Katz centrality

Katz centrality is a measure of node influence in a network that extends eigenvector centrality by introducing an attenuation factor that discounts the contribution of longer paths. Introduced by Leo Katz in 1953, it resolves the problem that eigenvector centrality is undefined for directed acyclic graphs and dominated by cycles in general graphs. The attenuation parameter — typically denoted α — controls how much weight is given to distant connections: as α approaches the reciprocal of the largest eigenvalue, Katz centrality converges to eigenvector centrality; as α approaches zero, it converges to degree centrality.

The mathematical formulation is x = (I − αA^T)⁻¹1, where A is the adjacency matrix and 1 is a vector of ones. This closed-form solution makes Katz centrality computationally attractive for large networks, though the choice of α remains a theoretical decision about the network\'s \'\'\'effective horizon\'\'\' — how far influence propagates before it becomes negligible.

Katz centrality is particularly useful in citation networks, where it captures the cumulative influence of a paper through direct and indirect citations without allowing cyclic amplification to dominate the ranking. It is also related to \'\'\'Bonacich power centrality\'\'\', which generalizes the same recursive structure with positive and negative edge weights.