Modularity of a community structure of a graph

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, resolution = 1, directed = TRUE, ...)

modularity_matrix(
  graph,
  membership = lifecycle::deprecated(),
  weights = NULL,
  resolution = 1,
  directed = TRUE
)

Arguments

x, graph: The input graph.
membership: Numeric vector, one value for each vertex, the membership vector of the community structure.
weights: If not NULL then a numeric vector giving edge weights.
resolution: The resolution parameter. Must be greater than or equal to 0. Set it to 1 to use the classical definition of modularity.
directed: Whether to use the directed or undirected version of modularity. Ignored for undirected graphs.
...: Additional arguments, none currently.

Value

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.

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 $$Q=\frac{1}{2m} \sum_{i,j} (A_{ij}-\gamma\frac{k_i k_j}{2m})\delta(c_i,c_j),$$ 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. For directed graphs, it is defined as $$Q = \frac{1}{m} \sum_{i,j} (A_{ij}-\gamma \frac{k_i^{out} k_j^{in}}{m})\delta(c_i,c_j).$$

The resolution parameter $\gamma$ allows weighting the random null model, which might be useful when finding partitions with a high modularity. Maximizing modularity with higher values of the resolution parameter typically results in more, smaller clusters when finding partitions with a high modularity. Lower values typically results in fewer, larger clusters. The original definition of modularity is retrieved when setting $\gamma$ to 1.

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, Physical Review E 2004, 70, 066111

Author

Gabor Csardi csardi.gabor@gmail.com

Examples


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)
#> [1] 0.5757576
modularity(g, membership(wtc))
#> [1] 0.5757576