@not_implemented_for('multigraph')
def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True):
    r"""Compute the Katz centrality for the graph G.


    Katz centrality is related to eigenvalue centrality and PageRank.
    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 the 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 :math:`\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.

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

    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("%d %0.2f"%(n,c))
    0 0.37
    1 0.60
    2 0.60
    3 0.37

    Notes
    ------
    This algorithm uses a direct linear solver to solve the above equation.
    The constant alpha should be strictly less than the inverse of largest
    eigenvalue of the adjacency matrix for there to be a solution.  When
    `\alpha = 1/\lambda_{max}` and `\beta=1` Katz centrality is the same as
    eigenvector centrality.

    References
    ----------
    .. [1] M. Newman, Networks: An Introduction. Oxford University Press,
       USA, 2010, p. 720.

    See Also
    --------
    katz_centrality
    eigenvector_centrality_numpy
    eigenvector_centrality
    pagerank
    hits
    """
    try:
        import numpy as np
    except ImportError:
        raise ImportError('Requires NumPy: http://scipy.org/')
    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 = G.nodes()
        try:
            b = np.ones((len(nodelist),1))*float(beta)
        except (TypeError,__HOLE__):
            raise nx.NetworkXError('beta must be a number')

    A=nx.adj_matrix(G, nodelist=nodelist)
    n = np.array(A).shape[0]
    centrality = np.linalg.solve( np.eye(n,n) - (alpha * A) , b)
    if normalized:
        norm = np.sign(sum(centrality)) * np.linalg.norm(centrality)
    else:
        norm = 1.0
    centrality=dict(zip(nodelist, map(float,centrality/norm)))
    return centrality


# fixture for nose tests
dataset/ETHPy150Open gkno/gkno_launcher/src/networkx/algorithms/centrality/katz.py/katz_centrality_numpy