Modularity of a community structure of a graph

Description

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

Usage

## S3 method for class 'igraph'
modularity(x, membership, weights = NULL, ...)
mod.matrix (graph, membership, weights = NULL) 

Arguments

Details

modularity calculates the modularity of a graph with respect to the given membership vector.

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=1/(2m) * sum(Aij-ki*kj/(2m)delta(ci,cj),i,j),

here m is the number of edges, Aij is the element of the A adjacency matrix in row i and column j, ki is the degree of i, kj is the degree of j, ci is the type (or component) of i, cj 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.

If edge weights are given, then these are considered as the element of the A adjacency matrix, and ki is the sum of weights of adjacent edges for vertex i.

mod.matrix calculates the modularity matrix. This is a dense matrix, and it is defined as the difference of the adjacency matrix and the configuration model null model matrix. In other words element M[i,j] is given as A[i,j]-d[i]d[j]/(2m), where A[i,j] is the (possibly weighted) adjacency matrix, d[i] is the degree of vertex i, and m is the number of edges (or the total weights in the graph, if it is weighed).

Value

For modularity a numeric scalar, the modularity score of the given configuration.

For mod.matrix a numeic square matrix, its order is the number of vertices in the graph.

Author(s)

Gabor Csardi csardi.gabor@gmail.com

References

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

See Also

walktrap.community, edge.betweenness.community, fastgreedy.community, spinglass.community for various community detection methods.

Examples

g <- graph.full(5) %du% graph.full(5) %du% graph.full(5)
g <- add.edges(g, c(1,6, 1,11, 6, 11))
wtc <- walktrap.community(g)
modularity(wtc)
modularity(g, membership(wtc))