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{\tilde{R}_G}{N^2},
\end{equation*}
where \(ω_{ji}\) is the effective resistance between nodes \(j\) and \(i\), and \(\tilde{R}_G\) 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.
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\end{thebibliography}
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References

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