Return 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, and \Phi a matrix with the Perron vector of P 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
nodelist : list, optional
weight : string or None, optional (default=’weight’)
walk_type : string or None, optional (default=None)
alpha : real
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Returns: | L : NumPy array
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Raises: | NetworkXError :
NetworkXNotImplemnted :
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See also
Notes
Only implemented for DiGraphs
References
[R293] | Fan Chung (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics, 9(1), 2005 |