networkx.linalg.modularitymatrix.directed_modularity_matrix

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

Returns the directed modularity matrix of G.

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

More specifically, the element B_ij of B is defined as

\[B_{ij} = A_{ij} - k_i^{out} k_j^{in} / m\]

where \(k_i^{in}\) is the in degree of node i, and \(k_j^{out}\) is the out degree of node j, with m 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
  • G (DiGraph) – A NetworkX DiGraph

  • 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.

Returns

B – The modularity matrix of G.

Return type

Numpy matrix

Examples

>>> import networkx as nx
>>> G = nx.DiGraph()
>>> G.add_edges_from(((1,2), (1,3), (3,1), (3,2), (3,5), (4,5), (4,6),
...                   (5,4), (5,6), (6,4)))
>>> B = nx.directed_modularity_matrix(G)

Notes

NetworkX defines the element A_ij of the adjacency matrix as 1 if there is a link going from node i to node j. Leicht and Newman use the opposite definition. This explains the different expression for B_ij.

See also

to_numpy_matrix(), modularity_spectrum(), adjacency_matrix(), modularity_matrix()

References

1

E. A. Leicht, M. E. J. Newman, “Community structure in directed networks”, Phys. Rev Lett., vol. 100, no. 11, p. 118703, 2008.