Warning
This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.
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 : NumPy array
Normalized Laplacian of G.
Raises : NetworkXError
If NumPy cannot be imported
NetworkXNotImplemnted
If G is not a DiGraph
See also
Notes
Only implemented for DiGraphs
References
[R319] Fan Chung (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics, 9(1), 2005