The Katz centrality of a graph vertex is a walk-counting centrality measure in which walks of length 
are weighted by a factor 
. Equivalently, for a graph with adjacency
matrix 
,
the vector of Katz centralities satisfies
 |
for parameters 
and 
for which the corresponding linear system is nonsingular.
Katz centrality was introduced as a status index for sociometric data, with attenuated contributions from increasingly long walks (Katz 1953). It is used when indirect
ties or endorsements should count, but should contribute less as the walks connecting
them become longer. It is closely related to eigenvector
centrality: when 
and 
is the reciprocal of a suitable eigenvalue, the homogeneous
equation becomes an eigenvector equation. With 
and smaller 
, Katz centrality instead counts attenuated walks from
an exogenous baseline.
Katz centrality is implemented in the Wolfram Language as KatzCentrality[g, alpha] and KatzCentrality[g, alpha, beta], and precomputed symbolic values for many named graphs can be obtained using GraphData[graph, "KatzCentralities"].
