IGraph/M

the igraph interface for Mathematica
0.4 (April 2, 2020)

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 0.4 (April 2, 2020)
igraph 0.9.0-pre+34199c62 (Apr  2 2020)
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 12.1 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"]

Many included graph creation functions implement random graph models. These use igraph’s (not Mathematica’s) random number generator, which can be re-seeded using IGSeedRandom[].

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:

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}

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:

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:

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:

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,
 Complement[VertexList[g], AdjacencyList[g, center, 4], {center}]
 ]

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

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

IGKaryTree

?IGKaryTree

The available options are:

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 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 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]
 ]

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:

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:

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 uses the Havel–Hakimi algorithm (undirected case) or Kleitman–Wang algorithm (directed case) to construct a graph with the given degree sequence. These algorithms work by selecting a 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. The order in which vertices are selected is controlled by the Method option.

To randomly sample multiple realizations of a degree sequence, use IGDegreeSequenceGame.

Available Method option values:

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}
IGRealizeDegreeSequence[degseq]

N@GraphAssortativity[%]
-0.524347
IGRealizeDegreeSequence[degseq, Method -> "LargestFirst"]

N@GraphAssortativity[%]
0.904728

Create a directed graph.

g = IGBarabasiAlbertGame[50, 1]

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

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

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]

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["Gr`HOk"]

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

IGFromNauty[":I`ESgTlVF"]

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:

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:

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:

Available Method options:

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:

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}]
{8, 8, 8, 8, 7, 9, 9, 6, 9, 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:

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:

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:

Available Method option values:

The built-in BarabasiAlbertGraphDistribution is equivalent to using \(A=0\) and DirectedEdges -> False in IGBarabasiAlbertGame.

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 \[UndirectedEdge] 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:

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:

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

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:

IGAdjacencyMatrixPlot[%]

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:

The available options are:

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

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:

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:

IGGeometricGame

?IGGeometricGame

Available options:

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:

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:

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]}
{{4, 3, 5, 1, 3, 3, 2, 1, 5, 3}, {2, 2, 4, 4, 3, 3, 4, 3, 3, 2}}
rg = IGRewireEdges[g, 1, "Out"];
{VertexInDegree[rg], VertexOutDegree[rg]}
{{3, 7, 4, 1, 1, 2, 3, 3, 2, 4}, {2, 2, 4, 4, 3, 3, 4, 3, 3, 2}}

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:

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:

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

Betweenness

?IGBetweenness

?IGBetweennessEstimate

?IGEdgeBetweenness

?IGEdgeBetweennessEstimate

Weighted graphs are supported by all betweenness functions in IGraph/M.

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 \(i\).

Available options:

Available Method options for vertex betweenness calculations:

Compare the "Fast" and "Precise" methods:

g = GridGraph[{30, 30}];
Timing[Max@IGBetweenness[g, Method -> "Precise"]]
{0.195409, 18980.5}
Timing[Max@IGBetweenness[g, Method -> "Fast"]]
{0.027016, 18980.5}

For this large grid graph, the "Fast" method no longer gives accurate results:

g = GridGraph[{40, 40}];
Timing[Max@IGBetweenness[g, Method -> "Precise"]]
{0.701559, 45701.7}
Timing[Max@IGBetweenness[g, Method -> "Fast"]]
{0.085084, 78169.7}

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:

Weighted betweenness calculations may be affected by numerical precision when non-integer weights are used. Betweenness computations count shortest paths, which means that the total weight of different paths must be compared for equality. Equality testing with floating point numbers is unreliable. This can be demonstrated on an example which is purposefully constructed to be problematic.

An unweighted grid graph has many shortest paths between the same pair of vertices.

g = GridGraph[{6, 6}];

Let us construct non-integer weights which can add up to (equal) integers in many different ways.

weights = 1/RandomInteger[{1, 5}, EdgeCount[g]];

Let us now associate weights with edges and order the edges in two different random ways. Betweenness should not depend on edge ordering, so graphs constructed from both should have the same betweenness values.

asc1 = Association@RandomSample@Thread[EdgeList[g] -> weights];
asc2 = Association@RandomSample@Thread[EdgeList[g] -> weights];
g1 = Graph[Keys[asc1], EdgeWeight -> Values[asc1]];
g2 = Graph[Keys[asc2], EdgeWeight -> Values[asc2]];
KeySort@AssociationThread[VertexList[g1], IGBetweenness[g1]] -
 KeySort@AssociationThread[VertexList[g2], IGBetweenness[g2]]
<|1 -> 0., 2 -> 0., 3 -> 0., 4 -> 0., 5 -> 0., 6 -> 0., 7 -> 0., 
8 -> 0., 9 -> 0., 10 -> -7.10543*10^-15, 11 -> 0., 12 -> 0., 
13 -> 0., 14 -> 0., 15 -> -2.84217*10^-14, 16 -> -1.42109*10^-14, 
17 -> 0., 18 -> 0., 19 -> 0., 20 -> 0., 21 -> -2.84217*10^-14, 
22 -> 0., 23 -> 0., 24 -> 0., 25 -> 0., 26 -> 0., 27 -> 0., 28 -> 0.,
 29 -> 0., 30 -> 0., 31 -> 0., 32 -> 0., 33 -> 0., 34 -> 0., 
35 -> 0., 36 -> 0.|>
Max@Abs[%]
2.84217*10^-14

Yet the results differ. This is because IGraph/M works with floating point numbers. Even when different sums of exact weights happen to be equal, floating point calculations will give slightly different results.

To obtain a reliable result, we must use integer weights. The weights in these example were inverses of {1, 2, 3, 4, 5}. Multiplying these by their least common multiple will always yield an integer.

LCM @@ {1, 2, 3, 4, 5}
60
asc1 = Association@RandomSample@Thread[EdgeList[g] -> 60 weights];
asc2 = Association@RandomSample@Thread[EdgeList[g] -> 60 weights];
g1 = Graph[Keys[asc1], EdgeWeight -> Values[asc1]];
g2 = Graph[Keys[asc2], EdgeWeight -> Values[asc2]];
KeySort@AssociationThread[VertexList[g1], IGBetweenness[g1]] -
 KeySort@AssociationThread[VertexList[g2], IGBetweenness[g2]]
<|1 -> 0., 2 -> 0., 3 -> 0., 4 -> 7.10543*10^-15, 5 -> 0., 6 -> 0., 
7 -> 0., 8 -> 0., 9 -> 7.10543*10^-15, 10 -> 7.10543*10^-15, 
11 -> 0., 12 -> 0., 13 -> 0., 14 -> 0., 15 -> 2.84217*10^-14, 
16 -> 4.26326*10^-14, 17 -> 0., 18 -> 0., 19 -> 0., 20 -> 0., 
21 -> 2.84217*10^-14, 22 -> 7.10543*10^-15, 23 -> 0., 24 -> 0., 
25 -> 0., 26 -> 0., 27 -> 0., 28 -> 0., 29 -> 0., 30 -> 0., 31 -> 0.,
 32 -> 0., 33 -> 0., 34 -> 0., 35 -> 0., 36 -> 0.|>
Chop@Max@Abs[%]
0

Now IGBetweenness gives the same result regardless of edge ordering.

Closeness

?IGCloseness

?IGClosenessEstimate

The normalized closeness centrality of a vertex is the inverse average shortest path length to other vertices.

Weighted graphs are supported.

Available options:

igraph’s closeness calculation differs from Mathematica’s in that when there is no path connecting two vertices, the total number of vertices is used as their distance (a number larger than any other distance in an unweighted graph). Mathematica computes closeness separately within each connected component.

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"]
 ]

Indeterminate is returned for a single-vertex graph.

IGCloseness@IGEmptyGraph[1]
{Indeterminate}

PageRank

?IGPageRank

?IGPersonalizedPageRank

Weighted graphs and multigraphs are supported.

The default damping factor is 0.85.

The following Method options are available:

Eigenvector centrality

?IGEigenvectorCentrality

Weighted graphs are supported.

The available options are:

Kleinberg’s hub and authority scores

?IGHubScore

?IGAuthorityScore

Weighted graphs are supported.

The available options are:

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]
0.315526
IGClosenessCentralization[g]
0.366524
IGDegreeCentralization[g, SelfLoops -> False]
0.206761
IGEigenvectorCentralization[g]
0.868027

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.

Graph[g,
  EdgeShapeFunction -> 
   GraphElementData[{"CurvedEdge", "Curvature" -> 1.5}]
  ] // IGVertexMap[{#, 0} &, 
  VertexCoordinates -> IGTopologicalOrdering /* Ordering]

When the graph contains cycles, and a complete topological sort cannot be performed, only a partial result is returned.

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

{1, 2}

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]
{2 \[DirectedEdge] 4, 2 \[DirectedEdge] 7, 8 \[DirectedEdge] 5, 
8 \[DirectedEdge] 6}
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]
{5, 7, 3, 6, 10, 9, 4, 2, 1, 8}
With[{am = AdjacencyMatrix[ag]},
 ArrayPlot /@ {am, am[[perm, perm]]}
 ]

References

Chordal graphs

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 \[UndirectedEdge] 4, 4 \[UndirectedEdge] 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. The visiting order is animated below:

seq = InversePermutation@IGMaximumCardinalitySearch[g]
{7, 6, 2, 9, 8, 4, 5, 3, 10, 1}
Table[
  HighlightGraph[
   Graph[g, VertexLabels -> "Name"],
   Take[seq, -i]
   ],
  {i, 1, 10}
  ] // ListAnimate

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:

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:

In the following graph, vertex 4 has only one neighbour. Thus its local clustering coefficient will be computed as either 0 or indeterminate depending on the setting for "ExcludeIsolates".

{1., 1., 0.333333, 0.}

{1., 1., 0.333333, Indeterminate}

IGAverageLocalClusteringCoefficient

?IGAverageLocalClusteringCoefficient

The available options are:

With "ExcludeIsolates" -> True, the local clustering coefficient of vertex 4 will be excluded from the calculation of the average.

{0.583333, 0.777778}

IGWeightedClusteringCoefficient

?IGWeightedClusteringCoefficient

IGWeightedClusteringCoefficient computes the weighted local clustering coefficient. This function expects a weighted graph as input.

The available options are:

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]
{3.75, 4., 4.5, 4.42857, 4.12, Indeterminate, 4.42857, 4.25}

An equivalent implementation of IGAverageDegreeConnectivity is:

Transpose[{VertexDegree[g], IGAverageNeighborDegree[g]}] //
  
  GroupBy[#, First -> Last, Mean] & //
 
 Lookup[#, Range@Max@VertexDegree[g], Indeterminate] &
{3.75, 4., 4.5, 4.42857, 4.12, Indeterminate, 4.42857, 4.25}

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:

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

If the graph is unconnected, vertex pairs between which there is no path are excluded from the calculation. This is different from the behaviour of MeanGraphDistance[], which returns in this case.

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, 0 is returned.

IGGirth@IGShorthand["1-2"]
0

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:

2
?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:

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}, 4 -> {3, 4, 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}, 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"
 ]

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:

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:

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}]
{0.547222, 0.366667, 0.429167}

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"]}
{{0.547222, 0.366667, 0.429167, 0.284722, 0., 0.597222, 0.277778, 
 0.166667, 0.375, 0.569444}, {0., 0.25, 0.429167, 0., 0., 0.75, 0.5, 
 0.5, 0., 0.75}, {0.547222, 0.291667, 0., 1., 0., 0., 0.125, 0.5, 
 0.5, 0.}}

Ignore edge directions when computing shortest paths.

IGLocalEfficiency[g, DirectedEdges -> False]
{0.833333, 0.666667, 0.683333, 0.305556, 0., 0.833333, 0.638889, \
0.833333, 0.722222, 0.833333}

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]}
{0.21824, 0.21824}

Use only the out-neighbourhood while computing the local efficiency.

IGAverageLocalEfficiency[g, "Out"]
0.171157

Bipartite graphs

?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[
    Thread[IGVertexProp["Type"][g] -> VertexList[g]],
    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

?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 IGZeroDiagonal[am.am\[Transpose]], where am is the adjacency matrix of the graph.

?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 IGZeroDiagonal[am\[Transpose].am], where am is the adjacency matrix of the graph.

?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.

?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.

?IGInverseLogWeightedSimilarity

The inverse log-weighted similarity of two vertices is the number of their common neighbours, weighted by the inverse logarithm of their degrees. 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.

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

References

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 connected by convention.

IGConnectedQ@IGEmptyGraph[0]
True

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] 10, 5 \[DirectedEdge] 10, 6 \[DirectedEdge] 4, 
 8 \[DirectedEdge] 10}, {1 \[DirectedEdge] 10, 4 \[DirectedEdge] 2, 
 5 \[DirectedEdge] 10, 8 \[DirectedEdge] 10}, {1 \[DirectedEdge] 10, 
 2 \[DirectedEdge] 10, 5 \[DirectedEdge] 10, 8 \[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] 10, 5 \[DirectedEdge] 10, 6 \[DirectedEdge] 4, 
 8 \[DirectedEdge] 10}, {1 \[DirectedEdge] 10, 4 \[DirectedEdge] 2, 
 5 \[DirectedEdge] 10, 8 \[DirectedEdge] 10}, {1 \[DirectedEdge] 10, 
 2 \[DirectedEdge] 10, 5 \[DirectedEdge] 10, 8 \[DirectedEdge] 10}}

If only a single minimum cut is needed, use IGMinimumCut:

IGMinimumCut[g, 1, 10]
{1 \[DirectedEdge] 10, 2 \[DirectedEdge] 10, 5 \[DirectedEdge] 10, 
8 \[DirectedEdge] 10}

The size (total weight) of the cut can be obtained with IGMinimumCutValue:

IGMinimumCutValue[g, 1, 10]
21.

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 \[UndirectedEdge] 3, 2 \[UndirectedEdge] 3, 
 1 \[UndirectedEdge] 2}, {1 \[UndirectedEdge] 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 \[UndirectedEdge] 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]
{8, 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]
{8, 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

Trees

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]
14

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.

{}

IGFromPrufer

?IGFromPrufer

IGToPrufer

?IGToPrufer

Spanning trees

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]
1.89734

Find a maximum spanning tree by negating the weights before running the algorithm.

IGSpanningTree[IGEdgeMap[Minus, EdgeWeight, g], 
 VertexCoordinates -> pts]

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 \[UndirectedEdge] 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

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 \[UndirectedEdge] 9, 5 \[UndirectedEdge] 10, 3 \[UndirectedEdge] 8,
 2 \[UndirectedEdge] 6, 1 \[UndirectedEdge] 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:

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 0.4, only size 3 and 4 motifs are supported, and 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

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:

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, 1.41453, Indeterminate, \
Indeterminate, Indeterminate, 1.41686, 0.309827, Indeterminate, \
Indeterminate, Indeterminate, 0.630358, 0.64798, 0., Indeterminate, \
0.014674, 0., 0., 6.17143, 0., 0., Indeterminate, Indeterminate, \
1.37886, 0.307329, 0.119888, Indeterminate, Indeterminate, 0.673385, \
0., 0.0178433, 0., Indeterminate, Indeterminate, 0., 0., 0., 0., \
Indeterminate, 0.203776, 0.707195, 0., 0.204099, 0., 0.181575, \
0.0479911, 0., 0., 0., 0., 0., 1.51458, 0., 0.0731707, 0., 0., 0., \
0., 0., 0., 0., Indeterminate, 0., 0., 0., 32.6408, 0., 0.183007, 0., \
0., 0., 0., 0.163265, 0., 0., 1.32195, 0., 0., 0., 0., 0., 0., 0., \
0., 0., 0., 0., 0., 0., 0., 0., 0.297946, 0.0921878, 0.801547, 0., \
0.0211334, 0., 0.217483, 0., 0.0414765, 0., 0., 0., 0., 0., 0., 0., \
0., 0., 0.528455, 0., 0.181818, 0., 0., 0., 0., 0., 0., 0., \
Indeterminate, 0.276486, 0., 0., 0., 2.11212, 0., 0., 41.1096, 0., \
0.513158, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \
1.72414, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.0714286, \
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., \
0., 0., 0., 0., 0., 0., 0., 0., 0., Indeterminate}
largeRatios = Select[ratios, # > 5 &]
{6.17143, 32.6408, 41.1096}

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
{0.680767, {Indeterminate, Indeterminate, 382909, Indeterminate, 1774,
  4629, 32984, 500, 298, 957, 7939, 4, 26, 923, 107, 6478}}

Sample 12.5% of motifs, i.e. a fraction of \(0.5^3\).

IGMotifs[bigG, 3, 1 - 0.5 {1, 1, 1}] // AbsoluteTiming
{10.17, {Indeterminate, Indeterminate, 13816444, Indeterminate, 
 345860, 542100, 7069751, 53710, 36824, 245943, 671106, 827, 5284, 
 126012, 14205, 827329}}

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:

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.

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]
838

IGMotifsTotalCount is effectively equivalent to (but much faster than) the following:

Count[Subsets[VertexList[g], {4}], 
 subset_ /; WeaklyConnectedGraphQ@Subgraph[g, subset]]
838

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
588.325

Estimate the count of connected subgraphs by considering a random subset of 15 vertices (out of a total of 20).

IGMotifsTotalCountEstimate[g, 4, 15]
577

Use the first 15 vertices tot estimate the count.

IGMotifsTotalCountEstimate[g, 4, Range[15]]
1113

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.

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:

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 \[UndirectedEdge] 2, 2 \[UndirectedEdge] 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 \[UndirectedEdge] 3, 
 2 \[UndirectedEdge] 3}, {1 \[UndirectedEdge] 3, 
 2 \[UndirectedEdge] 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

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 \[UndirectedEdge] 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 \[UndirectedEdge] 3]

g2 = EdgeAdd[PathGraph[Range[5], VertexLabels -> "Name"], 
  4 \[UndirectedEdge] 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 \[UndirectedEdge] 3 and 3 \[UndirectedEdge] 4 are treated as the same edge.

colors1 = Counts[Sort /@ EdgeList[g1]]
<|1 \[UndirectedEdge] 2 -> 1, 2 \[UndirectedEdge] 3 -> 2, 
3 \[UndirectedEdge] 4 -> 1, 4 \[UndirectedEdge] 5 -> 1|>
colors2 = Counts[Sort /@ EdgeList[g2]]
<|1 \[UndirectedEdge] 2 -> 1, 2 \[UndirectedEdge] 3 -> 1, 
3 \[UndirectedEdge] 4 -> 2, 4 \[UndirectedEdge] 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

LAD

The LAD library was developed by Christine Solnon. It is capable of finding subgraphs in a larger graph.

The LAD algorithm only supports simple graphs.

?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]
<|16 -> 5, 8 -> 7, 10 -> 10, 2 -> 1, 17 -> 7, 19 -> 1, 3 -> 1, 9 -> 1,
 7 -> 3, 11 -> 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:

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 \[UndirectedEdge] 2, 1 \[UndirectedEdge] 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 \[UndirectedEdge] 2, 2 \[UndirectedEdge] 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 \[UndirectedEdge] 2 \[UndirectedEdge] 5 \[UndirectedEdge] 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:

mw = Graph[{"1" \[UndirectedEdge] "2", "1" \[UndirectedEdge] "3", 
    "1" \[UndirectedEdge] "4", "1" \[UndirectedEdge] "5", 
    "1" \[UndirectedEdge] "6", "2" \[UndirectedEdge] "3", 
    "2" \[UndirectedEdge] "4", "2" \[UndirectedEdge] "5", 
    "2" \[UndirectedEdge] "7", "3" \[UndirectedEdge] "4", 
    "3" \[UndirectedEdge] "6", "3" \[UndirectedEdge] "7", 
    "4" \[UndirectedEdge] "5", "4" \[UndirectedEdge] "6", 
    "4" \[UndirectedEdge] "7", "5" \[UndirectedEdge] "6", 
    "5" \[UndirectedEdge] "7", "5" \[UndirectedEdge] "21", 
    "6" \[UndirectedEdge] "7", "7" \[UndirectedEdge] "8", 
    "7" \[UndirectedEdge] "11", "7" \[UndirectedEdge] "14", 
    "7" \[UndirectedEdge] "19", "8" \[UndirectedEdge] "9", 
    "8" \[UndirectedEdge] "11", "8" \[UndirectedEdge] "14", 
    "9" \[UndirectedEdge] "10", "10" \[UndirectedEdge] "12", 
    "10" \[UndirectedEdge] "13", "11" \[UndirectedEdge] "12", 
    "11" \[UndirectedEdge] "14", "12" \[UndirectedEdge] "16", 
    "13" \[UndirectedEdge] "16", "14" \[UndirectedEdge] "15", 
    "15" \[UndirectedEdge] "16", "17" \[UndirectedEdge] "18", 
    "17" \[UndirectedEdge] "19", "17" \[UndirectedEdge] "20", 
    "18" \[UndirectedEdge] "20", "18" \[UndirectedEdge] "21", 
    "19" \[UndirectedEdge] "20", "19" \[UndirectedEdge] "22", 
    "19" \[UndirectedEdge] "23", "20" \[UndirectedEdge] "21", 
    "21" \[UndirectedEdge] "22", "21" \[UndirectedEdge] "23", 
    "22" \[UndirectedEdge] "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" \[UndirectedEdge] "Jennie", 
    "Pauline" \[UndirectedEdge] "Ann", 
    "Jennie" \[UndirectedEdge] "Ann", 
    "Jennie" \[UndirectedEdge] "Michael", 
    "Michael" \[UndirectedEdge] "Ann", 
    "Holly" \[UndirectedEdge] "Jennie", 
    "Jennie" \[UndirectedEdge] "Lee", 
    "Michael" \[UndirectedEdge] "Lee", 
    "Harry" \[UndirectedEdge] "Bert", "Harry" \[UndirectedEdge] "Don",
     "Don" \[UndirectedEdge] "Bert", "Gery" \[UndirectedEdge] "Russ", 
    "Russ" \[UndirectedEdge] "Bert", 
    "Michael" \[UndirectedEdge] "John", 
    "Gery" \[UndirectedEdge] "John", "Russ" \[UndirectedEdge] "John", 
    "Holly" \[UndirectedEdge] "Pam", "Pam" \[UndirectedEdge] "Carol", 
    "Holly" \[UndirectedEdge] "Carol", 
    "Holly" \[UndirectedEdge] "Bill", 
    "Bill" \[UndirectedEdge] "Pauline", 
    "Bill" \[UndirectedEdge] "Michael", 
    "Bill" \[UndirectedEdge] "Lee", "Harry" \[UndirectedEdge] "Steve",
     "Steve" \[UndirectedEdge] "Don", 
    "Steve" \[UndirectedEdge] "Bert", 
    "Gery" \[UndirectedEdge] "Steve", 
    "Russ" \[UndirectedEdge] "Steve", 
    "Pam" \[UndirectedEdge] "Brazey", 
    "Brazey" \[UndirectedEdge] "Carol", "Pam" \[UndirectedEdge] "Pat",
     "Brazey" \[UndirectedEdge] "Pat", 
    "Carol" \[UndirectedEdge] "Pat", "Holly" \[UndirectedEdge] "Pat", 
    "Gery" \[UndirectedEdge] "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]
{{"J. Rothberg", "L. Giot", "P. Uetz", "G. Cagney", "T. Mansfield", 
 "R. Judson", "J. Knight", "D. Lockshon", "V. Narayan", 
 "M. Srinivasan", "P. Pochart", "A. Qureshiemili", "Y. Li", 
 "B. Godwin", "D. Conover", "T. Kalbfleisch", "G. Vijayadamodar", 
 "M. Yang", "M. Johnston", "S. Fields"}}

Cliques in directed graphs

The clique finder in IGraph/M ignores edge directions.

g = RandomGraph[{10, 60}, DirectedEdges -> True]

IGMaximalCliques[g]

{{10, 2, 8, 7, 9, 4, 3}, {10, 2, 8, 7, 9, 6}, {2, 9, 5, 7, 1, 4, 
 3}, {2, 9, 5, 7, 1, 6}, {2, 9, 5, 7, 8, 4, 3}, {2, 9, 5, 7, 8, 6}}

To find cliques in directed graphs, convert them to undirected and keep mutual (bidirectional) edges only.

IGMaximalCliques@IGUndirectedGraph[g, "Mutual"]
{{10, 6}, {10, 3, 9}, {2, 1, 9, 4}, {3, 1, 5}, {3, 1, 9}, {5, 7, 
 8}, {6, 7, 8}, {6, 1}, {8, 9}}

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:

Compute a minimum clique cover of a random graph.

g = RandomGraph[{10, 20}]

IGCliqueCover[g]
{{1, 7}, {2, 8}, {3, 5}, {4, 6, 9, 10}}

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, 7}, {2, 8}, {3, 5}, {4, 6, 9, 10}}

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] &, g, 40]

… or to visualize dynamic graph processes such as adding edges to the graph one by one:

g = IGLayoutKamadaKawai@
   Graph[Range[25], {1 \[UndirectedEdge] 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.

Drawing trees

IGLayoutReingoldTilford[] and IGLayoutReingoldTilfordCircular[] are designed for laying out trees. The following options are available:

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:

IGLayoutBipartite[IGBipartiteGameGNP[10, 10, 0.2], 
 VertexLabels -> "Name"]

By default, a partitioning is computed automatically.

g = Graph[{1 \[UndirectedEdge] 2, 3 \[UndirectedEdge] 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 = RandomGraph[BarabasiAlbertGraphDistribution[30, 1]];
layouts = 
  Graph[#[g], PlotLabel -> #, LabelStyle -> 7] & /@ {IGLayoutCircle, 
    IGLayoutDavidsonHarel, IGLayoutDrL, IGLayoutDrL3D, 
    IGLayoutFruchtermanReingold, IGLayoutFruchtermanReingold3D, 
    IGLayoutGEM, IGLayoutGraphOpt, IGLayoutKamadaKawai, 
    IGLayoutKamadaKawai3D, IGLayoutRandom, IGLayoutReingoldTilford, 
    IGLayoutReingoldTilfordCircular, IGLayoutSphere, 
    IGLayoutBipartite, IGLayoutPlanar};
Multicolumn[layouts]

Visualise a polyhedral graph with all layouts.

g = GraphData["DodecahedralGraph"];
layouts = 
  Graph[#[g], PlotLabel -> #, LabelStyle -> 7] & /@ {IGLayoutCircle, 
    IGLayoutDavidsonHarel, IGLayoutDrL, IGLayoutDrL3D, 
    IGLayoutFruchtermanReingold, IGLayoutFruchtermanReingold3D, 
    IGLayoutGEM, IGLayoutGraphOpt, IGLayoutKamadaKawai, 
    IGLayoutKamadaKawai3D, IGLayoutRandom, IGLayoutSphere, 
    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. 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. For a given partitioning, it can be computed using IGModularity. Community detection functions 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", "Michael", "Larry", "Nancy", 
 "Nora", "Julia"}, {"John", "Dorothy", "David", "Arlene", 
 "Rudy"}, {"Carol", "Ben", "Oscar", "Felicia"}}

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

Max[cl["Modularity"]]
0.374089

IGModularity

?IGModularity

IGModularity[graph, communities] is equivalent to GraphAssortativity[graph, communities, "Normalized" -> False].

IGCompareCommunities

?IGCompareCommunities

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:

Special properties returned with the result:

References

IGCommunitiesFluid

?IGCommunitiesFluid

IGCommunitiesFluid[] implements the fluid communities algorithm.

Reference

IGCommunitiesGreedy

?IGCommunitiesGreedy

IGCommunitiesGreedy[] implements greedy optimization of modularity.

Weighted graphs are supported.

Reference

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:

References

IGCommunitiesLabelPropagation

?IGCommunitiesLabelPropagation

Weighted graphs are supported.

References

IGCommunitiesLeadingEigenvector

?IGCommunitiesLeadingEigenvector

Weighted graphs are supported.

Available option values:

References

IGCommunitiesMultilevel

?IGCommunitiesMultilevel

IGCommunitiesMultilevel[] implements the Louvain community detection method.

Weighted graphs are supported.

References

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:

Other available options:

Special properties returned with the result :

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"]
 ]

References

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:

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

Special properties returned with the result:

References

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:

References

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:

These are present for hierarchical clustering methods:

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

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" \[UndirectedEdge] "Myriel", 
"Valjean" \[UndirectedEdge] "Mlle Baptistine", 
"Valjean" \[UndirectedEdge] "Mme. Magloire", 
"Gavroche" \[UndirectedEdge] "Valjean", 
"Gavroche" \[UndirectedEdge] "Javert", 
"Thenardier" \[UndirectedEdge] "Fantine", 
"Bamatabois" \[UndirectedEdge] "Javert", 
"Bossuet" \[UndirectedEdge] "Valjean", 
"Montparnasse" \[UndirectedEdge] "Valjean", 
"Gueulemer" \[UndirectedEdge] "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 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]
{1, 4, 2, 3, 1, 1, 3, 3, 3, 4, 4, 2, 1, 4, 2, 2, 4, 3, 2, 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@
 MapThread[{GeoStyling[{Opacity[0.5], #2}], 
    Polygon[#1]} &, {countries, 
   ColorData[97] /@ IGVertexColoring[graph]}]

\(k\)-colouring

?IGKVertexColoring

?IGKEdgeColoring

These functions find a colouring with \(k\) or fewer colours. They work by transforming the colouring into a satisfiability problem and using SatisfiabilityInstances.

The available option values are:

The default setting for "ForcedColoring" is "MaxDegreeClique".

The Moser spindle is not 3-colourable, so no solution is returned.

moser = GraphData["MoserSpindle"];
IGKVertexColoring[moser, 3]
{}

Find a 4-colouring of the Moser spindle …

IGKVertexColoring[moser, 4]
{{4, 1, 3, 1, 3, 2, 2}}

… and visualize it.

Graph[moser, GraphStyle -> "BasicBlack", VertexSize -> Large] // 
 IGVertexMap[ColorData[112], 
  VertexStyle -> (First@IGKVertexColoring[#, 4] &)]

Find a 4-edge-colouring of the Petersen graph.

PetersenGraph[GraphStyle -> "ThickEdge", EdgeStyle -> Opacity[2/3]] //
  IGEdgeMap[ColorData[112], 
  EdgeStyle -> (First@IGKEdgeColoring[#, 4] &)]

The following examples illustrate the use of the "ForcedColoring" option. The 6th order Mycielski graph has chromatic number 6. A 6-colouring is easily found even with "ForcedColoring" -> None.

g = GraphData[{"Mycielski", 6}];
IGKVertexColoring[g, 6, "ForcedColoring" -> None] // Timing
{0.045122, {{6, 4, 5, 5, 4, 4, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 
  3, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 
  2, 2, 2, 2, 2, 1}}}

However, showing that the graph is not 5-colourable takes considerably longer.

TimeConstrained[IGKVertexColoring[g, 5, "ForcedColoring" -> None], 5]
$Aborted

Forcing colours in the appropriate way reduces the computation time significantly.

IGKVertexColoring[g, 5, 
  "ForcedColoring" -> "MaxDegreeClique"] // Timing
{0.395202, {}}

Minimum colouring

?IGMinimumVertexColoring

?IGMinimumEdgeColoring

IGMinimumVertexColoring and IGMinimumEdgeColoring find minimum colourings of graphs, i.e. they find a colouring with the fewest possible number of colours. The current implementation tries successively larger \(k\)-colourings until it is successful.

IGMinimumVertexColoring and IGMinimumEdgeColoring can use the same "ForcedColoring" option values as IGKVertexColoring and IGKEdgeColoring.

WheelGraph[7, GraphStyle -> "BasicBlack", VertexSize -> Large] // 
 IGVertexMap[ColorData[97], VertexStyle -> IGMinimumVertexColoring]

Find a colouring of a large graph.

IGMinimumVertexColoring@RandomGraph[{100, 400}]
{2, 4, 3, 2, 3, 4, 1, 1, 3, 1, 4, 3, 3, 2, 2, 3, 4, 2, 1, 4, 2, 3, 1, \
2, 2, 1, 3, 3, 3, 4, 2, 3, 3, 3, 1, 1, 4, 2, 4, 2, 4, 1, 3, 1, 3, 3, \
1, 3, 1, 4, 2, 2, 2, 4, 4, 1, 1, 4, 4, 1, 2, 4, 4, 4, 4, 3, 4, 4, 2, \
4, 4, 2, 4, 3, 3, 1, 3, 1, 2, 1, 3, 2, 2, 2, 4, 3, 2, 1, 3, 2, 1, 3, \
2, 1, 2, 3, 3, 3, 1, 4}

Implement a multipartite graph layout: vertex colouring is equivalent to partitioning the vertices of the graph into groups such that all connections run between different groups, and never within the same group. The colours can be thought of as the indices of groups. IGMembershipToPartitions can be used to convert from a group-index (i.e. membership) representation to a partition representation.

multipartiteLayout[g_?GraphQ, separation : _?NumericQ : 1.5, 
   opt : OptionsPattern[]] := 
  Module[{n, partitions, partitionSizes, vertexCoordinates},
   partitions = IGMembershipToPartitions[g]@IGMinimumVertexColoring[g];
   partitionSizes = Length /@ partitions;
   n = Length[partitions];
   vertexCoordinates = 
    With[{hl = N@Sin[Pi/n], 
      ir = separation If[n == 2, 1/2, N@Cos[Pi/n]]},
     Catenate@Table[
       RotationTransform[2 Pi/n k][{#, ir} & /@ 
         Subdivide[-hl, hl, partitionSizes[[k]] - 1]],
       {k, 1, n}
       ]
     ];
   IGReorderVertices[Catenate[partitions], g, 
    VertexCoordinates -> vertexCoordinates, opt]
   ];

Lay out a bipartite graph.

g = IGBipartiteGameGNM[10, 10, 30];
multipartiteLayout[g]

Lay out multipartite graphs.

g = RandomGraph[{40, 40}];
multipartiteLayout[g]

g = RandomGraph[{40, 160}];
multipartiteLayout[g, GraphStyle -> "BasicBlack", 
 EdgeStyle -> Opacity[0.2]]

Compute a minimum colouring of a triangulation. It can be shown, e.g. based on Brooks’s theorem, that any triangulation of a polygon is 3-colourable.

mesh = DelaunayMesh[RandomReal[1, {20, 2}], 
   MeshCellStyle -> {1 -> Black}];
col = IGMinimumVertexColoring@IGMeshCellAdjacencyGraph[mesh, 2];
SetProperty[{mesh, {2, All}}, MeshCellStyle -> ColorData[97] /@ col]

Find a minimum edge colouring of a graph.

Graph[
  GraphData["SixteenCellGraph"],
  GraphStyle -> "ThickEdge", EdgeStyle -> Opacity[2/3]
  ] //
 IGEdgeMap[
  ColorData[104],
  EdgeStyle -> IGMinimumEdgeColoring
  ]

Chromatic number

?IGChromaticNumber

?IGChromaticIndex

The chromatic number of a graph is the smallest number of colours needed to colour its vertices. The chromatic index, or edge chromatic number, is the smallest number of colours needed to colour its edges.

Find the chromatic number and chromatic index of a graph.

g = GraphData["IcosahedralGraph"]

{IGChromaticNumber[g], IGChromaticIndex[g]}
{4, 5}

The implementation of IGChromaticNumber and IGChromaticIndex is effectively the following:

{Max@IGMinimumVertexColoring[g], Max@IGMinimumEdgeColoring[g]}
{4, 5}

Perfect graphs

?IGPerfectQ

IGPerfectQ tests if a graph is perfect. The clique number and the chromatic number is the same for every induced subgraph of a perfect graph.

The current implementation of IGPerfectQ uses the strong perfect graph theorem: it checks that neither the graph nor its complement have a graph hole of odd length.

g = GraphData[{"GeneralizedQuadrangle", {2, 1}}]

IGPerfectQ[g]
True

The clique number and the chromatic number is the same for every induced subgraph.

AllTrue[
 Subgraph[g, #] & /@ Subsets@VertexList[g],
 IGCliqueNumber[#] == IGChromaticNumber[#] &
 ]
True

Utility functions

?IGVertexColoringQ

IGVertexColoringQ checks whether neighbouring vertices of a graph are assigned different colours.

The colours may be given as a list, with the same ordering as VertexList[graph].

True

False

The colours may also be given as an association from vertices to colours.

True

Any expression may be used for the colours, not only integers.

True

Processes on graphs

Random walks

IGRandomWalk

?IGRandomWalk

IGRandomWalk[] takes a random walk over a directed or undirected graph. If the graph is weighted, the next edge to traverse is selected with probability proportional to its weight.

The available options are:

Traversing self-loops in different directions is considered as distinct probabilities in an undirected graph. Thus vertices 1 and 3 are visited more often in the below graphs than vertex 2:

g = Graph[{1 \[UndirectedEdge] 1, 1 \[UndirectedEdge] 2, 
   2 \[UndirectedEdge] 3, 3 \[UndirectedEdge] 3}, 
  VertexLabels -> "Name"]

IGRandomWalk[g, 1, 100000] // Counts // KeySort
<|1 -> 37776, 2 -> 25095, 3 -> 37129|>

This is consistent with their degrees:

VertexDegree[g]
{3, 2, 3}

Convert the graph to a directed version to traverse self-loops only in one direction.

dg = DirectedGraph[g]

{VertexOutDegree[dg], VertexInDegree[dg]}
{{2, 2, 2}, {2, 2, 2}}
IGRandomWalk[dg, 1, 100000] // Counts // KeySort
<|1 -> 33205, 2 -> 33218, 3 -> 33577|>

If the walker gets stuck, a list shorter than steps will be returned. This may happen in a non-connected directed graph, or in a single-vertex graph component.

IGRandomWalk[IGEmptyGraph[1], 1, 10]
{1}
IGRandomWalk[Graph[{1 -> 2}], 1, 10]
{1, 2}

How much time does a random walker spend on each node of a network?

g = IGBarabasiAlbertGame[50, 2, DirectedEdges -> False]

counts = Counts@IGRandomWalk[g, First@VertexList[g], 10000] /@ 
  VertexList[g]
{204, 379, 839, 271, 313, 641, 197, 304, 682, 198, 319, 265, 86, 145, \
152, 158, 217, 84, 325, 105, 165, 112, 100, 288, 264, 118, 105, 83, \
200, 98, 176, 168, 104, 174, 94, 181, 90, 100, 87, 160, 137, 99, 116, \
169, 163, 109, 87, 117, 137, 115}

The exact answer can be computed as the leading eigenvector of the process’s stochastic matrix:

Compare the exact answer with a finite sample:

Random walk on a square grid.

grid = IGSquareLattice[{50, 50}];
counts = Counts@IGRandomWalk[grid, 1, 5000];
Graph[grid,
 VertexStyle ->
  Prepend[
   Normal[ColorData["SolarColors"] /@ Normalize[counts, Max]],
   Black (* colour of unvisited nodes, i.e. default colour *)
   ],
 EdgeShapeFunction -> None,
 Background -> Black
 ]

The fraction of nodes reached after \(n\) steps on a grid and a comparable random regular graph.

nodesReached[graph_] := 
 Length@Union@IGRandomWalk[graph, 1, VertexCount[graph]]/
  VertexCount[graph]
grid = IGSquareLattice[{50, 50}, "Periodic" -> True];
regular = IGKRegularGame[50^2, 4];
Table[
   {nodesReached[grid], nodesReached[regular]},
   {5000}
   ] // Transpose // Histogram

Generate random spanning trees using loop erased random walks.

randomSpanningTree[graph_?GraphQ] :=
 
 Module[{visited = <||>, i = 2, k = 1, 
   batchSize = 2 VertexCount[graph], walk},
  walk = IGRandomWalk[graph, RandomChoice@VertexList[graph], 
    batchSize];
  visited[walk[[1]]] = True;
  While[k < VertexCount[graph],
      (* register a traversed edge only when it leads to a yet \
unvisited vertex *)
      If[! TrueQ[visited[walk[[i]]]],
       Sow[walk[[i - 1]] \[UndirectedEdge] walk[[i]]];
       visited[walk[[i]]] = True;
       k++
       ];
      i++;
      (* if the walk has not yet visited all vertices, keep walking *)

            If[i > Length[walk],
       walk = 
        Join[walk, Rest@IGRandomWalk[graph, Last[walk], batchSize]]
       ];
      ] // Reap // Last // First
  ]

By taking random spanning trees of spatially embedded planar graphs, we can generate mazes.

Take a sample of a large graph using a random walk. The following graph is too large to easily visualize, but visualizing a random-walk-based sample immediately shows signs of a community structure.

g = ExampleData[{"NetworkGraph", "AstrophysicsCollaborations"}];
{VertexCount[g], VertexCount@IGGiantComponent[g]}
{16706, 14845}
Subgraph[g, 
 IGRandomWalk[g, RandomChoice@VertexList@IGGiantComponent[g], 200]]

CommunityGraphPlot[%]

IGRandomEdgeWalk and IGRandomEdgeIndexWalk

?IGRandomEdgeWalk

?IGRandomEdgeIndexWalk

IGRandomEdgeWalk takes a random walk on a graph and returns the list of traversed edges. If the graph is weighted, the next edge to traverse is selected with probability proportional to its weight.

The available options are:

Take a random walk on a De Bruijn graph, and retrieve the traversed edges.

g = IGDeBruijnGraph[3, 3];
IGRandomEdgeWalk[g, RandomChoice@VertexList[g], 20]
{21 \[DirectedEdge] 9, 9 \[DirectedEdge] 25, 25 \[DirectedEdge] 20, 
20 \[DirectedEdge] 4, 4 \[DirectedEdge] 11, 11 \[DirectedEdge] 4, 
4 \[DirectedEdge] 12, 12 \[DirectedEdge] 9, 9 \[DirectedEdge] 27, 
27 \[DirectedEdge] 26, 26 \[DirectedEdge] 23, 23 \[DirectedEdge] 14, 
14 \[DirectedEdge] 15, 15 \[DirectedEdge] 16, 16 \[DirectedEdge] 19, 
19 \[DirectedEdge] 2, 2 \[DirectedEdge] 5, 5 \[DirectedEdge] 13, 
13 \[DirectedEdge] 11, 11 \[DirectedEdge] 4}

IGRandomEdgeIndexWalk returns the list of indices of the traversed edges instead. This makes it useful for working with multigraphs, as it allows distinguishing between parallel edges.

As an example application, let us consider the following set of affine transformations:

trafos = {a11, a21, b21, a12, a22};

Let us visualize them by showing an initial (black) triangle and its (red) transformation.

These transformations describe the mutual self-similarity structure of two fractal curves, according to the following directed graph. Each edge of the graph corresponds to a transformation.

Let us compute a random walk on this graph, and iteratively apply transformations to the point {0, 0} according to the traversed edges.

walk = IGRandomEdgeIndexWalk[graph, 1, 20000];
pts = Rest@FoldList[#2[#1] &, {0., 0.}, trafos[[walk]]];

The resulting list of points will approximate the union of the two fractal curves.

Image@Graphics[{AbsolutePointSize[1], Point[pts]}]

The two curves can be separated by filtering points according to which graph vertex the corresponding directed edge targets. For example, if the point was generated by a transformation corresponding to 1 \[DirectedEdge] 2, it will belong to curve 2.

targets = Last /@ EdgeList[graph]
{1, 1, 1, 2, 2}
Image@Graphics[{AbsolutePointSize[1], 
   Point@Pick[pts, targets[[walk]], 1]}]

Image@Graphics[{AbsolutePointSize[1], 
   Point@Pick[pts, targets[[walk]], 2]}]

The technique described here is taken from “Generating self-affine tiles and their boundaries” by Mark McClure.

Epidemic models

IGSIRProcess

?IGSIRProcess

IGSIRProcess simulates a stochastic version of the well known SIR model of disease spreading. In this model, each node of the network may be in one of three states: susceptible, infected or recovered, denoted by \(S\), \(I\) and \(R\), respectively. A susceptible node with \(k\) infected neighbours becomes infected with rate \(k \beta\), while an infected node recovers with rate \(\gamma\). At the start of the simulation, a random node is chosen to be infected. The simulation runs until no more infected nodes are left.

When performing a single simulation, IGSIRProcess returns a TimeSeries expression of {s, i, r} values. When multiple runs are requested, the resulting time series are combined into a TemporalData expression.

g = IGWattsStrogatzGame[100, 0.05];

Perform a single SIR simulation:

ts = IGSIRProcess[g, {5, 1}]

Plot the results with a legend:

ListLinePlot[ts, PlotLegends -> ts["ComponentNames"]]

Plot only the number of infected nodes:

(* In Mathematica 12.0 and later,
   ts["PathComponent", "I"] can also be used. *)
ListLinePlot[
 ts["PathComponent", 2],
 AxesLabel -> {"time", "infected"}, PlotStyle -> ColorData[97][2]]

Find the number of susceptible, infected and recovered nodes at a specific time point:

ts[1.0]
{17., 53., 30.}

The ResamplingMethod of the TimeSeries object is set to 0th order interpolation, therefore the last value is used beyond the last available time point.

ts[10]

{0., 0., 100.}

Perform 100 simulations simultaneously:

td = IGSIRProcess[g, {5, 1}, 100]

Plot the median number of susceptible, infected and recovered nodes:

Show[
    ListLinePlot[#, PlotStyle -> GrayLevel[0, 0.1], 
     PlotRange -> {0, VertexCount[g]}],
    Quiet@Plot[Median[#[t]], {t, 0, 4}, PlotStyle -> Red]
    ] & /@ td["PathComponents"] // GraphicsColumn

The sum of the three components, \(S+I+R\), always equals the total number of graph nodes.

First@Normal@Total[td["PathComponents"]] // Short

In the next example, we compare epidemic spreading on a periodic grid, i.e. a network that only has spatially local connections, with a rewired version of the same network which also includes long range links. We rewire 5% of links while ensuring that the graph stays connected.

g1 = IGSquareLattice[{30, 30}, "Periodic" -> True];
g2 = IGTryUntil[IGConnectedQ][IGRewireEdges[g1, 0.05]];

Generate 1000 simulations for each network.

r1 = IGSIRProcess[g1, {1, 1}, 1000];
r2 = IGSIRProcess[g2, {1, 1}, 1000];

Plot the histogram of the total duration of the epidemic.

Histogram[{r1["LastTimes"], r2["LastTimes"]}, 
 ChartLegends -> {"grid", "rewired"}]

Plot the fraction of recovered nodes at the end of the epidemic.

tmax = Max[r1["MaximumTime"], r2["MaximumTime"]];
Histogram[
 {r1["PathComponent", 3]["SliceData", tmax]/VertexCount[g1],
   r2["PathComponent", 3]["SliceData", tmax]/VertexCount[g2]} // Quiet,
 {0, 1, 0.02},
 ChartLegends -> {"grid", "rewired"}
 ]

Planar graphs

A graph is said to be planar if it can be drawn in the plane without edge crossings.

A useful concept when working with planar graphs is their combinatorial embedding. A combinatorial embedding of a graph is a counter-clockwise ordering of the incident edges around each vertex. IGraph/M represents combinatorial embeddings as associations from vertices to an ordering of their neighbours. Currently, only embeddings of simple graphs are supported.

Some of the planar graph functionality makes use of the LEMON Graph Library.

IGPlanarQ

?IGPlanarQ

IGPlanarQ[graph] checks if a graph is planar using the Boyer–Myrvold algorithm.

IGPlanarQ@GraphData[{"Apollonian", 6}]
True
IGPlanarQ@CompleteGraph[5]
False

IGPlanarQ[embedding] checks if a combinatorial embedding is planar. The following are both embeddings of the \(K_4\) complete graph. However, only the first one is planar.

emb1 = <|1 -> {2, 3, 4}, 2 -> {1, 4, 3}, 3 -> {2, 4, 1}, 
   4 -> {3, 2, 1}|>;
emb2 = <|1 -> {2, 4, 3}, 2 -> {4, 3, 1}, 3 -> {1, 2, 4}, 
   4 -> {3, 1, 2}|>;
IGPlanarQ /@ {emb1, emb2}
{True, False}

The second embedding generates only 2 faces instead of 4, which can be embedded on a torus, but not in the plane (or on a sphere).

Length /@ IGFaces /@ {emb1, emb2}
{4, 2}

Unlike the built-in PlanarGraphQ, IGPlanarQ considers the null graph to be planar.

{IGPlanarQ@IGEmptyGraph[], PlanarGraphQ@IGEmptyGraph[]}
{True, True}

IGMaximalPlanarQ

?IGMaximalPlanarQ

A simple graph is maximal planar if no new edges can be added to it without breaking planarity. Maximal planar graphs are sometimes called triangulated graphs or triangulations.

The 3-cycle is maximal planar.

IGMaximalPlanarQ[CycleGraph[3]]
True

The 4-cycle is not because a chord can be added to it without breaking planarity.

IGMaximalPlanarQ[CycleGraph[4]]
False
IGPlanarQ[EdgeAdd[CycleGraph[4], 1 \[UndirectedEdge] 3]]
True

Apollonian graphs are maximal planar.

g = GraphData[{"Apollonian", 2}]

IGMaximalPlanarQ[g]
True

All faces of a maximal planar graph are triangles.

Length /@ IGFaces[g]
{3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

Therefore the edge count \(E\) and the face count \(F\) of a maximal planar graph on more than 2 vertices satisfy \(2E=3F\). Each edge is incident to two faces and each face is incident to three edges.

{2 EdgeCount[g], 3 Length@IGFaces[g]}
{30, 30}

IGOuterplanarQ

?IGOuterplanarQ

IGOuterplanarQ[graph] checks if a graph is outerplanar, i.e. if it can be drawn in the plane without edge crossings and with all vertices being on the outer face.

Outerplanar graphs are also called circular planar. They can be drawn without edge crossings and all vertices on a circle. See the documentation of IGOuterplanarEmbedding for an example.

False

True

IGOuterplanarQ[embedding] checks if a combinatorial embedding is outerplanar. Not all planar embeddings of an outerplanar graph are also outerplanar embeddings.

Consider the following outerplanar graph …

g = IGShorthand["0-1-2-3-4-2,1-4"]

IGOuterplanarQ[g]
True

… and two of its embeddings:

emb1 = <|0 -> {1}, 1 -> {2, 0, 4}, 2 -> {1, 3, 4}, 3 -> {2, 4}, 
   4 -> {3, 1, 2}|>;
emb2 = <|0 -> {1}, 1 -> {0, 2, 4}, 2 -> {1, 3, 4}, 3 -> {2, 4}, 
   4 -> {3, 1, 2}|>;

They are both planar, but only the second one is outerplanar.

IGPlanarQ /@ {emb1, emb2}
{True, True}
IGOuterplanarQ /@ {emb1, emb2}
{False, True}
Graph[g, VertexCoordinates -> IGEmbeddingToCoordinates[#]] & /@ {emb1,
   emb2}

IGKuratowskiEdges

?IGKuratowskiEdges

IGKuratowskiEdges finds a Kuratowski subgraph of a non-planar graph. The subgraph is returned as a set of edges. If the graph is planar, {} is returned.

According to Kuratowski’s theorem, any non-planar graph contains a subgraph homeomorphic to the \(K_5\) complete graph or the \(K_{3,3}\) complete bipartite graph. This is called a Kuratowski subgraph.

Generate a random graph, which is non-planar with high probability.

g = RandomGraph[{20, 40}]

IGPlanarQ[g]
False

Compute a set of edges belonging to a Kuratowski subgraph.

kur = IGKuratowskiEdges[g]
{19 \[UndirectedEdge] 20, 15 \[UndirectedEdge] 18, 
14 \[UndirectedEdge] 15, 13 \[UndirectedEdge] 19, 
13 \[UndirectedEdge] 15, 10 \[UndirectedEdge] 18, 
10 \[UndirectedEdge] 16, 9 \[UndirectedEdge] 20, 
9 \[UndirectedEdge] 18, 5 \[UndirectedEdge] 14, 
5 \[UndirectedEdge] 9, 3 \[UndirectedEdge] 19, 
3 \[UndirectedEdge] 16, 3 \[UndirectedEdge] 5}

Highlight the Kuratowski subgraph.

HighlightGraph[g, Graph[kur]]

Display the Kuratowski subgraph on its own.

Graph[kur]

By smoothening the Kuratowski subgraph, we obtain either \(K_5\) or \(K_{3,3}\).

IGSmoothen@Graph[kur]

IGHomeomorphicQ[Graph[kur], #] & /@ {CompleteGraph[5], 
  CompleteGraph[{3, 3}]}
{False, True}

For planar graphs, {} is returned.

IGKuratowskiEdges@CycleGraph[5]
{}

IGFaces

?IGFaces

IGFaces returns the faces of a planar graph, or the faces corresponding to a specific (not necessarily planar) embedding. The faces are represented by a counter-clockwise ordering of vertices. The current implementation ignores self-loops and multi-edges.

The faces of a planar graph are unique if the graph is 3-vertex-connected. This can be checked using KVertexConnectedGraphQ.

g = GraphData["DodecahedralGraph"]

IGFaces[g]
{{1, 14, 9, 10, 15}, {1, 15, 4, 8, 16}, {1, 16, 7, 3, 14}, {2, 5, 11, 
 12, 6}, {2, 6, 20, 18, 13}, {2, 13, 17, 19, 5}, {3, 7, 11, 5, 
 19}, {3, 19, 17, 9, 14}, {4, 15, 10, 18, 20}, {4, 20, 6, 12, 8}, {7,
  16, 8, 12, 11}, {9, 17, 13, 18, 10}}
KVertexConnectedGraphQ[g, 3]
True

If the graph is not connected and has \(C\) connected components, then \(C-1\) faces will be redundant.

g = IGDisjointUnion[{CycleGraph[3], CycleGraph[3]}, 
  VertexLabels -> Automatic]

IGFaces[g]
{{{1, 1}, {1, 2}, {1, 3}}, {{1, 1}, {1, 3}, {1, 2}}, {{2, 1}, {2, 
  2}, {2, 3}}, {{2, 1}, {2, 3}, {2, 2}}}

In the above-drawn arrangement, the outer faces of the two triangles are the same face. However, one triangle could have been drawn inside of the other. Then the inner face of one would be the same as the outer face of the other. Thus the choice of faces to be eliminated as redundant is arbitrary, and is left up to the user.

IGFaces can also be used with a non-planar combinatorial embedding. The below embeddings both belong to the 4-vertex complete graph, however, only the first is planar.

emb1 = <|1 -> {2, 3, 4}, 2 -> {1, 4, 3}, 3 -> {2, 4, 1}, 
   4 -> {3, 2, 1}|>;
emb2 = <|1 -> {2, 4, 3}, 2 -> {4, 3, 1}, 3 -> {1, 2, 4}, 
   4 -> {3, 1, 2}|>;
IGFaces[emb1]
{{1, 2, 3}, {1, 3, 4}, {1, 4, 2}, {2, 4, 3}}
IGFaces[emb2]
{{1, 2, 3, 1, 4, 3, 2, 4}, {1, 3, 4, 2}}

Determine the genus \(g\) of an embedding belonging to a connected graph based on its face count \(F\), vertex count \(V\), and edge count \(E\), using the formula for the Euler characteristic \(2g-2=\chi =V-E+F\).

genus[emb_?
   IGEmbeddingQ] := (2 + Total[Length /@ emb]/2 - Length[emb] - 
    Length@IGFaces[emb])/2
genus /@ {emb1, emb2}
{0, 1}

IGDualGraph

?IGDualGraph

IGDualGraph returns a dual graph of a planar graph, or the dual corresponding to a specific embedding. The ordering of the dual graph’s vertices is consistent with the result of IGFaces.

Limitations:

The dual of a simple 3-vertex-connected graph is simple and unique, thus such graphs are not affected by the above limitations.

TableForm[
 Table[{CompleteGraph[k], IGDualGraph@CompleteGraph[k]}, {k, 1, 4}],
 TableHeadings -> {None, {"graph", "dual"}}
 ]

Currently, if the input is a graph, it must be planar.

IGDualGraph[CompleteGraph[5]]

$Failed

If the input is a combinatorial embedding, it does not need to be planar.

emb = <|1 -> {2, 4, 3}, 2 -> {4, 3, 1}, 3 -> {1, 2, 4}, 
   4 -> {3, 1, 2}|>;
IGPlanarQ[emb]
False
IGDualGraph[emb]

Find the dual of a square lattice graph. The dual graph also includes the outer face as a vertex.

IGSquareLattice[{5, 5}]

IGDualGraph[%]

The dual is unique if the graph is 3-vertex-connected. This can be verified using KVertexConnectedGraphQ. In this case, IGDualGraph@IGDualGraph[g] is isomorphic to g.

g = GraphData["IcosahedralGraph"]

dg = IGDualGraph[g]

IGIsomorphicQ[dg, GraphData["DodecahedralGraph"]]
True
IGIsomorphicQ[IGDualGraph[dg], g]
True

If the graph is not connected, the dual of each component is effectively computed separately.

IGDualGraph@IGDisjointUnion[{CycleGraph[3], CycleGraph[3]}]

IGEmbeddingQ

?IGEmbeddingQ

IGEmbeddingQ checks if an embedding is valid, and whether it belongs to a graph without self-loops and multi-edges.

This is a valid combinatorial embedding of the graph 1 \[UndirectedEdge] 3 \[UndirectedEdge] 2.

IGEmbeddingQ[<|1 -> {3}, 2 -> {3}, 3 -> {1, 2}|>]
True

The following embeddings do not belong to simple (i.e. loop free and multi-edge free) graphs:

IGEmbeddingQ[<|1 -> {3}, 2 -> {3, 3}, 3 -> {1, 2, 2}|>]
False
IGEmbeddingQ[<|1 -> {1, 2}, 2 -> {1}|>]
False

The following embedding is not valid because it does not contain the arc 2 \[DirectedEdge] 1 but it does contain 1 \[DirectedEdge] 2.