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

"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"

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+++ b/env/lib/python3.9/site-packages/networkx/algorithms/centrality/second_order.py	Mon Mar 22 18:12:50 2021 +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 ([e-mail])
+""" 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