igraph (version 1.0.1)

modularity.igraph: 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 igraph
modularity(x, membership, weights = NULL, ...)
modularity_matrix(graph, membership, weights = NULL)

Arguments

x,graph

The input graph.

membership

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

weights

If not NULL then a numeric vector giving edge weights.

…

Additional arguments, none currently.

Value

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

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

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 $$Q=\frac{1}{2m} \sum_{i,j} (A_{ij}-\frac{k_ik_j}{2m})\delta(c_i,c_j),$$ 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.

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

modularity_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 \(M_{ij}\) is given as \(A_{ij}-d_i d_j/(2m)\), where \(A_{ij}\) 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).

References

Clauset, A.; Newman, M. E. J. & Moore, C. Finding community structure in very large networks, Phyisical Review E 2004, 70, 066111

See Also

cluster_walktrap, cluster_edge_betweenness, cluster_fast_greedy, cluster_spinglass for various community detection methods.

Examples

# NOT RUN {
g <- make_full_graph(5) %du% make_full_graph(5) %du% make_full_graph(5)
g <- add_edges(g, c(1,6, 1,11, 6, 11))
wtc <- cluster_walktrap(g)
modularity(wtc)
modularity(g, membership(wtc))
# }