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# Source code for networkx.algorithms.centrality.katz

    """Katz centrality."""

    import math

    import networkx as nx
    from networkx.utils import not_implemented_for

    __all__ = ["katz_centrality", "katz_centrality_numpy"]

    @not_implemented_for("multigraph")
    @nx._dispatchable(edge_attrs="weight")
    def katz_centrality(
        G,
        alpha=0.1,
        beta=1.0,
        max_iter=1000,
        tol=1.0e-6,
        nstart=None,
        normalized=True,
        weight=None,
    ):
        r"""Compute the Katz centrality for the nodes of the graph G.

        Katz centrality computes the centrality for a node based on the centrality
        of its neighbors. It is a generalization of the eigenvector centrality. The
        Katz centrality for node $i$ is

        .. math::

            x_i = \alpha \sum_{j} A_{ij} x_j + \beta,

        where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$.

        The parameter $\beta$ controls the initial centrality and

        .. math::

            \alpha < \frac{1}{\lambda_{\max}}.

        Katz centrality computes the relative influence of a node within a
        network by measuring the number of the immediate neighbors (first
        degree nodes) and also all other nodes in the network that connect
        to the node under consideration through these immediate neighbors.

        Extra weight can be provided to immediate neighbors through the
        parameter $\beta$.  Connections made with distant neighbors
        are, however, penalized by an attenuation factor $\alpha$ which
        should be strictly less than the inverse largest eigenvalue of the
        adjacency matrix in order for the Katz centrality to be computed
        correctly. More information is provided in [1]_.

        Parameters
        ----------
        G : graph
          A NetworkX graph.

        alpha : float, optional (default=0.1)
          Attenuation factor

        beta : scalar or dictionary, optional (default=1.0)
          Weight attributed to the immediate neighborhood. If not a scalar, the
          dictionary must have a value for every node.

        max_iter : integer, optional (default=1000)
          Maximum number of iterations in power method.

        tol : float, optional (default=1.0e-6)
          Error tolerance used to check convergence in power method iteration.

        nstart : dictionary, optional
          Starting value of Katz iteration for each node.

        normalized : bool, optional (default=True)
          If True normalize the resulting values.

        weight : None or string, optional (default=None)
          If None, all edge weights are considered equal.
          Otherwise holds the name of the edge attribute used as weight.
          In this measure the weight is interpreted as the connection strength.

        Returns
        -------
        nodes : dictionary
           Dictionary of nodes with Katz centrality as the value.

        Raises
        ------
        NetworkXError
           If the parameter `beta` is not a scalar but lacks a value for at least
           one node

        PowerIterationFailedConvergence
            If the algorithm fails to converge to the specified tolerance
            within the specified number of iterations of the power iteration
            method.

        Examples
        --------
        >>> import math
        >>> G = nx.path_graph(4)
        >>> phi = (1 + math.sqrt(5)) / 2.0  # largest eigenvalue of adj matrix
        >>> centrality = nx.katz_centrality(G, 1 / phi - 0.01)
        >>> for n, c in sorted(centrality.items()):
        ...     print(f"{n} {c:.2f}")
        0 0.37
        1 0.60
        2 0.60
        3 0.37

        See Also
        --------
        katz_centrality_numpy
        eigenvector_centrality
        eigenvector_centrality_numpy
        :func:`~networkx.algorithms.link_analysis.pagerank_alg.pagerank`
        :func:`~networkx.algorithms.link_analysis.hits_alg.hits`

        Notes
        -----
        Katz centrality was introduced by [2]_.

        This algorithm it uses the power method to find the eigenvector
        corresponding to the largest eigenvalue of the adjacency matrix of ``G``.
        The parameter ``alpha`` should be strictly less than the inverse of largest
        eigenvalue of the adjacency matrix for the algorithm to converge.
        You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest
        eigenvalue of the adjacency matrix.
        The iteration will stop after ``max_iter`` iterations or an error tolerance of
        ``number_of_nodes(G) * tol`` has been reached.

        For strongly connected graphs, as $\alpha \to 1/\lambda_{\max}$, and $\beta > 0$,
        Katz centrality approaches the results for eigenvector centrality.

        For directed graphs this finds "left" eigenvectors which corresponds
        to the in-edges in the graph. For out-edges Katz centrality,
        first reverse the graph with ``G.reverse()``.

        References
        ----------
        .. [1] Mark E. J. Newman:
           Networks: An Introduction.
           Oxford University Press, USA, 2010, p. 720.
        .. [2] Leo Katz:
           A New Status Index Derived from Sociometric Index.
           Psychometrika 18(1):39–43, 1953
           https://link.springer.com/content/pdf/10.1007/BF02289026.pdf
        """
        if len(G) == 0:
            return {}

        nnodes = G.number_of_nodes()

        if nstart is None:
            # choose starting vector with entries of 0
            x = {n: 0 for n in G}
        else:
            x = nstart

        try:
            b = dict.fromkeys(G, float(beta))
        except (TypeError, ValueError, AttributeError) as err:
            b = beta
            if set(beta) != set(G):
                raise nx.NetworkXError(
                    "beta dictionary must have a value for every node"
                ) from err

        # make up to max_iter iterations
        for _ in range(max_iter):
            xlast = x
            x = dict.fromkeys(xlast, 0)
            # do the multiplication y^T = Alpha * x^T A + Beta
            for n in x:
                for nbr in G[n]:
                    x[nbr] += xlast[n] * G[n][nbr].get(weight, 1)
            for n in x:
                x[n] = alpha * x[n] + b[n]

            # check convergence
            error = sum(abs(x[n] - xlast[n]) for n in x)
            if error < nnodes * tol:
                if normalized:
                    # normalize vector
                    try:
                        s = 1.0 / math.hypot(*x.values())
                    except ZeroDivisionError:
                        s = 1.0
                else:
                    s = 1
                for n in x:
                    x[n] *= s
                return x
        raise nx.PowerIterationFailedConvergence(max_iter)

    @not_implemented_for("multigraph")
    @nx._dispatchable(edge_attrs="weight")
    def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True, weight=None):
        r"""Compute the Katz centrality for the graph G.

        Katz centrality computes the centrality for a node based on the centrality
        of its neighbors. It is a generalization of the eigenvector centrality. The
        Katz centrality for node $i$ is

        .. math::

            x_i = \alpha \sum_{j} A_{ij} x_j + \beta,

        where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$.

        The parameter $\beta$ controls the initial centrality and

        .. math::

            \alpha < \frac{1}{\lambda_{\max}}.

        Katz centrality computes the relative influence of a node within a
        network by measuring the number of the immediate neighbors (first
        degree nodes) and also all other nodes in the network that connect
        to the node under consideration through these immediate neighbors.

        Extra weight can be provided to immediate neighbors through the
        parameter $\beta$.  Connections made with distant neighbors
        are, however, penalized by an attenuation factor $\alpha$ which
        should be strictly less than the inverse largest eigenvalue of the
        adjacency matrix in order for the Katz centrality to be computed
        correctly. More information is provided in [1]_.

        Parameters
        ----------
        G : graph
          A NetworkX graph

        alpha : float
          Attenuation factor

        beta : scalar or dictionary, optional (default=1.0)
          Weight attributed to the immediate neighborhood. If not a scalar the
          dictionary must have an value for every node.

        normalized : bool
          If True normalize the resulting values.

        weight : None or string, optional
          If None, all edge weights are considered equal.
          Otherwise holds the name of the edge attribute used as weight.
          In this measure the weight is interpreted as the connection strength.

        Returns
        -------
        nodes : dictionary
           Dictionary of nodes with Katz centrality as the value.

        Raises
        ------
        NetworkXError
           If the parameter `beta` is not a scalar but lacks a value for at least
           one node

        Examples
        --------
        >>> import math
        >>> G = nx.path_graph(4)
        >>> phi = (1 + math.sqrt(5)) / 2.0  # largest eigenvalue of adj matrix
        >>> centrality = nx.katz_centrality_numpy(G, 1 / phi)
        >>> for n, c in sorted(centrality.items()):
        ...     print(f"{n} {c:.2f}")
        0 0.37
        1 0.60
        2 0.60
        3 0.37

        See Also
        --------
        katz_centrality
        eigenvector_centrality_numpy
        eigenvector_centrality
        :func:`~networkx.algorithms.link_analysis.pagerank_alg.pagerank`
        :func:`~networkx.algorithms.link_analysis.hits_alg.hits`

        Notes
        -----
        Katz centrality was introduced by [2]_.

        This algorithm uses a direct linear solver to solve the above equation.
        The parameter ``alpha`` should be strictly less than the inverse of largest
        eigenvalue of the adjacency matrix for there to be a solution.
        You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest
        eigenvalue of the adjacency matrix.

        For strongly connected graphs, as $\alpha \to 1/\lambda_{\max}$, and $\beta > 0$,
        Katz centrality approaches the results for eigenvector centrality.

        For directed graphs this finds "left" eigenvectors which corresponds
        to the in-edges in the graph. For out-edges Katz centrality,
        first reverse the graph with ``G.reverse()``.

        References
        ----------
        .. [1] Mark E. J. Newman:
           Networks: An Introduction.
           Oxford University Press, USA, 2010, p. 173.
        .. [2] Leo Katz:
           A New Status Index Derived from Sociometric Index.
           Psychometrika 18(1):39–43, 1953
           https://link.springer.com/content/pdf/10.1007/BF02289026.pdf
        """
        import numpy as np

        if len(G) == 0:
            return {}
        try:
            nodelist = beta.keys()
            if set(nodelist) != set(G):
                raise nx.NetworkXError("beta dictionary must have a value for every node")
            b = np.array(list(beta.values()), dtype=float)
        except AttributeError:
            nodelist = list(G)
            try:
                b = np.ones((len(nodelist), 1)) * beta
            except (TypeError, ValueError, AttributeError) as err:
                raise nx.NetworkXError("beta must be a number") from err

        A = nx.adjacency_matrix(G, nodelist=nodelist, weight=weight).todense().T
        n = A.shape[0]
        centrality = np.linalg.solve(np.eye(n, n) - (alpha * A), b).squeeze()

        # Normalize: rely on truediv to cast to float, then tolist to make Python numbers
        norm = np.sign(sum(centrality)) * np.linalg.norm(centrality) if normalized else 1
        return dict(zip(nodelist, (centrality / norm).tolist()))