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
Returns:	L – Normalized Laplacian of G.
Return type:	NumPy array
Raises:	`NetworkXError` – If NumPy cannot be imported `NetworkXNotImplemnted` – If G is not a DiGraph

Notes

Only implemented for DiGraphs