directed_laplacian_matrix(G, nodelist=None, weight='weight', walk_type=None, alpha=0.95)
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 :None:None:`I` is the identity matrix, :None:None:`P` is the transition matrix of the graph, and :None:None:`\Phi` a matrix with the Perron vector of :None:None:`P` in the diagonal and zeros elsewhere .
Depending on the value of walk_type, :None:None:`P` can be the transition matrix induced by a random walk, a lazy random walk, or a random walk with teleportation (PageRank).
Only implemented for DiGraphs
A NetworkX graph
The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes().
The edge data key used to compute each value in the matrix. If None, then each edge has weight 1.
If None, :None:None:`P` is selected depending on the properties of the graph. Otherwise is one of 'random', 'lazy', or 'pagerank'
(1 - alpha) is the teleportation probability used with pagerank
Normalized Laplacian of G.
Returns the directed Laplacian matrix of G.
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