Katz centrality
Katz centrality
, also known as Katz prestige, is a generalization of eigenvector centrality [2, 3]. In the original formulation, the influence of a node \(i\) is measured by a weighted sum of all powers of the adjacency matrix \(A\), with an attenuation factor \(α\):
\begin{equation*}
c_{\mathrm{Katz}}(i) = \sum_{k=1}^{\infty} \sum_{j=1}^{N} α^k \, (A^k)_{ij},
\end{equation*}
or, equivalently, in vector form:
\begin{equation*}
\mathbf{c}_{\mathrm{Katz}} = (I - α A)^{-1} \mathbf{1},
\end{equation*}
where \(\mathbf{1}\) is the vector of all ones. The series converges only if the attenuation factor \(α\) is smaller than the reciprocal of the largest eigenvalue of the adjacency matrix \(A\), i.e., \(α < 1 / λ_{\max}\), where \(λ_{\max}\) denotes the principal eigenvalue of \(A\).
Katz centrality is closely related to Bonacich’s power centrality \(c_{\mathrm{power}}\) [4], which is defined as
\begin{equation*}
c_{\mathrm{power}} = \sum_{k=0}^{\infty} \sum_{j=1}^{N} α^k \, (A^{k+1})_{ij}.
\end{equation*}
The two measures are directly proportional, with Bonacich's power centrality expressed in terms of Katz centrality as
\begin{equation*}
c_{\mathrm{power}} = α \, c_{\mathrm{Katz}}.
\end{equation*}
A further generalization, often referred to as
alpha centrality
or
Bonacich alpha centrality
, introduces an exogenous factor \(β\) and allows \(α\) to take negative values to model negative influence [4, 5, 6]. The centrality of node \(i\) is then defined by
\begin{equation*}
c_{α}(i) = α \sum_{j=1}^{N} a_{ij} \, c_{α}(j) + β,
\end{equation*}
where \(α\) and \(β\) are positive constants. This formulation balances the contribution from the eigenvector-like term with the constant \(β\). In most current implementations, \(α\) is set to \(0.1\) and \(β\) to \(1.0\). Bonacich and Lloyd [5] demonstrated that Katz and alpha centralities differ only by a constant \(β\) under the condition
\(α < \frac{1}{λ_{\max}}\), with \(λ_{\max}\) being the principal eigenvalue of \(A\).