Collective network entanglement (CNE)
Collective network entanglement
(CNE) is an induced, entropy-based centrality measure that quantifies the role of individual nodes in preserving the functional diversity of a network [2]. The CNE score \(c_{CNE}(i)\) of node \(i \in \mathcal{N}\) is defined as the change in von Neumann entropy caused by the detachment of the node and its incident edges:
\[
c_{CNE}(i) = \left( S_β(G^*_i) + S_β(G_i)\right) - S_β(G),
\]
where \(G_i\) denotes the subgraph obtained by removing node \(i\), \(G^*_i\) is the star graph of node \(i\), and \(S_β(G)\) is the von Neumann entropy of \(G\). The entropy is computed from a density matrix \(ρ\) derived from the network Laplacian \(L(G)\):
\[
ρ = \frac{e^{-β L(G)}}{\mathrm{tr}(e^{-β L(G)})}, \qquad
S_β(G) = -\mathrm{tr}(ρ \log ρ),
\]
with \(β > 0\) as a diffusion time parameter controlling the scale of information propagation.
At very short diffusion times (\(β \to 0\)), CNE reduces to degree centrality, reflecting a node's immediate neighborhood. At very long times (\(β \to \infty\)), it captures a node's contribution to the overall network connectivity. Ghavasieh et al. [2] demonstrate that \(β = β_c\) provides an appropriate timescale for network disintegration, as confirmed in experiments on social, biological, and transportation networks.