Zeta vector centrality
Zeta vector centrality , also referred to as node displacement [2] or topological centrality [3], identifies the most effective “spreader” node in a network [4]. Inspired by electrical flows in a resistor network, Van Mieghem et al. define the best conducting node \(i\) in a graph \(G\) as the node that minimizes the diagonal element \(Q^{\dagger}_{ii}\) of the pseudoinverse \(Q^{\dagger}\) of the weighted Laplacian matrix of \(G\):\begin{equation*}c_{\rm Zeta}(i) = Q^{\dagger}_{ii} = \frac{1}{N} \sum_{j=1}^{N} ω_{ji} - \frac{R̃_G}{N^2},\end{equation*}where \(ω_{ji}\) is the effective resistance between nodes \(j\) and \(i\), and \(R̃_G\) is the effective graph resistance. In other words, \(c_{\rm Zeta}(i)\) represents the average effective resistance from node \(i\) to all other nodes, minus the mean effective resistance of the graph.The node with minimal \(c_{\rm Zeta}(i)\) is the best electrical spreader, having the lowest energy or potential and the strongest connectivity to the rest of the network. Alternative interpretations of \(Q^{\dagger}_{ii}\) include detour overheads in random walks or the average connectedness of nodes when the network splits, highlighting the node’s role in global communication [3]. The ranking of nodes based on \(Q^{\dagger}_{ii}\) is equivalent to node displacement [2], computed as\begin{equation*}\Delta x_i = \sqrt{\frac{Q^{\dagger}_{ii}}{β k}},\end{equation*}where \(k\) is a common spring constant and \(β\) is the inverse temperature, linking network topology to physical interpretations of displacement.