Contribution centrality
Contribution centrality
is a spectral centrality measure based on structural dissimilarity [2]. The contribution centrality of node \(i\), denoted as \(c_{\mathrm{Contr}}(i)\), is proportional to the sum of the centralities of its neighboring nodes, weighted by their topological contributions. Formally,
\begin{equation*}
c_{\mathrm{Contr}}(i) = \frac{1}{λ} \sum_{j=1}^{N} a_{ij} D_{ij} c_{\mathrm{Contr}}(j),
\end{equation*}
where \(a_{ij}\) is the adjacency matrix element and \(D_{ij}\) is a structural dissimilarity measure defined as
\begin{equation*}
D_{ij} = 1 - \frac{|\mathcal{N}(i) \cap \mathcal{N}(j)|}{|\mathcal{N}(i) \cup \mathcal{N}(j)|},
\end{equation*}
with \(\mathcal{N}(i)\) denoting the set of neighbors of node \(i\).
The centrality equation can be expressed as an eigenvalue problem. Defining \(A_D = A \circ D\), where \(\circ\) denotes the Hadamard (element-wise) product, the contribution centrality vector \(c_{\mathrm{Contr}}\) satisfies
\begin{equation*}
A_D \, c_{\mathrm{Contr}} = λ \, c_{\mathrm{Contr}},
\end{equation*}
with \(λ = λ_{\max}\) being the dominant eigenvalue of \(A_D\). The corresponding principal eigenvector \(c_{\mathrm{Contr}}\) gives the contribution centralities of the nodes, where larger values indicate a greater structural influence within the network.