Total communicability centrality (TCC)
The
total communicability centrality
(TCC) quantifies how effectively each node communicates with all other nodes in a network [2]. It extends the concept of subgraph centrality, which evaluates all possible closed walks of different lengths. The centrality of node \(i\), \(c_{\mathrm{TTC}}(i)\), is defined as the sum of weighted walks of all lengths starting at node \(i\) and ending at any node \(j\):
\begin{equation*}
c_{\mathrm{TTC}}(i) = \sum_{k=0}^{\infty} \sum_{j=1}^{N} \frac{(A^k)_{ij}}{k!}
= \sum_{j=1}^{N} [e^A]_{ij}
= \sum_{j=1}^{N} \sum_{l=1}^{N} v_j(i) v_l(i) e^{λ_j},
\end{equation*}
where \(A\) is the adjacency matrix, \(v_j(i)\) is the \(i\)-th component of the eigenvector \(v_j\) corresponding to eigenvalue \(λ_j\) of \(A\). Benzi and Klymko [2] show that for networks with a large spectral gap (\(λ_1 \gg λ_2\)), the TTC and subgraph centralities converge to the eigenvector centrality, and they illustrate graph types for which TTC and subgraph centrality yield identical node rankings.