modularity_matrix(G, nodelist=None, weight=None)[source]

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.

  • G (Graph) – A NetworkX graph

  • nodelist (list, 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().

  • weight (string 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.


B – The modularity matrix of G.

Return type

Numpy matrix


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

See also

to_numpy_array(), modularity_spectrum(), adjacency_matrix(), directed_modularity_matrix()



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