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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, 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).

  • 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

L – Normalized Laplacian of G.

Return type:

NumPy array


Only implemented for DiGraphs


[1]Fan Chung (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics, 9(1), 2005