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)
x, graph |
The input graph. |
membership |
Numeric vector, one value for each vertex, the membership vector of the community structure. |
weights |
If not |
... |
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=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.
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[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).
For modularity a numeric scalar, the modularity score of the
given configuration.
For modularity_matrix a numeric square matrix, its order is the number of
vertices in the graph.
Gabor Csardi csardi.gabor@gmail.com
Clauset, A.; Newman, M. E. J. & Moore, C. Finding community structure in very large networks, Physical Review E 2004, 70, 066111
cluster_walktrap,
cluster_edge_betweenness,
cluster_fast_greedy, cluster_spinglass,
cluster_louvain and cluster_leiden for
various community detection methods.
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))
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