9, 2, 11, 4, 8, 7, 12, 10}, {3, 5, 9, 1, 6, 11, 2, 7, 12, 4, 8, 10}, {3, 6, 1, 9, 5, 2, 11, 8, 4, 12, 7, 10}, {3, 9, 5, 6, 1, 11, 2, 12, 7, 8, 4, 10}, {4, 1, 7, 2, 10, 5, 8, 3, 11, 6, 12, 9}, {4, 2, 10, 1, 7, 8, 5, 6, 12, 3, 11, 9}, {4, 7, 1, 10, 2, 5, 8, 11, 3, 12, 6, 9}, {4, 10, 2, 7, 1, 8, 5, 12, 6, 11, 3, 9}, {5, 1, 7, 3, 11, 4, 9, 2, 10, 6, 12, 8}, {5, 3, 11, 1, 7, 9, 4, 6, 12, 2, 10, 8}, {5, 7, 1, 11, 3, 4, 9, 10, 2, 12, 6, 8}, {5, 11, 3, 7, 1, 9, 4, 12, 6, 10, 2, 8}, {6, 2, 3, 8, 9, 1, 12, 4, 5, 10, 11, 7}, {6, 3, 2, 9, 8, 1, 12, 5, 4, 11, 10, 7}, {6, 8, 9, 2, 3, 12, 1, 10, 11, 4, 5, 7}, {6, 9, 8, 3, 2, 12, 1, 11, 10, 5, 4, 7}, {7, 4, 5, 10, 11, 1, 12, 2, 3, 8, 9, 6}, {7, 5, 4, 11, 10, 1, 12, 3, 2, 9, 8, 6}, {7, 10, 11, 4, 5, 12, 1, 8, 9, 2, 3, 6}, {7, 11, 10, 5, 4, 12, 1, 9, 8, 3, 2, 6}, {8, 2, 10, 6, 12, 4, 9, 1, 7, 3, 11, 5}, {8, 6, 12, 2, 10, 9, 4, 3, 11, 1, 7, 5}, {8, 10, 2, 12, 6, 4, 9, 7, 1, 11, 3, 5}, {8, 12, 6, 10, 2, 9, 4, 11, 3, 7, 1, 5}, {9, 3, 11, 6, 12, 5, 8, 1, 7, 2, 10, 4}, {9, 6, 12, 3, 11, 8, 5, 2, 10, 1, 7, 4}, {9, 11, 3, 12, 6, 5, 8, 7, 1, 10, 2, 4}, {9, 12, 6, 11, 3, 8, 5, 10, 2, 7, 1, 4}, {10, 4, 8, 7, 12, 2, 11, 1, 6, 5, 9, 3}, {10, 7, 12, 4, 8, 11, 2, 5, 9, 1, 6, 3}, {10, 8, 4, 12, 7, 2, 11, 6, 1, 9, 5, 3}, {10, 12, 7, 8, 4, 11, 2, 9, 5, 6, 1, 3}, {11, 5, 9, 7, 12, 3, 10, 1, 6, 4, 8, 2}, {11, 7, 12, 5, 9, 10, 3, 4, 8, 1, 6, 2}, {11, 9, 5, 12, 7, 3, 10, 6, 1, 8, 4, 2}, {11, 12, 7, 9, 5, 10, 3, 8, 4, 6, 1, 2}, {12, 8, 9, 10, 11, 6, 7, 2, 3, 4, 5, 1}, {12, 9, 8, 11, 10, 6, 7, 3, 2, 5, 4, 1}, {12, 10, 11, 8, 9, 7, 6, 4, 5, 2, 3, 1}, {12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}} Permuting the adjacency matrix with any of these leaves it invariant. perms = PermutationList[#, VertexCount[g1]] & /@ GroupElements[group]; Equal @@ (AdjacencyMatrix[g1][[#, #]] & /@ perms) True Bliss works by computing a canonical labelling of vertices. Then isomorphism can be tested for by comparing the canonically relabelled graphs. IGBlissCanonicalGraph[g1] === IGBlissCanonicalGraph[g2] True IGBlissCanonicalGraph returns graphs in a consistent format so that two graphs are isomorphic if and only if their canonical graphs will compare equal with ===. Note that in Mathematica, graphs may not always compare equal even if they have the same vertex and edge lists. The corresponding permutation and labelling are IGBlissCanonicalPermutation[g1] {12, 11, 9, 10, 8, 7, 6, 5, 3, 4, 2, 1} IGBlissCanonicalLabeling[g1] <|1 -> 12, 2 -> 11, 3 -> 9, 4 -> 10, 5 -> 8, 6 -> 7, 7 -> 6, 8 -> 5, 9 -> 3, 10 -> 4, 11 -> 2, 12 -> 1|> Notice that the canonical labelling is simply AssociationThread[VertexList[g1], IGBlissCanonicalPermutation[g1]] <|1 -> 12, 2 -> 11, 3 -> 9, 4 -> 10, 5 -> 8, 6 -> 7, 7 -> 6, 8 -> 5, 9 -> 3, 10 -> 4, 11 -> 2, 12 -> 1|> Also notice that it is a mapping from g1 to IGBlissCanonicalGraph[g1]: MemberQ[ IGVF2FindIsomorphisms[g1, IGBlissCanonicalGraph[g1]], IGBlissCanonicalLabeling[g1] ] True The canonical graph returned by IGBlissCanonicalGraph always has vertices labelled by the integers 1, 2, … It can also be used to filter duplicates from a list of graphs For example, let us generate all possible adjacency matrices of 3-vertex simple directed graphs. (* fills nondiagonal entries of n by n matrix from vector *) toMat[vec_, n_] := SparseArray@ Partition[Flatten@Riffle[Partition[vec, n], 0, {1, -1, 2}], n] There are 2^(3 2) = 2^6 = 64 such matrices. graphs = AdjacencyGraph[toMat[#, 3], DirectedEdges -> True] & /@ IntegerDigits[Range[2^6] - 1, 2, 6]; But only 16 of them correspond to non-isomorphic graphs DeleteDuplicatesBy[graphs, IGBlissCanonicalGraph] // Length 16 When IGBlissCanonicalGraph is given a vertex coloured graph, it will encode the colours into a vertex property named "Color". This allows distinguishing between graphs whose canonical graphs are identical in structure, but differ in colouring. Take for example the following coloured graphs: g = Graph[{1 <-> 2, 2 <-> 3}, VertexSize -> Large, GraphStyle -> "BasicBlack"]; colg1 = Graph[g, Properties -> {1 -> {"color" -> 1}, 2 -> {"color" -> 3}, 3 -> {"color" -> 2}}]; colg2 = Graph[g, Properties -> {1 -> {"color" -> 1}, 2 -> {"color" -> 3}, 3 -> {"color" -> 1}}]; Visualize them for clarity: IGVertexMap[ColorData[97], VertexStyle -> IGVertexProp["color"]] /@ {colg1, colg2} The vertex and edge lists of their canonical graphs are identical: cang1 = IGBlissCanonicalGraph[{colg1, "VertexColors" -> "color"}]; cang2 = IGBlissCanonicalGraph[{colg2, "VertexColors" -> "color"}]; VertexList /@ {cang1, cang2} EdgeList /@ {cang1, cang2} {{1, 2, 3}, {1, 2, 3}} {{1 <-> 3, 2 <-> 3}, {1 <-> 3, 2 <-> 3}} But they differ in colouring, and therefore do not compare equal: IGVertexPropertyList[cang1] {"Color", VertexCoordinates, VertexShape, VertexShapeFunction, \ VertexSize, VertexStyle} IGVertexProp["Color"] /@ {cang1, cang2} {{1, 2, 3}, {1, 1, 3}} cang1 === cang2 False The performance of Bliss functions may depend significantly on the choice of splitting heuristics. g = LineGraph@GraphData[{"Hadamard", {24, 6}}]; timings = {#, First@Timing@ IGBlissAutomorphismGroup[g, "SplittingHeuristics" -> #]} & /@ {"First", "FirstSmallest", "FirstLargest", "FirstMaximallyConnected", "FirstSmallestMaximallyConnected", "FirstLargestMaximallyConnected"}; TableForm[timings, TableHeadings -> {None, {"Splitting heuristics", "Timing (s)"}}] #### Additional examples Let us visualize the vertex equivalence classes induced by a graph’s automorphism group. Two vertices are considered equivalent if there is an automorphism that maps one into the other. With[{g = GraphData[{"Mycielski", 4}]}, HighlightGraph[g, GroupOrbits@IGBlissAutomorphismGroup[g], VertexSize -> Large, GraphStyle -> "BasicBlack"] ] Visualize the edge equivalence classes of a polyhedron, induced by its skeleton’s automorphism group. mesh = PolyhedronData["TruncatedOctahedron", "BoundaryMeshRegion"] With[{g = IGMeshGraph[mesh, VertexStyle -> Black]}, HighlightGraph[g, EdgeList[g][[#]] & /@ GroupOrbits@IGBlissAutomorphismGroup@LineGraph[g] ] ] #### References • T. Junttila, P. Kaski, Engineering an Efficient Canonical Labeling Tool for Large and Sparse Graphs, 2007 Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments, doi:10.1137/1.9781611972870.13. ## VF2▴ ?IGVF2* VF2 supports vertex coloured and edge coloured graphs. A colour specification consists of one or more of the "VertexColors" and "EdgeColors" options. Allowed formats for these options are a list of integers, an association assigning integers to the vertices/edges, or None. When using associations, it is not necessarily to specify a colour for each vertex/edge. The omitted ones are assumed to have colour 0. The VF2 algorithm only supports simple graphs. The following graph has two automorphisms: {1, 2} and {2, 1}. g = Graph[{1 <-> 2}]; IGVF2IsomorphismCount[g, g] 2 If we colour one of the vertices, the permutation {2, 1} becomes forbidden, so only one automorphism remains. IGVF2IsomorphismCount[{g, "VertexColors" -> {1, 2}}, {g, "VertexColors" -> {1, 2}}] 1 Multigraphs are not directly supported for isomorphism checking, but we can map the multigraph isomorphism problem into an edge-coloured graph isomorphism one by designating the multiplicity of each edge as its colour. g1 = EdgeAdd[PathGraph[Range[5], VertexLabels -> "Name"], 2 <-> 3] g2 = EdgeAdd[PathGraph[Range[5], VertexLabels -> "Name"], 4 <-> 3] IGVF2IsomorphicQ[g1, g2] Failed Since g1 and g2 are undirected, we need to bring their edges into a sorted canonical form before counting them. This ensures that 4 <-> 3 and 3 <-> 4 are treated as the same edge. colors1 = Counts[Sort /@ EdgeList[g1]] <|1 <-> 2 -> 1, 2 <-> 3 -> 2, 3 <-> 4 -> 1, 4 <-> 5 -> 1|> colors2 = Counts[Sort /@ EdgeList[g2]] <|1 <-> 2 -> 1, 2 <-> 3 -> 1, 3 <-> 4 -> 2, 4 <-> 5 -> 1|> IGVF2IsomorphicQ[{Graph@Keys[colors1], "EdgeColors" -> colors1}, {Graph@Keys[colors2], "EdgeColors" -> colors2}] True IGIsomorphicQ and IGSubisomorphicQ check multigraph isomorphism in a similar way, based on edge colouring. #### References • L. P. Cordella, P. Foggia, C. Sansone, and M. Vento, IEEE Trans. Pattern Anal. Mach. Intell. 26, 1367 (2004). ## LAD▴ The LAD library was developed by Christine Solnon. It is capable of finding subgraphs in a larger graph. The LAD algorithm does not support multi-edges. ?IGLAD* With the "Induced" -> True option LAD will search for induced subgraphs. True False True Highlight subgraphs in a grid graph. g = GridGraph[{3, 3}]; HighlightGraph[g, Subgraph[g, #], GraphHighlightStyle -> "Thick"] & /@ Union[Sort@*Values /@ IGLADFindSubisomorphisms[GridGraph[{2, 2}], g]] Count how many times each vertex of a graph appears at the apex of the following subgraph (motif): Generate a directed random graph to do the counting in. g = RandomGraph[{20, 120}, DirectedEdges -> True] IGShorthand provides a concise way to input this subgraph. motif = IGShorthand["2<->1<->3"] This motif has a two-fold symmetry, as revealed by its automorphism group. We divide the final counts by two. Counts@Lookup[ IGLADFindSubisomorphisms[motif, g, "Induced" -> True], 1 ]/IGBlissAutomorphismCount[motif] <|13 -> 2, 18 -> 2, 20 -> 4, 12 -> 1, 16 -> 1|> Check that a graph is claw-free. clawFreeQ[graph_?UndirectedGraphQ] := Not@IGLADSubisomorphicQ[ StarGraph[4], (* claw graph *) graph, "Induced" -> True ] clawFreeQ /@ {GraphData["DodecahedralGraph"], GraphData["TruncatedPrismGraph"]} {False, True} #### References ## Isomorphism of coloured graphs▴ All three included isomorphism algorithms support vertex coloured graphs, and VF2 supports edge coloured graphs as well. A coloured graph is specified as {g, "VertexColors" -> …, "EdgeColors" -> …}, where both vertex and edge colour specifications are optional. Colours are represented by integers and may be specified in one of the following ways: • A list of integers, given in the same order as VertexList[g] (or EdgeList[g] if specifying edge colours). {Graph[{a, b}, {a <-> b}], "VertexColors" -> {1, 2}}. • An association assigning integers to vertices (or edges). Vertices (or edges) not present in the association are assumed to have colour 0. {Graph[{a <-> b}], "VertexColors" -> <|a -> 1, b -> 2|>}. • The name of a a vertex (or edge) property. Vertices (or edges) without an assigned property value are assumed to have colour 0. {Graph[{Property[a, "color" -> 1], Property[b, "color" -> 2]}, {a <-> b}], "VertexColors" -> "color"} • "VertexColors" -> None indicates no colouring. Example. Define a graph along with the colours of its vertices. g = CycleGraph[4]; vcols = <| 1 -> 1, 2 -> 1, 3 -> 2, 4 -> 2 |>; Visualize it. Graph[g, VertexStyle -> Normal[ColorData[24] /@ vcols], VertexSize -> Medium, VertexLabels -> Placed["Name", Center] ] Compute its automorphism group, taking vertex colours into account. IGBlissAutomorphismGroup[{g, "VertexColors" -> vcols}] PermutationGroup[{Cycles[{{1, 2}, {3, 4}}]}] The functions in this section test for properties related to a graph’s automorphism group. The summary table below illustrates the functions on a set of graphs which all have different properties. graphs = {StarGraph[4], IGSquareLattice[{2, 3}, "Periodic" -> True], HypercubeGraph[3], GraphData[{"Rook", {4, 4}}], GraphData["ShrikhandeGraph"], GraphData["HoltGraph"], GraphData["Tutte12Cage"], GraphData[{"Paulus", {25, 1}}]}; functions = <| "regular" -> IGRegularQ, "strongly regular" -> IGStronglyRegularQ, "distance regular" -> IGDistanceRegularQ, "vertex transitive" -> IGVertexTransitiveQ, "edge transitive" -> IGEdgeTransitiveQ, "arc transitive" -> IGEdgeTransitiveQ@*DirectedGraph, "distance transitive" -> IGDistanceTransitiveQ |>; TableForm[ Through[Values[functions][#]] & /@ graphs, TableHeadings -> {Show[#, ImageSize -> 50] & /@ graphs, Keys[functions]}, TableDirections -> Row ] // Style[#, "Text"] & ### IGRegularQ▴ ?IGRegularQ IGRegularQ checks if a graph is regular. All vertices of a regular graph have the same degrees. In regular directed graphs, the in- and out-degrees are also equal to each other. IGRegularQ[IGSquareLattice[{3, 4}, "Periodic" -> True]] True Check if a graph is $$k$$-regular for $$k=2$$ and $$k=3$$. IGRegularQ[CycleGraph[10], 2] True IGRegularQ[CycleGraph[10], 3] False The null graph is considered 0-regular. IGRegularQ[IGEmptyGraph[]] True Check if a directed graph is regular. IGRegularQ[CycleGraph[5, DirectedEdges -> True]] True IGRegularQ considers self-loops and multi-edges when computing vertex degrees. True ### IGStronglyRegularQ▴ ?IGStronglyRegularQ IGStronglyRegularQ checks if a graph is strongly regular. A strongly regular graph is a regular graph where each pair of connected vertices have the same number of common neighbours, $$\lambda$$, and each pair of unconnected vertices also have the same number of common neighbours, $$\mu$$. IGStronglyRegularQ@GraphData["ShrikhandeGraph"] True Hypercube graphs and 3 and higher dimensions are not strongly regular, even though they are regular. IGStronglyRegularQ /@ HypercubeGraph /@ Range[2, 4] {True, False, False} Some authors exclude empty and complete graph from the definition, as they satisfy these conditions trivially. IGStronglyRegularQ returns True for these. IGStronglyRegularQ /@ {IGEmptyGraph[5], IGCompleteGraph[6]} {True, True} It also returns True for graphs on 0, 1 and 2 vertices. IGStronglyRegularQ /@ IGCompleteGraph /@ Range[0, 2] {True, True, True} Currently, IGStronglyRegularQ does not support directed graphs. IGStronglyRegularQ@Graph[{1 -> 2}] Failed ### IGStronglyRegularParameters▴ ?IGStronglyRegularParameters IGStronglyRegularParameters returns the parameters $$(v,k,\lambda ,\mu )$$ of a strongly regular graph. $$v$$ is the number of vertices, $$k$$ the degree of the vertices, $$\lambda$$ the number of common neighbours of connected vertices and $$\mu$$ the number of common neighbours of unconnected vertices. IGStronglyRegularParameters[PetersenGraph[]] {10, 3, 0, 1} IGStronglyRegularParameters[CycleGraph[5]] {5, 2, 0, 1} The parameters of a strongly regular graph satisfy the equation $$(v-k-1)\mu =k(k-\lambda -1)$$. {v, k, lambda, mu} = IGStronglyRegularParameters[GraphData[{"Paley", 101}]] {101, 50, 24, 25} (v - k - 1) mu == k (k - lambda - 1) True $$\lambda$$ and $$\mu$$ are not well-defined for empty and complete graphs, respectively. In these cases, 0 is returned. IGStronglyRegularParameters /@ {IGEmptyGraph[5], IGCompleteGraph[6]} {{5, 0, 0, 0}, {6, 5, 4, 0}} For non-strongly-regular graphs, {} is returned. IGStronglyRegularParameters[HypercubeGraph[3]] {} ### IGDistanceRegularQ▴ ?IGDistanceRegularQ IGDistanceRegularGraph checks if a graph is distance regular. IGDistanceRegularQ@HypercubeGraph[5] True IGDistanceRegularQ@IGSquareLattice[{2, 5}, "Periodic" -> True] False A distance regular graph with a diameter of 2 is also strongly regular. g = GraphData[{"Paley", 13}] {IGDiameter[g], IGDistanceRegularQ[g], IGStronglyRegularQ[g]} {2, True, True} The Shrikhande graph is the smallest graph that is distance regular, but not distance transitive. Through[{IGDistanceRegularQ, IGDistanceTransitiveQ}[ GraphData["ShrikhandeGraph"]]] {True, False} A disconnected graph is distance regular if its components are distance regular and they are co-spectral. The following graphs are co-spectral: components = {GraphData[{"Rook", {4, 4}}], GraphData["ShrikhandeGraph"]} Eigenvalues /@ AdjacencyMatrix /@ components {{6, -2, -2, -2, -2, -2, -2, -2, -2, -2, 2, 2, 2, 2, 2, 2}, {6, -2, -2, -2, -2, -2, -2, -2, -2, -2, 2, 2, 2, 2, 2, 2}} They are both distance regular with the same intersection array. IGIntersectionArray /@ components {{{6, 3}, {1, 2}}, {{6, 3}, {1, 2}}} Thus their disjoint union is also distance regular. IGDistanceRegularQ@IGDisjointUnion[components] True All distance transitive graphs are also distance regular, but the reverse is not true. IGDistanceTransitiveQ /@ components {True, False} IGDistanceRegularQ does not currently support directed graphs or non-simple graphs. IGDistanceRegularQ[Graph[{1 -> 2}]] Failed IGDistanceRegularQ[ Graph[{1 <-> 2, 1 <-> 2}]] Failed ### IGIntersectionArray▴ ?IGIntersectionArray IGIntersectionArray@GraphData["IcosahedralGraph"] {{5, 2, 1}, {1, 2, 5}} IGIntersectionArray@GraphData["SuzukiGraph"] {{416, 315}, {1, 96}} IGIntersectionArray@CycleGraph[6] {{2, 1, 1}, {1, 1, 2}} For non-distance-regular graphs, {} is returned. IGIntersectionArray[GridGraph[{3, 3}]] {} IGIntersectionArray does not currently support directed graphs. IGIntersectionArray[Graph[{1 -> 2}]] Failed ### IGVertexTransitiveQ▴ ?IGVertexTransitiveQ IGVertexTransitiveQ checks if a graph is vertex transitive, i.e. if any vertex can be mapped into any other by some automorphism of the graph. True False All Cayley graphs are vertex transitive. cg = CayleyGraph@IGBlissAutomorphismGroup@IGLCF[{2, -1, 2}, 3] IGVertexTransitiveQ[cg] True ### IGEdgeTransitiveQ▴ ?IGEdgeTransitiveQ IGEdgeTransitiveQ checks if a graph is edge transitive, i.e. if any edge can be mapped into any other by some automorphism of the graph. False True The Folkman graph is not vertex transitive but it is edge transitive. Through[{IGVertexTransitiveQ, IGEdgeTransitiveQ}@ GraphData["FolkmanGraph"]] {False, True} IGEdgeTransitiveQ takes into account edge directions. IGEdgeTransitiveQ@Graph[{1 -> 2, 2 -> 3}] False IGEdgeTransitiveQ@Graph[{1 -> 2, 3 -> 2}] True Arc transitivity in an undirected graph refers to edge transitivity when each undirected edge is replaced by two opposite directed edges. arcTransitiveQ[graph_?UndirectedGraphQ] := IGEdgeTransitiveQ@DirectedGraph[graph] Some graphs are edge transitive, but not arc transitive. IGEdgeTransitiveQ@GraphData[{"Bouwer", {2, 4, 15}}] True arcTransitiveQ@GraphData[{"Bouwer", {2, 4, 15}}] False Most graphs are edge transitive if their line graphs are vertex transitive. The exceptions are disjoint unions of the 3-star and 3-cycle. These two graphs have the same line graph, but they are not isomorphic. {IGEdgeTransitiveQ[g], IGVertexTransitiveQ@LineGraph[g]} {False, True} ### IGSymmetricQ▴ ?IGSymmetricQ IGSymmetricQ checks if a graph is both vertex transitive and edge transitive. Note that this property is distinct from being arc transitive, which is the definition used for “symmetric” by some authors. IGSymmetricQ[GraphData["DodecahedralGraph"]] True Make a table of symmetric graphs up to size 7: Grid[ Table[ Graph[#, ImageSize -> 50, PlotTheme -> "Business"] & /@ Select[GraphData /@ GraphData[k], IGSymmetricQ], {k, 1, 7} ], Frame -> All, ItemSize -> All] Some authors use the term symmetric graph to refer to arc transitive graphs. Arc transitivity can be checked using IGEdgeTransitiveQ@DirectedGraph[#] &. All arc-transitive graphs are both vertex- and edge-transitive, but the reverse is not true. The smallest graph that is both vertex- and edge-transitive, but not arc-transitive, is the 27-vertex Doyle graph, also known as the Holt graph. doyle = GraphData["DoyleGraph"] {IGVertexTransitiveQ[doyle], IGEdgeTransitiveQ[doyle]} {True, True} IGEdgeTransitiveQ@DirectedGraph[doyle] False ### IGDistanceTransitiveQ▴ ?IGDistanceTransitiveQ IGDistanceTransitiveQ checks if a graph is distance transitive. In a distance transitive graph, any two ordered pairs of vertices which are the same distance apart can be mapped into each other by some automorphism. All Platonic graphs are distance transitive. IGMeshGraph@PolyhedronData[#, "BoundaryMeshRegion"] & /@ PolyhedronData["Platonic"] IGDistanceTransitiveQ /@ % {True, True, True, True, True} Some graphs are symmetric, but not distance transitive. g = GraphData[{"Circulant", {10, {1, 4}}}] {IGSymmetricQ[g], IGDistanceTransitiveQ[g]} {True, False} IGDistanceTransitiveQ does not exclude non-connected graphs. True IGDistanceTransitiveQ works with directed graphs. g = With[{n = 11}, RelationGraph[ MemberQ[Rest@Union@Mod[Range[n]^2, n], Mod[#1 - #2, n]] &, Range[n] - 1] ] IGDistanceTransitiveQ[g] True The following directed graph is vertex transitive, but not distance transitive. False ## Homeomorphism▴ ?IGHomeomorphicQ IGHomeomorphicQ tests if two graphs are homeomorphic, i.e. whether they have the same topological structure. Two graphs $$G_1$$ and $$G_2$$ are homeomorphic if there is an isomorphism from a subdivision of $$G_1$$ to a subdivision of $$G_2$$. is effectively implemented as . The following graphs are homeomorphic. True They smoothen to the same graph. Any two cycle graphs are homeomorphic. IGHomeomorphicQ[CycleGraph[5], CycleGraph[9]] True A cycle and a path graph are not homeomorphic. IGHomeomorphicQ[CycleGraph[5], PathGraph@Range[5]] False A triangular and a square lattice on the same number of vertices are, in general, topologically different. IGHomeomorphicQ[IGSquareLattice[{3, 3}], IGTriangularLattice[{3, 3}]] False When testing empirical graphs for equivalence, it is often useful to remove tree-like components. For example, the face-face and the face-edge adjacency graphs of a geometric mesh are equivalent, save for the tree-like components. mesh = IGLatticeMesh["SnubSquare", {3, 3}]; ffg = IGMeshCellAdjacencyGraph[mesh, 2, VertexCoordinates -> Automatic] feg = IGMeshCellAdjacencyGraph[mesh, 1, 2, VertexCoordinates -> Automatic] IGHomeomorphicQ[feg, ffg] False feg = VertexDelete[feg, IGTreelikeComponents[feg]] ffg = VertexDelete[ffg, IGTreelikeComponents[ffg]] IGHomeomorphicQ[feg, ffg] True ## Other functions▴ ### IGSelfComplementaryQ▴ ?IGSelfComplementaryQ A graph is called self-complementary if it is isomorphic with its complement. The 4-vertex path graph is self-complementary. IGSelfComplementaryQ[PathGraph@Range[4]] True Find all 3-vertex self-complementary directed graphs. Select[IGData[{"AllDirectedGraphs", 3}], IGSelfComplementaryQ] ### IGColoredSimpleGraph▴ ?IGColoredSimpleGraph IGColoredSimpleGraph is a helper function that encodes a non-simple graph (i.e a graph with self-loops or multi-edges) into an edge- and vertex-colored simple graph. The coloured simple graph can be used directly as an input to coloured isomorphism checking functions such as IGVF2IsomorphicQ. The vertex colours are computed as the multiplicity of self-loops at each vertex. The edge colours are computed as the multiplicities or non-loop edges. The following graphs are not simple and cannot be used with IGVF2IsomorphicQ directly. IGVF2IsomorphicQ[g1, g2] Failed IGColoredSimpleGraph can encode them as coloured graphs. Its output can be supplied directly to IGVF2IsomorphicQ. IGColoredSimpleGraph[g1] Now can can determine that g1 is isomorphic to g2, but not to g3. IGVF2IsomorphicQ[IGColoredSimpleGraph[g1], IGColoredSimpleGraph[g2]] True IGVF2IsomorphicQ[IGColoredSimpleGraph[g1], IGColoredSimpleGraph[g3]] False When searching for subgraphs in multigraphs with this method, be aware that a match occurs only if the edge multiplicities are the same. This sort of matching is useful e.g. in substructure search chemistry, where a double bond must only match another double bond, but not a single one. False True Use IGSubisomorphicQ to match any subgraph. True # Maximum flow and minimum cut ## Maximum flow▴ ### IGMaximumFlowValue▴ ?IGMaximumFlowValue IGMaximumFlowValue is equivalent to IGMinimumCutValue except that it uses the EdgeCapacity property instead of EdgeWeight. Edge capacities are taken from the EdgeCapacity property. {3.5, 2, 1, 2.5, 5, 1, 3.5, 4} IGMaximumFlowValue[g, 1, 4] 3.5 ### IGMaximumFlowMatrix▴ ?IGMaximumFlowMatrix Element $$F_{ij}$$ of the flow matrix is the flow through the edge connecting the $$i$$th node to the $$j$$th one. In an undirected graph, $$F_{ij}=-F_{ij}$$. Edge capacities are taken from the EdgeCapacity property. Let us take a directed graph with edge capacities set … IGEdgeProp[EdgeCapacity][g] {3.5, 2, 1, 2.5, 5, 1, 3.5, 4} … and compute the maximum flow between two of its vertices. flowMat = IGMaximumFlowMatrix[g, 1, 4] The result is returned as a sparse matrix containing the flows through each edge. MatrixForm[flowMat] If the input is an undirected graph, the flow matrix contains entries of opposing sign for the two directions along each edge. IGMaximumFlowMatrix[UndirectedGraph[g], 1, 4] // MatrixForm ## Minimum edge cuts▴ ### IGMinimumCut▴ ?IGMinimumCut IGMinimumCut finds a single minimum edge cut in a weighted graph. To find all minimum cuts between two given vertices, use IGFindMinimumCuts. ### IGMinimumCutValue▴ ?IGMinimumCutValue Unlike IGEdgeConnectivity, IGMinimumCutValue takes weights into account. IGMinimumCutValue[ Graph[{1 <-> 2, 2 <-> 3}, EdgeWeight -> {3.5, 5.6}]] 3.5 The minimum cut value of the null graph and singleton graph are returned as 0 and ∞, respectively. IGMinimumCutValue /@ {IGEmptyGraph[0], IGEmptyGraph[1]} {0., ∞} ### IGGomoryHuTree▴ ?IGGomoryHuTree The Gomory–Hu tree is a weighted tree that encodes the minimum cuts between all pairs of vertices of an undirected graph. The Gomory–Hu tree has the same vertices as the graph it characterizes. The minimum cut between an $$s$$-$$t$$ pair of the graph has the same size as smallest edge weight on the path from $$s$$ to $$t$$ in the Gomory–Hu tree. Weighted graphs are supported. t = IGGomoryHuTree[g, EdgeLabels -> "EdgeWeight", VertexShapeFunction -> "Name" ] The path from 1 to 9 is 1 <-> 2 <-> 5 <-> 9 and has the weights {3, 5, 4}. The smallest one, 3, is the minimum value of a cut separating 1 from 9. {IGMinimumCutValue[g, 1, 9], IGMinimumCutValue[t, 1, 9]} {3., 3.} ## Cohesive blocks▴ ?IGCohesiveBlocks The following examples are based on the ones in the R/igraph documentation. This is the network from the Moody-White paper: • J. Moody and D. R. White. Structural cohesion and embeddedness: A hierarchical concept of social groups. American Sociological Review, 68(1):103–127, Feb 2003. mw = Graph[{"1" <-> "2", "1" <-> "3", "1" <-> "4", "1" <-> "5", "1" <-> "6", "2" <-> "3", "2" <-> "4", "2" <-> "5", "2" <-> "7", "3" <-> "4", "3" <-> "6", "3" <-> "7", "4" <-> "5", "4" <-> "6", "4" <-> "7", "5" <-> "6", "5" <-> "7", "5" <-> "21", "6" <-> "7", "7" <-> "8", "7" <-> "11", "7" <-> "14", "7" <-> "19", "8" <-> "9", "8" <-> "11", "8" <-> "14", "9" <-> "10", "10" <-> "12", "10" <-> "13", "11" <-> "12", "11" <-> "14", "12" <-> "16", "13" <-> "16", "14" <-> "15", "15" <-> "16", "17" <-> "18", "17" <-> "19", "17" <-> "20", "18" <-> "20", "18" <-> "21", "19" <-> "20", "19" <-> "22", "19" <-> "23", "20" <-> "21", "21" <-> "22", "21" <-> "23", "22" <-> "23"}, VertexLabels -> "Name"]; {blocks, cohesion} = IGCohesiveBlocks[mw] {{{"1", "2", "3", "4", "5", "6", "7", "21", "8", "11", "14", "19", "9", "10", "12", "13", "16", "15", "17", "18", "20", "22", "23"}, {"1", "2", "3", "4", "5", "6", "7", "21", "19", "17", "18", "20", "22", "23"}, {"7", "8", "11", "14", "9", "10", "12", "13", "16", "15"}, {"1", "2", "3", "4", "5", "6", "7"}, {"7", "8", "11", "14"}}, {1, 2, 2, 5, 3}} CommunityGraphPlot[mw, Rest@blocks, CommunityRegionStyle -> Table[Directive[Opacity[0.5], ColorData[96][i]], {i, Length[blocks] - 1}]] cohesion {1, 2, 2, 5, 3} Science camp network: sc = Graph[{"Pauline" <-> "Jennie", "Pauline" <-> "Ann", "Jennie" <-> "Ann", "Jennie" <-> "Michael", "Michael" <-> "Ann", "Holly" <-> "Jennie", "Jennie" <-> "Lee", "Michael" <-> "Lee", "Harry" <-> "Bert", "Harry" <-> "Don", "Don" <-> "Bert", "Gery" <-> "Russ", "Russ" <-> "Bert", "Michael" <-> "John", "Gery" <-> "John", "Russ" <-> "John", "Holly" <-> "Pam", "Pam" <-> "Carol", "Holly" <-> "Carol", "Holly" <-> "Bill", "Bill" <-> "Pauline", "Bill" <-> "Michael", "Bill" <-> "Lee", "Harry" <-> "Steve", "Steve" <-> "Don", "Steve" <-> "Bert", "Gery" <-> "Steve", "Russ" <-> "Steve", "Pam" <-> "Brazey", "Brazey" <-> "Carol", "Pam" <-> "Pat", "Brazey" <-> "Pat", "Carol" <-> "Pat", "Holly" <-> "Pat", "Gery" <-> "Pat"}, VertexLabels -> "Name"]; {blocks, cohesion} = IGCohesiveBlocks[sc] {{{"Pauline", "Jennie", "Ann", "Michael", "Holly", "Lee", "Harry", "Bert", "Don", "Gery", "Russ", "John", "Pam", "Carol", "Bill", "Steve", "Brazey", "Pat"}, {"Harry", "Bert", "Don", "Steve"}, {"Holly", "Pam", "Carol", "Brazey", "Pat"}, {"Pauline", "Jennie", "Ann", "Michael", "Lee", "Bill"}}, {2, 3, 3, 3}} CommunityGraphPlot[sc, Rest@blocks, CommunityRegionStyle -> ColorData[96], ImageSize -> Large] # Cliques and independent vertex sets ?IG*Clique* A clique is a fully connected subgraph. An independent vertex set is a subset of a graph’s vertices with no connections between them. ## Counting cliques▴ Mathematica’s FindClique function only finds maximal cliques. IGraph/M provides functions for finding or counting all cliques, i.e. complete subgraphs, of a graph. g = ExampleData[{"NetworkGraph", "CoauthorshipsInNetworkScience"}]; {VertexCount[g], EdgeCount[g]} {1589, 2742} Simply counting cliques is much more memory efficient (and faster) than returning all of them. IGCliqueSizeCounts[g] {1589, 2742, 3764, 7159, 17314, 39906, 78055, 126140, 167993, 184759, \ 167960, 125970, 77520, 38760, 15504, 4845, 1140, 190, 20, 1} BarChart[%, ChartLabels -> Range@Length[%]] IGMaximalCliqueSizeCounts[g] {128, 221, 195, 108, 52, 19, 3, 8, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1} BarChart[%, ChartLabels -> Range@Length[%]] IGLargestCliques[g] {{"V. Narayan", "L. Giot", "J. Rothberg", "S. Fields", "M. Johnston", "M. Yang", "G. Vijayadamodar", "T. Kalbfleisch", "D. Conover", "B. Godwin", "Y. Li", "A. Qureshiemili", "P. Pochart", "M. Srinivasan", "D. Lockshon", "J. Knight", "R. Judson", "T. Mansfield", "G. Cagney", "P. Uetz"}} ## Cliques in directed graphs▴ The clique finder in IGraph/M ignores edge directions. g = RandomGraph[{10, 60}, DirectedEdges -> True] IGMaximalCliques[g] {{7, 2, 8, 6, 5, 3}, {2, 1, 10, 8, 6, 5, 4, 3}, {2, 1, 10, 8, 6, 5, 4, 9}} To find cliques in directed graphs, convert them to undirected and keep mutual (bidirectional) edges only. IGMaximalCliques@IGUndirectedGraph[g, "Mutual"] {{2, 7}, {2, 3, 4}, {6, 1, 10}, {7, 5}, {5, 4, 3}, {5, 4, 10, 1}, {8, 4, 3}, {8, 4, 9}, {9, 1, 10, 4}} ## Clique cover▴ ?IGCliqueCover ?IGCliqueCoverNumber A clique cover of a graph is a partitioning of its vertices such that each partition forms a clique. IGCliqueCover finds a minimum clique cover, i.e. a partitioning into a smallest number of cliques. The clique cover number of a graph is the smallest number of cliques that can be used to cover its vertices. Available Method option values are: • "Minimum" finds a minimum clique cover. • "Heuristic" is much faster, but the result is not typically a minimum cover. Compute a minimum clique cover of a random graph. g = RandomGraph[{10, 20}] IGCliqueCover[g] {{1, 2, 6, 7}, {3, 5, 9}, {4, 10}, {8}} Visualize the clique cover. HighlightGraph[g, IGCliqueCover[g], VertexSize -> Large] Find the clique cover number without returning a cover. IGCliqueCoverNumber[g] 4 The clique cover problem is equivalent to the colouring of the complement graph. IGCliqueCover is effectively implemented as IGMembershipToPartitions[g]@IGMinimumVertexColoring@GraphComplement[g] {{1, 2, 6, 7}, {3, 5, 9}, {4, 10}, {8}} For difficult problems, it may be useful to use IGMinimumVertexColoring or IGVertexColoring directly instead of IGCliqueCover, and tune their options to achieve better performance. See the "ForcedColoring" option of IGMinimumVertexColoring on how to do this. ## Reconstruct bipartite graph of co-occurrence network▴ g = ExampleData[{"NetworkGraph", "LesMiserables"}] ExampleData[{"NetworkGraph", "LesMiserables"}, "LongDescription"] "Coappearance network of characters in the novel Les Miserables. \ EdgeWeight describes the number of coappearance." The maximal cliques of the graph can approximate the scenes in which characters appear together. cliques = IGMaximalCliques[g]; We can construct a bipartite graph of connections between potential scenes and characters IGLayoutBipartite[ Graph@Catenate[ Thread /@ Thread[Range@Length[cliques] <-> cliques]], VertexSize -> 0.5, ImageSize -> 220 ] // IGVertexMap[Placed[#, If[IntegerQ[#], Before, After]] &, VertexLabels -> VertexList] ## Graphlet decomposition▴ Note: The term “graphlet” is used for multiple unrelated concepts in the literature. This section deals with decomposing weighted graphs into cliques. If you are looking to count induced subgraphs, see the IGMotifs function. ?IGGraphlets ?IGGraphletBasis ?IGGraphletProject g = IGShorthand["A,B,D,E,C, A-B-C-A, C-E-D-B, D-C, E-B", EdgeWeight -> {2, 3, 2, 4, 4, 1, 4, 1}, EdgeLabels -> "EdgeWeight", VertexLabels -> None, VertexShapeFunction -> "Name", PerformanceGoal -> "Quality", GraphLayout -> "CircularEmbedding" ] basis = IGGraphletBasis[g] <|{"A", "B", "C"} -> 2., {"B", "D", "E", "C"} -> 1., {"B", "C"} -> 3., {"D", "E", "C"} -> 4.|> IGGraphletProject[g, Keys[basis]] <|{"A", "B", "C"} -> 0.925543, {"B", "D", "E", "C"} -> 0.861478, {"B", "C"} -> 2.90881*10^-253, {"D", "E", "C"} -> 1.13842| > IGGraphlets[g] <|{"D", "E", "C"} -> 1.13842, {"A", "B", "C"} -> 0.925543, {"B", "D", "E", "C"} -> 0.861478, {"B", "C"} -> 2.90881*10^-253|> #### References # Layout algorithms The following functions are available: ?IGLayout* If you are looking for the Sugiyama layout from igraph, try the built-in GraphLayout -> "LayeredDigraphEmbedding", or LayeredGraphPlot. These are also based on the Sugiyama algorithm. ## Common options and examples▴ Layout functions also take any standard Graph option. Many layout algorithms take the following options: "MaxIterations" controls either the maximum number of iterations performed by the algorithm or the exact number of iterations, depending on the specific algorithm and settings. The option name is the same for all functions to make it easier to interchange them when visualizing dynamic graphs. "Align" -> True aligns the output horizontally. Examples: {IGLayoutFruchtermanReingold[IGSquareLattice[{2, 4}](*, "Align" -> True is the default *)], IGLayoutFruchtermanReingold[IGSquareLattice[{2, 4}], "Align" -> False]} "Continue" -> True allows using existing vertex coordinates as starting points for algorithms that update vertex positions incrementally. We can use this to visualize how the layout algorithms work … g = IGLayoutRandom@ RandomGraph[BarabasiAlbertGraphDistribution[100, 1]]; ListAnimate@ NestList[ IGLayoutGraphOpt[#, "Continue" -> True, "MaxIterations" -> 80, "Align" -> False] &, g, 40] … or to visualize dynamic graph processes such as adding edges to the graph one by one: g = IGLayoutKamadaKawai@ Graph[Range[25], {1 <-> 25}, VertexLabels -> "Name"]; ListAnimate@NestList[ IGLayoutKamadaKawai[ EdgeAdd[#, UndirectedEdge @@ RandomSample[VertexList[#], 2]], "MaxIterations" -> 15, "Continue" -> True, "Align" -> False] &, g, 30 ] Visualize a planar graph without edge crossings using the Davidson–Harel simulated annealing method, and taking starting coordinates from GraphLayout -> "PlanarEmbedding". g = Graph@GraphData[{"Fullerene", {60, 1}}, "EdgeList"] This layout avoids crossings, but it is not pleasing: Graph[g, GraphLayout -> "PlanarEmbedding"] We can post process it while avoiding the introduction of any edge crossings: IGLayoutDavidsonHarel[ IGVertexMap[# &, VertexCoordinates -> (Rescale@ GraphEmbedding[#, "PlanarEmbedding"] &), g], "Continue" -> True, "EdgeCrossingWeight" -> 1000 ] ## Weighted graphs▴ Several of the graph layout algorithms in igraph can take edge weights into accounts. How the weights are used during layout differs between them. • IGLayoutFruchtermanReingold multiplies the attraction between vertices by the weights. Thus higher weights result in shorter edges. • IGLayoutKamadaKawai produces longer edges for higher weights ## Constraining vertex positions▴ Graph layout functions which have a "Constraints" option allow fixing the position of some vertices, or constraining them into a box. This is an experimental feature that may change in the future. The value of the "Constraints" option must be an association from vertex names to vertex coordinates, or to bounding boxes. Fix the positions of three vertices and highlight them in red: IGLayoutFruchtermanReingold[ IGShorthand["1:2:3:4 - 1:2:3:4:5, 5-6-7, 7:8:9 - 7:8:9"], "Constraints" -> <|4 -> {-1, 0}, 7 -> {1, 0}, 6 -> {0, 1}|>, VertexStyle -> {4 -> Red, 7 -> Red, 6 -> Red}, Frame -> True, FrameTicks -> True, GridLines -> Automatic ] ## Drawing trees▴ IGLayoutReingoldTilford[] and IGLayoutReingoldTilfordCircular[] are designed for laying out trees or forests. The following options are available: • "RootVertices" allows specifying the root node(s). It must be a list, even if there is a single root node. Multiple root nodes are meant to be used with forests. The roots should be selected so that all vertices of the graph are reachable from them. "RootVertices" -> Automatic chooses roots automatically, preferring low eccentricity vertices in small graphs (fewer than 500 vertices) and high degree vertices in large graphs. • DirectedEdges -> False ignores edge directions. By default, directed graphs are laid out so that edges are pointing away from the root. • "Rotation" controls the orientation of the layout. It must be given in radians. The following options are unique to IGLayoutReingoldTilford[]: • "LeafDistance" sets the spacing between tree leaves. The default is 1. • "LayerHeight" sets the spacing between layers of the drawing. The default is 1. The same tree laid out in directed and undirected modes: t = IGTreeGame[12, DirectedEdges -> True]; {IGLayoutReingoldTilford[t], IGLayoutReingoldTilford[t, DirectedEdges -> False]} Lay out the tree radially, with successive layers places on circular shells: IGLayoutReingoldTilfordCircular[t, GraphStyle -> "Minimal", Prolog -> {Thin, Gray, Table[Circle[{0, 0}, r], {r, IGDiameter[t, "ByComponents" -> True]}]}] Use a left-to-right layout, with tightly spaced leaves: IGLayoutReingoldTilford[t, "Rotation" -> Pi/2, "LeafDistance" -> 1/3] In an undirected tree, any vertex may be chosen as the root: t = IGTreeGame[6]; Table[ IGLayoutReingoldTilford[t, "RootVertices" -> {r}, GraphStyle -> "DiagramGold"], {r, VertexList[t]} ] ## Drawing bipartite graphs▴ ?IGLayoutBipartite IGLayoutBipartite draws a bipartite graph, attempting to minimize the number of edge crossing using the Sugiyama algorithm. The available options are: • "Orientation" can be Horizontal or Vertical • "PartitionGap" controls the size of the gap between the two partitions • "VertexGap" controls the minimum size of the gap between vertices in a partition • MaxIterations controls the maximum number of iterations performed during edge crossing minimization. • "BipartitePartitions" can be used to explicitly specify the partitioning of the graph. IGLayoutBipartite[IGBipartiteGameGNP[10, 10, 0.2], VertexLabels -> "Name"] By default, a partitioning is computed automatically. g = Graph[{1 <-> 2, 3 <-> 4}, VertexLabels -> "Name"]; IGLayoutBipartite[g] The partitioning can also be specified explicitly. IGLayoutBipartite[g, "BipartitePartitions" -> {{2, 3}, {4, 1}}] Draw a bipartite layout with curved edges. ## Drawing large graphs▴ IGLayoutDrL is designed specifically for visualizing large graphs with high clustering. The following image is created using DrL and shows a 36000 node network of collaborations between condensed matter scientists. The image was generated using the following code: lg = ExampleData[{"NetworkGraph", "CondensedMatterCollaborations2005"}]; lg = IndexGraph@Subgraph[lg, First@ConnectedComponents[lg]]; c = IGCommunitiesMultilevel[lg] pts = GraphEmbedding@IGLayoutDrL[lg]; (* this takes a while *) figure = Graphics[ GraphicsComplex[pts, { {White, AbsoluteThickness[0.3], Opacity[0.05], Line[List @@@ EdgeList[lg]]}, {AbsolutePointSize[2], Opacity[0.7], MapIndexed[ {ColorData[45]@First[#2], Point[#1]} &, c["Communities"] ]} } ], Background -> Black ] shift-enter evaluation is disabled in the cell above to avoid running it accidentally. Running the code takes about 2-3 minutes on a modern computer. Copy the code to a new cell to try it. Create galleries of the various graph layouts available in IGraph/M. Visualise a tree graph with all layouts. g = IGBarabasiAlbertGame[32, 1, DirectedEdges -> False]; layouts = Graph[#[g], PlotLabel -> #, LabelStyle -> 7] & /@ {IGLayoutCircle, IGLayoutSphere, IGLayoutDavidsonHarel, IGLayoutDrL, IGLayoutDrL3D, IGLayoutFruchtermanReingold, IGLayoutFruchtermanReingold3D, IGLayoutGEM, IGLayoutGraphOpt, IGLayoutKamadaKawai, IGLayoutKamadaKawai3D, IGLayoutRandom, IGLayoutReingoldTilford, IGLayoutReingoldTilfordCircular, IGLayoutBipartite, IGLayoutPlanar}; Multicolumn[layouts] Visualise a polyhedral graph with all layouts. g = GraphData["DodecahedralGraph"]; layouts = Graph[#[g], PlotLabel -> #, LabelStyle -> 7] & /@ {IGLayoutCircle, IGLayoutSphere, IGLayoutDavidsonHarel, IGLayoutDrL, IGLayoutDrL3D, IGLayoutFruchtermanReingold, IGLayoutFruchtermanReingold3D, IGLayoutGEM, IGLayoutGraphOpt, IGLayoutKamadaKawai, IGLayoutKamadaKawai3D, IGLayoutRandom, IGLayoutPlanar, IGLayoutTutte}; Multicolumn[layouts] # Community detection The following functions are available: ?IGCommunities* ## Concepts▴ Modularity is defined for a given partitioning of a graph’s vertices into communities. For undirected graphs, it is defined as \[Q=\frac{1}{2m}\sum _{i,j} \left(A_{ij}-\frac{k_ik_j}{2m}\right)\delta _{c_ic_j},

where $$m$$ is the number of edges, $$A$$ is the adjacency matrix, $$k_i$$ is the degree of node $$i$$, and $$c_i$$ is the community that node $$i$$ belongs to. $$\delta _{ij}$$ is the Kronecker $$\delta$$ symbol. For weighted graphs, $$A$$ is the weighted adjacency matrix, $$k_i$$ are the sum of weights of edges incident on node $$i$$, and $$m$$ is the sum of all weights.

Modularity characterizes the tendency of vertices to connect more within their own group than with other groups, relative to a null model that considers vertex degrees to be fixed. For a given partitioning, it can be computed using IGModularity. Most community detection methods aim find a partitioning of the graph which results in high modularity.

## Basic usage and utility functions▴

Community detection functions return IGClusterData objects.

g = ExampleData[{"NetworkGraph", "FamilyGathering"}]

cl = IGCommunitiesGreedy[g]

The data available in the object can be queried using IGClusterData[…]["Properties"]. See the Examples section below for more information. In Mathematica 12.0 and later, Information can be used to get a quick human-readable summary.

cl["Properties"]
{"Algorithm", "Communities", "ElementCount", "Elements", \
"HierarchicalClusters", "Merges", "Modularity", "Properties", "Tree"}
cl["Communities"]
{{"Elisabeth", "James", "Anna", "Nancy"}, {"John", "Dorothy", "David",
"Arlene", "Rudy"}, {"Linda", "Michael", "Nora", "Julia"}, {"Larry",
"Carol", "Ben", "Oscar", "Felicia"}}
CommunityGraphPlot[g, cl["Communities"], ImageSize -> Medium]

IGModularity[g, cl]
0.454735

### IGClusterData▴

?IGClusterData

IGClusterData represents a partitioning of a graph into communities. This object cannot be created directly. It is returned by community detection functions. See the Examples section below for more information.

cl = IGCommunitiesLabelPropagation@
ExampleData[{"NetworkGraph", "FamilyGathering"}]

Query the available properties.

cl["Properties"]
{"Algorithm", "Communities", "ElementCount", "Elements", \
"Modularity", "Properties"}

Retrieve the communities.

cl["Communities"]
{{"Elisabeth", "James", "Anna", "Linda", "Larry", "Carol", "Nancy",
"David", "Ben", "Oscar", "Felicia", "Arlene", "Rudy"}, {"John",
"Dorothy"}, {"Michael", "Nora", "Julia"}}

When the "Modularity" property is available, Max[cl["Modularity"]] gives the modularity of the current partitioning.

Max[cl["Modularity"]]
0.188866

### IGModularity▴

?IGModularity

IGModularity computes the generalized modularity in undirected or directed graphs, taking weights into account. For undirected graphs, it is defined as

$Q=\frac{1}{2m}\sum _{i,j} \left(A_{ij}-\gamma \frac{k_ik_j}{2m}\right)\delta _{c_ic_j},$

where $$m$$ is the number of edges, $$A$$ is the adjacency matrix, $$k_i$$ is the degree of node $$i$$, and $$c_i$$ is the community that node $$i$$ belongs to. $$\delta _{ij}$$ is the Kronecker $$\delta$$ symbol. For weighted graphs, $$A$$ is the weighted adjacency matrix, $$k_i$$ are the sum of weights of edges incident on node $$i$$, and $$m$$ is the sum of all weights. In the undirected case, $$A_{ii}$$ is assumed to contain twice the number of self-loops on vertex $$i$$, or twice their total weight in the weighted case. $$\gamma$$ is a resolution parameter, with $$\gamma =1$$ yielding the standard modularity.

The directed generalization is

$Q=\frac{1}{m}\sum _{i,j} \left(A_{ij}-\gamma \frac{k_i^{\text{out}}k_j^{\text{in}}}{m}\right)\delta _{c_ic_j}.$

For simple graphs, IGModularity[graph, communities] is equivalent to GraphAssortativity[graph, communities, "Normalized" -> False]. However, in contrast to GraphAssortativity, IGModularity does take self-loops and multi-edges into account.

Available options are:

• "Resolution" -> γ sets the resolution parameter. The default is 1.

• DirectedEdges -> True takes into account edge directions in directed graphs. By default, they are ignored.

Compute the modularity of a stochastic block model graph having three partitions, each with 20 vertices.

IGModularity[g, Partition[VertexList[g], 20]]
0.456154

Setting the resolution parameter to zero computes the fraction of intra-community edges.

IGModularity[g, Partition[VertexList[g], 20], "Resolution" -> 0]
0.8125

### IGModularityMatrix▴

?IGModularityMatrix

IGModularityMatrix computes the generalized modularity matrix of a graph, defined as

$B_{ij}=A_{ij}-\gamma \frac{k_ik_j}{2m}$

for undirected graphs and as

$B_{ij}=A_{ij}-\gamma \frac{k_i^{\text{out}}k_j^{\text{in}}}{m}$

for directed ones. Here, $$A$$ represents the adjacency matrix, $$k_i$$ is the degree of vertex $$i$$, $$m$$ is the sum of edge weights and $$\gamma$$ is the resolution parameter. Just as with IGModularity, in undirected graphs $$A_{ii}$$ is assumed to contain twice the number of self-loops. This way, the result for an undirected graph is the same as for the corresponding directed one where all undirected edges are replaced by a pair of reciprocal directed ones.

Available options are:

• "Resolution" -> γ sets the resolution parameter. The default is 1.

• DirectedEdges -> True takes into account edge directions in directed graphs. By default, they are ignored.

The modularity matrix is used in spectral clustering algorithms, such as the one implemented by IGCommunitiesLeadingEigenvector. Communities can be separated based on spatial clustering of the points formed by the first few eigenvectors of the modularity matrix.

g = IGStochasticBlockModelGame[
0.05 + 0.5 IdentityMatrix[3], {10, 10, 10}]

ListPlot[
FindClusters@Transpose@Eigenvectors[IGModularityMatrix[g], 2],
PlotStyle -> PointSize[Large], AspectRatio -> Automatic]

### IGCompareCommunities▴

?IGCompareCommunities

Some of these measures are defined based on the entropy of a discrete random variable associated with a given clustering $$C$$ of vertices. Let $$p_i$$ be the probability that a randomly picked vertex would be part of cluster $$i$$. Then the entropy of the clustering is

$H(C)=-\sum _ip_i\ln p_i.$

Similarly, we can define the joint entropy of two clusterings $$C_1$$ and $$C_2$$ based on the probability $$p_{ij}$$ that a random vertex is part of cluster $$i$$ in the first clustering and cluster $$j$$ in the second one:

$H\left(C_1,C_2\right)=-\sum _{i,j}p_{ij}\ln p_{ij}.$

The mutual information of $$C_1$$ and $$C_2$$ is then $$\text{MI}\left(C_1,C_2\right)=H\left(C_1\right)+H\left(C_2\right)-H\left(C_1,C_2\right)\text{>=}0$$. A large mutual information indicates a high overlap between the two clusterings. The normalized mutual information, as computed by igraph, is

$\text{NMI}\left(C_1,C_2\right)=\frac{2\text{MI}\left(C_1,C_2\right)}{H\left(C_1\right)+H\left(C_2\right)}.$

It takes its value from the interval $$(0,1]$$, with 1 achieved when the two clusterings coincide.

The variation of information is defined as $$\text{VI}\left(C_1,C_2\right)=\left[H\left(C_1\right)-\text{MI}\left(C_1,C_2\right)\right]+\left[H\left(C_2\right)-\text{MI}\left(C_1,C_2\right)\right]=2H\left(C_1,C_2\right)-H\left(C_1\right)-H\left(C_2\right)$$. Lower values of the variation of information indicate a smaller difference between the two clusterings, with $$\text{VI}=0$$ achieved precisely when they coincide.

The Rand index is defined based on counting pairs of vertices that are within the same or different clusters in the two clusterings. Let $$a_1$$ and $$a_2$$ denote the number of pairs that are grouped together in $$C_1$$ and $$C_2$$, respectively. Then the Rand index is

$\text{RI}\left(C_1,C_2\right)=\frac{a_1+a_2}{\left( \begin{array}{c} n \\ 2 \\ \end{array} \right)},$

where $$\left( \begin{array}{c} n \\ 2 \\ \end{array} \right)$$ is the total number of pairs of $$n$$ vertices. The value of the Rand index varies between 0 and 1. The adjusted Rand index, $$\text{ARI}$$, corrects the value based on the Rand index expected after after a random rearrangement of the vertices, denoted $$\text{ERI}$$:

$\text{ARI} = \frac{\text{RI}-\text{ERI}}{1-\text{ERI}}.$

g = ExampleData[{"NetworkGraph", "FamilyGathering"}];
{cl1, cl2} = {IGCommunitiesGreedy[g], IGCommunitiesEdgeBetweenness[g]}

IGCompareCommunities[cl1, cl2]
<|"VariationOfInformation" -> 0.278001,
"NormalizedMutualInformation" -> 0.899283, "SplitJoinDistance" -> 2,
"UnadjustedRandIndex" -> 0.947712, "AdjustedRandIndex" -> 0.841942|>

## Community detection methods▴

### IGCommunitiesEdgeBetweenness▴

?IGCommunitiesEdgeBetweenness

IGCommunitiesEdgeBetweenness[] implements the Girvan–Newman algorithm.

Weighted graphs are supported. Weights are treated as “distances”, i.e. a large weight represents a weak connection.

Available option values:

• "ClusterCount", the number of communities to return. Default: Automatic.

Special properties returned with the result:

• "RemovedEdges" is the list of edges removed in each step of the algorithm.

• "Bridges" records the steps which resulted in splitting the graph into more components.

#### References

• M. Girvan and M. E. J. Newman: Community structure in social and biological networks, PNAS 99, 7821-7826 (2002).

### IGCommunitiesFluid▴

?IGCommunitiesFluid

IGCommunitiesFluid[] implements the fluid communities algorithm.

#### Reference

• F. Parés, D. Garcia-Gasulla, A. Vilalta, J. Moreno, E. Ayguadé, Jesús Labarta, U. Cortés, T. Suzumura: Fluid Communities: A Competitive, Scalable and Diverse Community Detection Algorithm, https://arxiv.org/abs/1703.09307

### IGCommunitiesGreedy▴

?IGCommunitiesGreedy

IGCommunitiesGreedy[] implements greedy optimization of modularity.

Weighted graphs are supported.

### IGCommunitiesInfoMAP▴

?IGCommunitiesInfoMAP

IGCommunitiesInfoMAP[] implements the InfoMAP algorithm.

It supports both edge weights and vertex weights.

The default number of trials is 10.

Special properties returned with the result:

• "CodeLength" is the code length of the partition.

#### References

• M. Rosvall and C. T. Bergstrom, Maps of information flow reveal community structure in complex networks, PNAS 105, 1118 (2008)

• M. Rosvall, D. Axelsson, and C. T. Bergstrom, The map equation, Eur. Phys. J. Special Topics 178, 13 (2009)

### IGCommunitiesLabelPropagation▴

?IGCommunitiesLabelPropagation

Weighted graphs are supported.

#### References

• Raghavan, U.N. and Albert, R. and Kumara, S.: Near linear time algorithm to detect community structures in large-scale networks. Phys. Rev. E 76, 036106. (2007).

?IGCommunitiesLeadingEigenvector

Weighted graphs are supported.

Available option values:

• "ClusterCount", the number of communities to return. May return fewer communities than requested. Default: Automatic.

#### References

• M. E. J. Newman: Finding community structure using the eigenvectors of matrices, Phys. Rev. E 74:036104 (2006).

### IGCommunitiesMultilevel▴

?IGCommunitiesMultilevel

IGCommunitiesMultilevel[] implements the Louvain community detection method.

Weighted graphs are supported.

#### References

• V. D. Blondel, J.-L. Guillaume, R. Lambiotte and E. Lefebvre: Fast unfolding of community hierarchies in large networks, J. Stat. Mech. P10008 (2008)

### IGCommunitiesLeiden▴

?IGCommunitiesLeiden

The Leiden algorithm is similar to the multilevel algorithm, often called the Louvain algorithm, but it is faster and yields higher quality solutions. It can optimize both modularity and the Constant Potts Model, which does not suffer from the resolution-limit (see preprint http://arxiv.org/abs/1104.3083).

The Leiden algorithm consists of three phases: (1) local moving of nodes, (2) refinement of the partition and (3) aggregation of the network based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. In the local move procedure in the Leiden algorithm, only nodes whose neighborhood has changed are visited. The refinement is done by restarting from a singleton partition within each cluster and gradually merging the subclusters. When aggregating, a single cluster may then be represented by several nodes (which are the subclusters identified in the refinement).

The Leiden algorithm provides several guarantees. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. At each iteration all clusters are guaranteed to be connected and well-separated. After an iteration in which nothing has changed, all nodes and some parts are guaranteed to be locally optimally assigned. Finally, asymptotically, all subsets of all clusters are guaranteed to be locally optimally assigned.

The Leiden method maximizes a quality measure (a generalization of modularity) defined as

$Q=\frac{1}{2m}\sum _{i,j}\left(A_{ij}-\gamma n_in_j\right)\delta _{c_ic_j}$

where $$m$$ is the sum of edge weights (number of edges if the graph is unweighted), $$A$$ is the weighted adjacency matrix, $$n_i$$ is the weight of vertex $$i$$, and $$c_i$$ is the community that vertex $$i$$ belongs to. $$\delta _{ij}$$ is the Kronecker $$\delta$$ symbol.

$$\gamma$$ is a resolution parameter that can be set with the "Resolution" option.

The function chooses the vertex weights automatically, according to the value of the VertexWeight option:

• VertexWeight -> "NormalizedStrength" (default) sets $$n_i=k_i/\sqrt{2m}$$, where $$k_i$$ is the strength (sum of incident edge weights) of vertex $$i$$. If $$\gamma =1$$, then the quality measure becomes equivalent to the modularity.

• VertexWeight -> "Constant" sets $$n_i=1$$. With this choice, it is recommended to set the resolution parameter $$\gamma$$ explicitly. A reasonable $$\gamma$$ value for unweighted graphs is the graph density.

• VertexWeight -> "VertexWeight" takes vertex weights from the VertexWeight graph property.

Other available options:

• "Resolution" -> γ sets the resolution parameter $$\gamma$$. The default is $$\gamma =1$$. With VertexWeight -> "NormalizedStrength", a reasonable value is 1. With VertexWeight -> "Constant", a reasonable value is the graph density.

• "Beta" -> β sets the randomness used in the refinement step when merging clusters. The default is $$\beta =0.01$$.

Special properties returned with the result :

• "Quality" is the value of the quality measure $$Q$$.

Examples:

g = Graph[ExampleData[{"NetworkGraph", "LesMiserables"}],
GraphStyle -> "BasicBlack", VertexSize -> 2];

With the default option values VertexWeight -> "NormalizedStrength" and "Resolution" -> 1, IGCommunitiesLeiden effectively uses the modularity as the quality measure.

cl = IGCommunitiesLeiden[g]

{cl["Quality"], IGModularity[g, cl]}
{0.566688, 0.566688}
HighlightGraph[g, cl["Communities"]]

A higher "Resolution" value results in more communities.

HighlightGraph[
g,
IGCommunitiesLeiden[g, "Resolution" -> 3]["Communities"]
]

With VertexWeight -> "Constant", it is recommended to set "Resolution" explicitly. A reasonable starting point is GraphDensity[g].

HighlightGraph[
g,
IGCommunitiesLeiden[g, VertexWeight -> "Constant",
"Resolution" -> 0.1]["Communities"]
]

### IGCommunitiesOptimalModularity▴

?IGCommunitiesOptimalModularity

Finds the clustering that maximizes modularity exactly. This algorithm is very slow.

Weighted graphs are supported.

### IGCommunitiesSpinGlass▴

?IGCommunitiesSpinGlass

Weighted graphs are supported.

Option values for Method are:

• "Original" only supports positive edge weights, but doesn’t check that the supplied weights are actually positive.

• "Negative" supports negative weights as well.

• Automatic selects "Negative" if negative weights are presents and "Original" otherwise.

Option values for "UpdateRule" are: "Simple", "Configuration"

Special properties returned with the result:

• "FinalTemperature" is the final temperature at the end of the algorithm.

#### References

• For Method -> "Original, see Joerg Reichardt and Stefan Bornholdt: Statistical Mechanics of Community Detection, http://arxiv.org/abs/cond-mat/0603718

• For Method -> "Negative", see V. A. Traag and Jeroen Bruggeman: Community detection in networks with positive and negative links, http://arxiv.org/abs/0811.2329

### IGCommunitiesWalktrap▴

?IGCommunitiesWalktrap

IGCommunitiesWalktrap[] finds communities using short random walks, exploiting the fact that random walks tend to stay within the same cluster.

Weighted graphs are supported.

The default number of steps is 4.

Available option values:

• "ClusterCount", the number of communities to return. Default: Automatic.

## Examples▴

g = ExampleData[{"NetworkGraph", "LesMiserables"}]

IGEdgeWeightedQ[g]
True

Community detection functions return IGClusterData objects.

cl1 = IGCommunitiesEdgeBetweenness[g, "ClusterCount" -> 7]
cl2 = IGCommunitiesWalktrap[g]

Various properties of these objects can be queried:

cl1["Communities"]
{{"Myriel", "Napoleon", "Mlle Baptistine", "Mme. Magloire",
"Countess De Lo", "Geborand", "Champtercier", "Cravatte", "Count",
"Old Man"}, {"Labarre", "Valjean", "Marguerite", "Mme. De R",
"Isabeau", "Gervais", "Bamatabois", "Perpetue", "Simplice",
"Scaufflaire", "Woman1", "Judge", "Champmathieu", "Brevet",
"Chenildieu", "Cochepaille"}, {"Tholomyes", "Listolier", "Fameuil",
"Blacheville", "Favourite", "Dahlia", "Zephine",
"Fantine"}, {"Mme. Thenardier", "Thenardier", "Cosette", "Javert",
"Boulatruelle", "Eponine", "Anzelma", "Woman2", "Gueulemer",
"Babet", "Claquesous", "Montparnasse", "Toussaint",
"Brujon"}, {"Fauchelevent", "Mother Innocent",
"Gribier"}, {"Pontmercy", "Gillenormand", "Magnon",
"Mlle Gillenormand", "Mme. Pontmercy", "Mlle Vaubois",
"Lt. Gillenormand", "Marius", "Baroness T"}, {"Jondrette",
"Mme. Burgon", "Gavroche", "Mabeuf", "Enjolras", "Combeferre",
"Prouvaire", "Feuilly", "Courfeyrac", "Bahorel", "Bossuet", "Joly",
"Grantaire", "Mother Plutarch", "Child1", "Child2",
"Mme. Hucheloup"}}

Visualize the detected communities in two different ways:

CommunityGraphPlot[g, cl1["Communities"]]

HighlightGraph[g, Subgraph[g, #] & /@ cl1["Communities"],
GraphHighlightStyle -> "DehighlightGray"]

Plot the adjacency matrix, reordered to show the community structure.

IGAdjacencyMatrixPlot[g, Catenate@cl1["Communities"]]

The available properties depend on which algorithm was used for community detection. The following are always present:

• "Properties" returns all available properties.

• "Algorithm" returns the algorithm used for community detection.

• "Communities" returns the list of communities.

• "Elements" returns the vertices of the graph.

• "ElementCount" returns the vertex count of the graph.

These are present for hierarchical clustering methods:

• "HierarchicalClusters" returns the clustering in a format compatible with the Hierarchical Clustering standard package. Note: Isolated vertices may not be included.

• "Merges" represents the hierarchical clustering as a sequence of element merges. Elements are represented by their integer indices, and higher indices are introduced for the subclusters formed by the merges. This format is similar to the one used by MATLAB and many other tools. Note: Isolated vertices may not be included.

• "Tree" gives a binary tree representation of the merges. Note: Isolated vertices may not be included.

Additionally, the following, and other, algorithm-specific properties may be present:

• "Modularity" is a list of modularities for each step of the algorithm, or a single-element list containing the modularity corresponding to the returned clustering. What constitutes a step depends on the particular algorithm.

The "RemovedEdges" property is specific to the "EdgeBetweenness" method, and isn’t present for "Walktrap".

cl1["Properties"]
{"Algorithm", "Bridges", "Communities", "EdgeBetweenness", \
"ElementCount", "Elements", "HierarchicalClusters", "Merges", \
"Properties", "RemovedEdges", "Tree"}
Take[cl1["RemovedEdges"], 10]
{"Valjean" <-> "Myriel",
"Valjean" <-> "Mlle Baptistine",
"Valjean" <-> "Mme. Magloire",
"Gavroche" <-> "Valjean",
"Gavroche" <-> "Javert",
"Thenardier" <-> "Fantine",
"Bamatabois" <-> "Javert",
"Bossuet" <-> "Valjean",
"Montparnasse" <-> "Valjean",
"Gueulemer" <-> "Gavroche"}
cl2["Properties"]
{"Algorithm", "Communities", "ElementCount", "Elements", \
"HierarchicalClusters", "Merges", "Modularity", "Properties", "Tree"}

Multiple properties may be retrieved at the same time.

cl2[{"Algorithm", "ElementCount"}]
{"Walktrap", 77}

Compare the two clusterings:

IGCompareCommunities[cl1, cl2]
<|"VariationOfInformation" -> 0.804544,
"NormalizedMutualInformation" -> 0.786844, "SplitJoinDistance" -> 29,
"UnadjustedRandIndex" -> 0.879699, "AdjustedRandIndex" -> 0.555464|>

Visualize the hierarchical clustering using the Hierarchical Clustering Package.

<< HierarchicalClustering
DendrogramPlot[cl1["HierarchicalClusters"],
LeafLabels -> (Rotate[#, Pi/2] &), ImageSize -> 750,
AspectRatio -> 1/2]

Hierarchical community structures can also be obtained as a vertex-weighted tree graph.

g = ExampleData[{"NetworkGraph", "ZacharyKarateClub"}];
cl = IGCommunitiesGreedy[g];
clusteringTree = cl["Tree"]

{GraphQ[clusteringTree], IGVertexWeightedQ[clusteringTree]}
{True, True}

This tree can be supplied as input to Dendrogram.

Dendrogram[clusteringTree, Left]

# Graph cycles

## Eulerian paths and cycles▴

An Eulerian path passes through each edge of a graph precisely once. An Eulerian cycle is a closed Eulerian path: its starting vertex is the same as its ending vertex. Eulerian paths are also known as Eulerian trails.

Note: As of IGraph/M 0.5, the Eulerian path functions are still experimental.

### IGEulerianQ▴

?IGEulerianQ

The following graph does not have an Eulerian path:

IGEulerianQ[graph]
False

Removing the edge A <-> D makes it Eulerian:

IGEulerianQ@EdgeDelete[graph, "A" <-> "D"]
True

But it will only have an Eulerian path, not an Eulerian cycle:

IGEulerianQ[
EdgeDelete[graph, "A" <-> "D"],
Closed -> True
]
False

One possible path is the following:

IGEulerianPathVertices[EdgeDelete[graph, "A" <-> "D"]]
{"B", "A", "B", "D", "C", "A", "C"}

### IGEulerianPath and IGEulerianPathVertices▴

?IGEulerianPath

?IGEulerianPathVertices

Find an Eulerian cycle through an icosidodecahedral graph:

g = GraphData["IcosidodecahedralGraph"];
cycle = IGEulerianPath[g, Closed -> True]
{1 <-> 5, 5 <-> 7, 7 <-> 8,
6 <-> 8, 2 <-> 6, 2 <-> 16,
6 <-> 16, 6 <-> 30,
8 <-> 30, 8 <-> 14,
7 <-> 14, 7 <-> 29,
5 <-> 29, 5 <-> 15,
1 <-> 15, 1 <-> 18,
18 <-> 20, 11 <-> 20,
11 <-> 12, 3 <-> 12,
3 <-> 9, 9 <-> 15,
15 <-> 22, 9 <-> 22,
9 <-> 17, 3 <-> 17,
3 <-> 27, 12 <-> 27,
12 <-> 13, 4 <-> 13,
4 <-> 10, 10 <-> 16,
16 <-> 23, 14 <-> 23,
14 <-> 22, 22 <-> 23,
10 <-> 23, 10 <-> 17,
4 <-> 17, 4 <-> 28,
2 <-> 28, 2 <-> 19,
19 <-> 21, 11 <-> 21,
11 <-> 13, 13 <-> 28,
19 <-> 28, 19 <-> 26,
21 <-> 26, 20 <-> 21,
20 <-> 25, 24 <-> 25,
24 <-> 26, 26 <-> 30,
24 <-> 30, 24 <-> 29,
25 <-> 29, 18 <-> 25,
18 <-> 27, 1 <-> 27}

Visualize it using colour hues:

HighlightGraph[g,
MapIndexed[Style[#1, Hue[First[#2]/Length[cycle]]] &, cycle],
GraphStyle -> "ThickEdge"
]

Direct the edges of the graph along the cycle:

Graph[
VertexList[g],
DirectedEdge @@@
Partition[IGEulerianPathVertices[g, Closed -> True], 2, 1],
VertexCoordinates -> GraphEmbedding[g]
]

# Graph colouring

The graph colouring problem is assigning “colours” or “labels” to the vertices of a graph so that no two adjacent vertices will have the same colour. Similarly, edge colouring assigns colours to edges so that adjacent edges never have the same colour.

IGraph/M represents colours with the integers 1, 2, …. Edge directions and self-loops are ignored.

## Fast heuristic colouring▴

?IGVertexColoring

?IGEdgeColoring

These function will find a colouring of the graph using a fast heuristic algorithm. The colouring may not be minimal. Edge directions are ignored.

Compute a vertex colouring of a Mycielski graph.

g = GraphData[{"Mycielski", 4}]

IGVertexColoring returns a list of integers, each representing the colour of the vertex that is in the same position in the vertex list.

IGVertexColoring[g]
{4, 3, 1, 1, 3, 1, 2, 2, 2, 2, 2}

Associate the colours with vertex names.

AssociationThread[VertexList[g], IGVertexColoring[g]]
<|1 -> 4, 2 -> 3, 3 -> 1, 4 -> 1, 5 -> 3, 6 -> 1, 7 -> 2, 8 -> 2,
9 -> 2, 10 -> 2, 11 -> 2|>

Visualize the colours using IGraph/M’s property mapping functionality. See the Property handling functions documentation section for more information.

Graph[g, VertexSize -> 1/3, EdgeStyle -> Gray] //
IGVertexMap[ColorData[97], VertexStyle -> IGVertexColoring]

Visualize an edge colouring of the same graph.

Graph[g, GraphStyle -> "ThickEdge", EdgeStyle -> Opacity[0.7],
VertexStyle -> Black] //
IGEdgeMap[ColorData[106], EdgeStyle -> IGEdgeColoring]

Compute a checkerboard-like colouring of a three-dimensional grid graph.

IGVertexMap[ColorData[97], VertexStyle -> IGVertexColoring,
Graph3D@GridGraph[{4, 4, 4}, VertexSize -> 0.8]]

Compute a colouring of a Voronoi mesh.

mesh = VoronoiMesh[RandomReal[1, {20, 2}]]

col = IGVertexColoring@IGMeshCellAdjacencyGraph[mesh, 2]
{2, 5, 4, 2, 1, 1, 2, 2, 4, 1, 3, 1, 2, 3, 3, 4, 2, 1, 3, 1}
SetProperty[{mesh, {2, All}}, MeshCellStyle -> ColorData[97] /@ col]

Compute a colouring of the map of African countries.

countries = CountryData["Africa"];
borderingQ[c1_, c2_] := MemberQ[c1["BorderingCountries"], c2]
graph = RelationGraph[borderingQ, countries];
GeoGraphics@
ColorData[97] /@ IGVertexColoring[graph]}]
## $$k$$-colouring▴
?IGKVertexColoring