modularity_matrix

modularity_matrix(G, nodelist=None, weight=None)

Returns the modularity matrix of G.

The modularity matrix is the matrix B = A - <A>, where A is the adjacency matrix and <A> is the average adjacency matrix, assuming that the graph is described by the configuration model.

More specifically, the element B_ij of B is defined as

\[A_{ij} - {k_i k_j \over 2 m}\]

where k_i is the degree of node i, and where m is the number of edges in the graph. When weight is set to a name of an attribute edge, Aij, k_i, k_j and m are computed using its value.

Parameters:

GGraph

A NetworkX graph

nodelistlist, optional

The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes().

weightstring or None, optional (default=None)

The edge attribute that holds the numerical value used for the edge weight. If None then all edge weights are 1.

Returns:

BNumpy array

The modularity matrix of G.

See also

to_numpy_array

modularity_spectrum

adjacency_matrix

directed_modularity_matrix

References

[1]

M. E. J. Newman, “Modularity and community structure in networks”, Proc. Natl. Acad. Sci. USA, vol. 103, pp. 8577-8582, 2006.

Examples

>>> k = [3, 2, 2, 1, 0]
>>> G = nx.havel_hakimi_graph(k)
>>> B = nx.modularity_matrix(G)
----

Additional backends implement this function

graphblas : OpenMP-enabled sparse linear algebra backend.