modularity: Modularity of a community structure of a graph

Description

This function calculates how modular is a given division of a graph into subgraphs.

Usage

modularity(graph, membership)

Arguments

graph

The input graph.

membership

Numeric vector, for each vertex it gives its community. The communities are numbered from zero.

Value

A numeric scalar, the modularity score of the given configuration.

Details

The modularity of a graph with respect to some division (or vertex types) measures how good the division is, or how separated are the different vertex types from each other. It defined as

Q = \frac{1}{2 m} \sum_{i, j} A_{i j} - \frac{k_{i} k_{j}}{2 m} δ (c_{i}, c_{j}),

here $m$ is the number of edges, $A_{ij}$ is the element of the $A$ adjacency matrix in row $i$ and column $j$, $k_i$ is the degree of $i$, $k_j$ is the degree of $j$, $c_i$ is the type (or component) of $i$, $c_j$ that of $j$, the sum goes over all $i$ and $j$ pairs of vertices, and $\delta(x,y)$ is 1 if $x=y$ and 0 otherwise.

References

MEJ Newman and M Girvan: Finding and evaluating community structure in networks. Physical Review E 69 026113, 2004.

Examples

Run this code

g <- graph.full(5) %du% graph.full(5) %du% graph.full(5)
g <- add.edges(g, c(0,5, 0,10, 5, 10))
wtc <- walktrap.community(g)
memb <- community.to.membership(g, wtc$merges, steps=12)
modularity(g, memb$membership)

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