networkx.linalg.laplacianmatrix.directed_laplacian_matrix¶

directed_laplacian_matrix
(G, nodelist=None, weight='weight', walk_type=None, alpha=0.95)[source]¶ Returns the directed Laplacian matrix of G.
The graph directed Laplacian is the matrix
\[L = I  (\Phi^{1/2} P \Phi^{1/2} + \Phi^{1/2} P^T \Phi^{1/2} ) / 2\]where
I
is the identity matrix,P
is the transition matrix of the graph, andPhi
a matrix with the Perron vector ofP
in the diagonal and zeros elsewhere.Depending on the value of walk_type,
P
can be the transition matrix induced by a random walk, a lazy random walk, or a random walk with teleportation (PageRank). Parameters
G (DiGraph) – 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=’weight’)) – The edge data key used to compute each value in the matrix. If None, then each edge has weight 1.
walk_type (string or None, optional (default=None)) – If None,
P
is selected depending on the properties of the graph. Otherwise is one of ‘random’, ‘lazy’, or ‘pagerank’alpha (real) – (1  alpha) is the teleportation probability used with pagerank
 Returns
L – Normalized Laplacian of G.
 Return type
NumPy matrix
Notes
Only implemented for DiGraphs
See also
References
 1
Fan Chung (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics, 9(1), 2005