GraphUtilities`

GraphUtilities`

CommunityModularity

As of Version 10, all the functionality of the GraphUtilities package is built into the Wolfram System. »

CommunityModularity[g,partition]

gives the community modularity of a partition.

CommunityModularity[g,assignment]

gives the community modularity of an assignment.

Details and Options

CommunityModularity functionality is now available in the built-in Wolfram Language function GraphAssortativity.
To use CommunityModularity, you first need to load the Graph Utilities Package using Needs["GraphUtilities`"].
A community in a network is a group of vertices such that there is a higher density of edges within the group than between them.
Given a graph , let the vertex set be partitioned into subsets such that each subset belongs to one community. The community modularity of this partition is defined as , where is the percentage of edges that have both ends in community , and is the percentage of edges that start from community . In other words, and .
The community modularity is a number less than or equal to 1. A large positive value indicates that the vertex partition gives significant community structure.
The following option can be given:
Weighted False whether edges with higher weights are preferred during matching

Examples

open allclose all

Basic Examples (2)Summary of the most common use cases

This defines a small graph:

In[1]:=1

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https://wolfram.com/xid/0v07q4w9a6tk87he-jcfcec

In[2]:=2

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https://wolfram.com/xid/0v07q4w9a6tk87he-6gkdid

In[3]:=3

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https://wolfram.com/xid/0v07q4w9a6tk87he-bjd3n5

Out[3]=3

This gives the community modularity, assuming a partition {{1,2,3},{4,5,6}}:

In[4]:=4

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https://wolfram.com/xid/0v07q4w9a6tk87he-k8rzul

Out[4]=4

CommunityModularity has been superseded by GraphAssortativity:

In[1]:=1

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https://wolfram.com/xid/0v07q4w9a6tk87he-qs9cop

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0v07q4w9a6tk87he-zzf40

Out[2]=2

Options (1)Common values & functionality for each option

Weighted (1)

This defines a graph with edge weights:

In[1]:=1

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https://wolfram.com/xid/0v07q4w9a6tk87he-8vmc22

In[2]:=2

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https://wolfram.com/xid/0v07q4w9a6tk87he-7mkkh0

In[3]:=3

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https://wolfram.com/xid/0v07q4w9a6tk87he-nskqab

Out[3]=3

The community modularity of partition {{1,2},{3,4}}, ignoring edge weights:

In[4]:=4

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https://wolfram.com/xid/0v07q4w9a6tk87he-r2ckeb

Out[4]=4

The community modularity of partition {{1,2},{3,4}}, taking into account edge weights:

In[5]:=5

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https://wolfram.com/xid/0v07q4w9a6tk87he-vtk5tg

Out[5]=5

When taking into account edge weights, partition {{1,4},{2,3}} has higher community modularity:

In[6]:=6

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https://wolfram.com/xid/0v07q4w9a6tk87he-8szl9n

Out[6]=6

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