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# Source code for networkx.algorithms.centrality.second_order

    """Copyright (c) 2015 – Thomson Licensing, SAS

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    NO EXPRESS OR IMPLIED LICENSES TO ANY PARTY'S PATENT RIGHTS ARE
    GRANTED BY THIS LICENSE.  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT
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    """

    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