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# directed_laplacian_matrix

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

  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:
    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)

      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.

    alpha real

      (1 - alpha) is the teleportation probability used with pagerank

  Returns:
    L NumPy matrix

      Normalized Laplacian of G.

  See also

  `laplacian_matrix`
  `normalized_laplacian_matrix`
  `directed_combinatorial_laplacian_matrix`

  Notes

  Only implemented for DiGraphs

  The result is always a symmetric matrix.

  This calculation uses the out-degree of the graph `G`. To use the in-degree for calculations instead, use `G.reverse(copy=False)` and take the transpose.

  References

  [1]

  Fan Chung (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics, 9(1), 2005