Skip to main content
`Ctrl`+`K`

# katz_centrality_numpy

katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True, weight=None)

  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

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

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

  The parameter \(\beta\) controls the initial centrality and

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

  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

  See also

  `katz_centrality`
  `eigenvector_centrality_numpy`
  `eigenvector_centrality`
  `pagerank()`
  `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

  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