## Mercurial > repos > shellac > sam_consensus_v3

### view env/lib/python3.9/site-packages/networkx/algorithms/centrality/second_order.py @ 0:4f3585e2f14b draft default tip

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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"

author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |

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"""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) """ Second order centrality measure.""" __all__ = ["second_order_centrality"] @not_implemented_for("directed") def second_order_centrality(G): """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. 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. """ try: import numpy as np except ImportError as e: raise ImportError("Requires NumPy: http://numpy.org/") from e 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, [np.sqrt(2 * np.sum(M[:, i]) - n * (n + 1)) for i in range(n)]) ) # eq 6