cugraph.centrality.katz_centrality#

cugraph.centrality.katz_centrality(G, alpha=None, beta=1.0, max_iter=100, tol=1e-06, nstart=None, normalized=True)[source]#

Compute the Katz centrality for the nodes of the graph G. This implementation is based on a relaxed version of Katz defined by Foster with a reduced computational complexity of O(n+m)

On a directed graph, cuGraph computes the out-edge Katz centrality score. This is opposite of NetworkX which compute the in-edge Katz centrality score by default. You can flip the NetworkX edges, using G.reverse, so that the results match cuGraph.

Parameters:

GcuGraph.Graph or networkx.Graph

cuGraph graph descriptor with connectivity information. The graph can contain either directed or undirected edges.

alphafloat, optional (default=None)

Attenuation factor defaulted to None. If alpha is not specified then it is internally calculated as 1/(degree_max) where degree_max is the maximum out degree.

NOTE:: The maximum acceptable value of alpha for convergence alpha_max = 1/(lambda_max) where lambda_max is the largest eigenvalue of the graph. Since lambda_max is always lesser than or equal to degree_max for a graph, alpha_max will always be greater than or equal to (1/degree_max). Therefore, setting alpha to (1/degree_max) will guarantee that it will never exceed alpha_max thus in turn fulfilling the requirement for convergence.

betafloat, optional (default=None)

Weight scalar added to each vertex’s new Katz Centrality score in every iteration. If beta is not specified then it is set as 1.0.

max_iterint, optional (default=100)

The maximum number of iterations before an answer is returned. This can be used to limit the execution time and do an early exit before the solver reaches the convergence tolerance.

tolfloat, optional (default=1.0e-6)

Set the tolerance the approximation, this parameter should be a small magnitude value. The lower the tolerance the better the approximation. If this value is 0.0f, cuGraph will use the default value which is 1.0e-6. Setting too small a tolerance can lead to non-convergence due to numerical roundoff. Usually values between 1e-2 and 1e-6 are acceptable.

nstartcudf.Dataframe, optional (default=None)

GPU Dataframe containing the initial guess for katz centrality.

nstart[‘vertex’]cudf.Series: Contains the vertex identifiers
nstart[‘values’]cudf.Series: Contains the katz centrality values of vertices

normalizednot supported

If True normalize the resulting katz centrality values

Returns:

dfcudf.DataFrame or Dictionary if using NetworkX

GPU data frame containing two cudf.Series of size V: the vertex identifiers and the corresponding katz centrality values.

df[‘vertex’]cudf.Series: Contains the vertex identifiers
df[‘katz_centrality’]cudf.Series: Contains the katz centrality of vertices

References

Foster, K.C., Muth, S.Q., Potterat, J.J. et al. Computational & Mathematical Organization Theory (2001) 7: 275. https://doi.org/10.1023/A:1013470632383

Katz, L. (1953). A new status index derived from sociometric analysis. Psychometrika, 18(1), 39-43.

Examples

>>> from cugraph.datasets import karate
>>> G = karate.get_graph(download=True)
>>> kc = cugraph.katz_centrality(G)