Katz Centrality (Centrality Measure)

In graph theory, the Katz centrality of a node is a measure of centrality in a network. It was introduced by Leo Katz in 1953 and is used to measure the relative degree of influence of an actor (or node) within a social network. Unlike typical centrality measures which consider only the shortest path (the geodesic) between a pair of actors, Katz centrality measures influence by taking into account the total number of walks between a pair of actors.

It is similar to Google’s PageRank and to the eigenvector centrality.

Measuring Katz centrality

A simple social network: the nodes represent people or actors and the edges between nodes represent some relationship between actors

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. Connections made with distant neighbors are, however, penalized by an attenuation factor \alpha . Each path or connection between a pair of nodes is assigned a weight determined by \alpha and the distance between nodes as \alpha ^{d} .

For example, in the figure on the right, assume that John’s centrality is being measured and that \alpha =0.5 . The weight assigned to each link that connects John with his immediate neighbors Jane and Bob will be (0.5)^{1}=0.5. Since Jose connects to John indirectly through Bob, the weight assigned to this connection (composed of two links) will be (0.5)^{2}=0.25. Similarly, the weight assigned to the connection between Agneta and John through Aziz and Jane will be (0.5)^{3}=0.125 and the weight assigned to the connection between Agneta and John through Diego, Jose and Bob will be (0.5)^{4}=0.0625.



Mathematical formulation
Let A be the adjacency matrix of a network under consideration. Elements (a_{ij}) of A are variables that take a value 1 if a node i is connected to node j and 0 otherwise. The powers of A indicate the presence (or absence) of links between two nodes through intermediaries. For instance, in matrix A^{3} , if element (a_{2,12})=1 , it indicates that node 2 and node 12 are connected through some first and second degree neighbors of node 2. If  C_{\mathrm {Katz} }(i) denotes Katz centrality of a node i, then mathematically:

C_{\mathrm {Katz} }(i)=\sum _{k=1}^{\infty }\sum _{j=1}^{n}\alpha ^{k}(A^{k})_{ji}
Note that the above definition uses the fact that the element at location (i,j) of the adjacency matrix A raised to the power k (i.e.  A^{k}) reflects the total number of k degree connections between nodes i and j . The value of the attenuation factor \alpha has to be chosen such that it is smaller than the reciprocal of the absolute value of the largest eigenvalue of the adjacency matrix A. In this case the following expression can be used to calculate Katz centrality:

{\overrightarrow {C}}_{\mathrm {Katz} }=((I-\alpha A^{T})^{-1}-I){\overrightarrow {I}}
HereI is the identity matrix, {\overrightarrow {I}} is an identity vector of size n (n is the number of nodes) consisting of ones. A^{T} denotes the transposed matrix of A and (I-\alpha A^{T})^{-1} denotes matrix inversion of the term (I-\alpha A^{T} ).

Following is the code for the calculation of the Katz Centrality of the graph and its various nodes.

def katz_centrality(G, alpha=0.1, beta=1.0,
                    max_iter=1000, tol=1.0e-6, 
                    nstart=None, normalized=True,
                    weight = 'weight'):
    """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 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. 


    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.

    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
      If None, all edge weights are considered equal.
      Otherwise holds the name of the edge attribute
      used as weight.

    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

     

    Notes
    -----
    
    This algorithm it uses the power method to find
    the eigenvector corresponding to the largest 
    eigenvalue of the adjacency matrix of G.
    The constant alpha should be strictly less than 
    the inverse of largest eigenvalue of the adjacency
    matrix for the algorithm to converge.
    The iteration will stop after max_iter iterations 
    or an error tolerance ofnumber_of_nodes(G)*tol 
     has been reached.

    When `\alpha = 1/\lambda_{max}` and `\beta=0`, 
    Katz centrality is the same as 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().

    
    """
    from math import sqrt

    if len(G) == 0:
        return {}

    nnodes = G.number_of_nodes()

    if nstart is None:

        # choose starting vector with entries of 0
        x = dict([(n,0) for n in G])
    else:
        x = nstart

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

    # make up to max_iter iterations
    for i 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
        err = sum([abs(x[n]-xlast[n]) for n in x])
        if err < nnodes*tol:
            if normalized:

                # normalize vector
                try:
                    s = 1.0/sqrt(sum(v**2 for v in x.values()))

                # this should never be zero?
                except ZeroDivisionError:
                    s = 1.0
            else:
                s = 1
            for n in x:
                x[n] *= s
            return x

    raise nx.NetworkXError('Power iteration failed to converge in '
                           '%d iterations.' % max_iter)

The above function is invoked using the networkx library and once the library is installed, you can eventually use it and the following code has to be written in python for the implementation of the katz centrality of a node.

>>> import networkx as nx
>>> 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("%d %0.2f"%(n,c))

The output of the above code is:


0 0.37
1 0.60
2 0.60
3 0.37

The above result is a dictionary depicting the value of katz centrality of each node. The above is an extension of my article series on the centrality measures. Keep networking!!!

References
http://networkx.readthedocs.io/en/networkx-1.10/index.html
https://en.wikipedia.org/wiki/Katz_centrality



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