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:
- GDiGraph
A NetworkX DiGraph
- 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 matrix
The modularity matrix of G.
See also
to_numpy_array
modularity_spectrum
adjacency_matrix
modularity_matrix
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.
References
[1]E. A. Leicht, M. E. J. Newman, “Community structure in directed networks”, Phys. Rev Lett., vol. 100, no. 11, p. 118703, 2008.
Examples
>>> 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)