directed_laplacian_matrix#

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

\[L = I - \frac{1}{2} \left (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} \right )\]

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 [1].

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:

GDiGraph: A NetworkX graph
nodelistlist, 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().
weightstring 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_typestring or None, optional (default=None): One of "random", "lazy", or "pagerank". If walk_type=None (the default), then a value is selected according to the properties of G: - walk_type="random" if G is strongly connected and aperiodic - walk_type="lazy" if G is strongly connected but not aperiodic - walk_type="pagerank" for all other cases.
alphareal: (1 - alpha) is the teleportation probability used with pagerank

Returns:

LNumPy matrix: Normalized Laplacian of G.