U 9%eD@s dZddlZdgZdddZdS) z Laplacian centrality measures. Nlaplacian_centralityTweightffffff?csddl}ddl}ddl}t|dkr.td|dkrvt||tt|kr^td|fdd|D} n t |}} | rt || |||} nt || | } ||jj| ddd } i} t|D]\} }t || jd}|| | |ddfdd|f}| t| dd| f}||||||jj|ddd }| |}|rx|| }|| |<q| S) av 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 `. If None, the transition matrix is selected depending on the properties of the graph. Otherwise can be `random`, `lazy`, or `pagerank`. alpha : real (default = 0.95) Optional parameter `alpha` used when calling :func:`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. 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. https://math.wvu.edu/~cqzhang/Publication-files/my-paper/INS-2012-Laplacian-W.pdf See Also -------- directed_laplacian_matrix laplacian_matrix rNz$null graph has no centrality definedz.nodelist has duplicate nodes or nodes not in Gcsg|]}|kr|qSr).0nZnodesetrg/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/networkx/algorithms/centrality/laplacian.py jsz(laplacian_centrality..T)Z eigvals_only)numpyscipyZ scipy.linalglennxZNetworkXPointlessConceptsetZ nbunch_iterZ NetworkXErrorlistZ is_directedZdirected_laplacian_matrixZlaplacian_matrixZtoarraypowerZlinalgZeighsum enumerateZarangeshaperemoveZdiagonalabsZ fill_diagonal)G normalizedZnodelistrZ walk_typealphanpspr ZnodesZ lap_matrixZ full_energyZlaplace_centralities_dictinodeZ all_but_iZA_2Znew_diagZ new_energyZ lapl_centrrr r s8V      )TNrNr)__doc__Znetworkxr__all__rrrrr s