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Diffstat (limited to '.venv/lib/python3.12/site-packages/networkx/algorithms/centrality/second_order.py')
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diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/centrality/second_order.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/centrality/second_order.py new file mode 100644 index 00000000..35583cd6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/centrality/second_order.py @@ -0,0 +1,141 @@ +"""Copyright (c) 2015 – Thomson Licensing, SAS + +Redistribution and use in source and binary forms, with or without +modification, are permitted (subject to the limitations in the +disclaimer below) provided that the following conditions are met: + +* Redistributions of source code must retain the above copyright +notice, this list of conditions and the following disclaimer. + +* Redistributions in binary form must reproduce the above copyright +notice, this list of conditions and the following disclaimer in the +documentation and/or other materials provided with the distribution. + +* Neither the name of Thomson Licensing, or Technicolor, nor the names +of its contributors may be used to endorse or promote products derived +from this software without specific prior written permission. + +NO EXPRESS OR IMPLIED LICENSES TO ANY PARTY'S PATENT RIGHTS ARE +GRANTED BY THIS LICENSE. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT +HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED +WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF +MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE +DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE +LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR +CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF +SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR +BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, +WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE +OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN +IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. +""" + +import networkx as nx +from networkx.utils import not_implemented_for + +# Authors: Erwan Le Merrer (erwan.lemerrer@technicolor.com) + +__all__ = ["second_order_centrality"] + + +@not_implemented_for("directed") +@nx._dispatchable(edge_attrs="weight") +def second_order_centrality(G, weight="weight"): + """Compute the second order centrality for nodes of G. + + The second order centrality of a given node is the standard deviation of + the return times to that node of a perpetual random walk on G: + + Parameters + ---------- + G : graph + A NetworkX connected and undirected graph. + + weight : string or None, optional (default="weight") + The name of an edge attribute that holds the numerical value + used as a weight. If None then each edge has weight 1. + + Returns + ------- + nodes : dictionary + Dictionary keyed by node with second order centrality as the value. + + Examples + -------- + >>> G = nx.star_graph(10) + >>> soc = nx.second_order_centrality(G) + >>> print(sorted(soc.items(), key=lambda x: x[1])[0][0]) # pick first id + 0 + + Raises + ------ + NetworkXException + If the graph G is empty, non connected or has negative weights. + + See Also + -------- + betweenness_centrality + + Notes + ----- + Lower values of second order centrality indicate higher centrality. + + The algorithm is from Kermarrec, Le Merrer, Sericola and Trédan [1]_. + + This code implements the analytical version of the algorithm, i.e., + there is no simulation of a random walk process involved. The random walk + is here unbiased (corresponding to eq 6 of the paper [1]_), thus the + centrality values are the standard deviations for random walk return times + on the transformed input graph G (equal in-degree at each nodes by adding + self-loops). + + Complexity of this implementation, made to run locally on a single machine, + is O(n^3), with n the size of G, which makes it viable only for small + graphs. + + References + ---------- + .. [1] Anne-Marie Kermarrec, Erwan Le Merrer, Bruno Sericola, Gilles Trédan + "Second order centrality: Distributed assessment of nodes criticity in + complex networks", Elsevier Computer Communications 34(5):619-628, 2011. + """ + import numpy as np + + n = len(G) + + if n == 0: + raise nx.NetworkXException("Empty graph.") + if not nx.is_connected(G): + raise nx.NetworkXException("Non connected graph.") + if any(d.get(weight, 0) < 0 for u, v, d in G.edges(data=True)): + raise nx.NetworkXException("Graph has negative edge weights.") + + # balancing G for Metropolis-Hastings random walks + G = nx.DiGraph(G) + in_deg = dict(G.in_degree(weight=weight)) + d_max = max(in_deg.values()) + for i, deg in in_deg.items(): + if deg < d_max: + G.add_edge(i, i, weight=d_max - deg) + + P = nx.to_numpy_array(G) + P /= P.sum(axis=1)[:, np.newaxis] # to transition probability matrix + + def _Qj(P, j): + P = P.copy() + P[:, j] = 0 + return P + + M = np.empty([n, n]) + + for i in range(n): + M[:, i] = np.linalg.solve( + np.identity(n) - _Qj(P, i), np.ones([n, 1])[:, 0] + ) # eq 3 + + return dict( + zip( + G.nodes, + (float(np.sqrt(2 * np.sum(M[:, i]) - n * (n + 1))) for i in range(n)), + ) + ) # eq 6 |