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# directed_laplacian_matrix¶

directed_laplacian_matrix(G, nodelist=None, weight='weight', walk_type=None, alpha=0.95)[source]

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 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 : NumPy array Normalized Laplacian of G. NetworkXError If NumPy cannot be imported NetworkXNotImplemnted If G is not a DiGraph