Source code for networkx.algorithms.centrality.laplacian

Laplacian centrality measures.
import networkx as nx

__all__ = ["laplacian_centrality"]

[docs] @nx._dispatchable(edge_attrs="weight") def laplacian_centrality( G, normalized=True, nodelist=None, weight="weight", walk_type=None, alpha=0.95 ): r"""Compute the Laplacian centrality for nodes in the graph `G`. The Laplacian Centrality of a node ``i`` is measured by the drop in the Laplacian Energy after deleting node ``i`` from the graph. The Laplacian Energy is the sum of the squared eigenvalues of a graph's Laplacian matrix. .. math:: C_L(u_i,G) = \frac{(\Delta E)_i}{E_L (G)} = \frac{E_L (G)-E_L (G_i)}{E_L (G)} E_L (G) = \sum_{i=0}^n \lambda_i^2 Where $E_L (G)$ is the Laplacian energy of graph `G`, E_L (G_i) is the Laplacian energy of graph `G` after deleting node ``i`` and $\lambda_i$ are the eigenvalues of `G`'s Laplacian matrix. This formula shows the normalized value. Without normalization, the numerator on the right side is returned. Parameters ---------- G : graph A networkx graph normalized : bool (default = True) If True the centrality score is scaled so the sum over all nodes is 1. If False the centrality score for each node is the drop in Laplacian energy when that node is removed. nodelist : list, optional (default = None) 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`) Optional parameter `weight` to compute the Laplacian matrix. 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) Optional parameter `walk_type` used when calling :func:`directed_laplacian_matrix <networkx.directed_laplacian_matrix>`. 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 (default = 0.95) Optional parameter `alpha` used when calling :func:`directed_laplacian_matrix <networkx.directed_laplacian_matrix>`. (1 - alpha) is the teleportation probability used with pagerank. Returns ------- nodes : dictionary Dictionary of nodes with Laplacian centrality as the value. Examples -------- >>> G = nx.Graph() >>> edges = [(0, 1, 4), (0, 2, 2), (2, 1, 1), (1, 3, 2), (1, 4, 2), (4, 5, 1)] >>> G.add_weighted_edges_from(edges) >>> sorted((v, f"{c:0.2f}") for v, c in laplacian_centrality(G).items()) [(0, '0.70'), (1, '0.90'), (2, '0.28'), (3, '0.22'), (4, '0.26'), (5, '0.04')] Notes ----- The algorithm is implemented based on [1]_ with an extension to directed graphs using the ``directed_laplacian_matrix`` function. Raises ------ NetworkXPointlessConcept If the graph `G` is the null graph. ZeroDivisionError If the graph `G` has no edges (is empty) and normalization is requested. References ---------- .. [1] Qi, X., Fuller, E., Wu, Q., Wu, Y., and Zhang, C.-Q. (2012). Laplacian centrality: A new centrality measure for weighted networks. Information Sciences, 194:240-253. See Also -------- :func:`~networkx.linalg.laplacianmatrix.directed_laplacian_matrix` :func:`~networkx.linalg.laplacianmatrix.laplacian_matrix` """ import numpy as np import scipy as sp if len(G) == 0: raise nx.NetworkXPointlessConcept("null graph has no centrality defined") if G.size(weight=weight) == 0: if normalized: raise ZeroDivisionError("graph with no edges has zero full energy") return {n: 0 for n in G} if nodelist is not None: nodeset = set(G.nbunch_iter(nodelist)) if len(nodeset) != len(nodelist): raise nx.NetworkXError("nodelist has duplicate nodes or nodes not in G") nodes = nodelist + [n for n in G if n not in nodeset] else: nodelist = nodes = list(G) if G.is_directed(): lap_matrix = nx.directed_laplacian_matrix(G, nodes, weight, walk_type, alpha) else: lap_matrix = nx.laplacian_matrix(G, nodes, weight).toarray() full_energy = np.power(sp.linalg.eigh(lap_matrix, eigvals_only=True), 2).sum() # calculate laplacian centrality laplace_centralities_dict = {} for i, node in enumerate(nodelist): # remove row and col i from lap_matrix all_but_i = list(np.arange(lap_matrix.shape[0])) all_but_i.remove(i) A_2 = lap_matrix[all_but_i, :][:, all_but_i] # Adjust diagonal for removed row new_diag = lap_matrix.diagonal() - abs(lap_matrix[:, i]) np.fill_diagonal(A_2, new_diag[all_but_i]) if len(all_but_i) > 0: # catches degenerate case of single node new_energy = np.power(sp.linalg.eigh(A_2, eigvals_only=True), 2).sum() else: new_energy = 0.0 lapl_cent = full_energy - new_energy if normalized: lapl_cent = lapl_cent / full_energy laplace_centralities_dict[node] = float(lapl_cent) return laplace_centralities_dict