Modularity of a community structure of a graph

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

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

modularity_matrix(graph, membership, weights = NULL)

The input graph.
Numeric vector, for each vertex it gives its community. The communities are numbered from one.
If not NULL then a numeric vector giving edge weights.
Additional arguments, none currently.

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).


  • 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.


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.

  • mod.matrix
  • modularity
  • modularity.igraph
  • modularity_matrix
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(g, membership(wtc))
Documentation reproduced from package igraph, version 1.0.0, License: GPL (>= 2)

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