- directed_combinatorial_laplacian_matrix(G, nodelist=None, weight='weight', walk_type=None, alpha=0.95)¶
Return the directed combinatorial Laplacian matrix of G.
The graph directed combinatorial Laplacian is the matrix\[L = \Phi - (\Phi P + P^T \Phi) / 2\]
Pis the transition matrix of the graph and
Phia matrix with the Perron vector of
Pin the diagonal and zeros elsewhere.
Depending on the value of walk_type,
Pcan be the transition matrix induced by a random walk, a lazy random walk, or a random walk with teleportation (PageRank).
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)
Pis selected depending on the properties of the graph. Otherwise is one of ‘random’, ‘lazy’, or ‘pagerank’
(1 - alpha) is the teleportation probability used with pagerank
- LNumPy matrix
Combinatorial Laplacian of G.
Only implemented for DiGraphs
Fan Chung (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics, 9(1), 2005