A local eigenvector centrality

TL;DR

The paper addresses localisation in eigenvector centrality by proposing local eigenvector centrality that combines local and global connectivity through prominent eigengaps of the adjacency matrix $A$, using a multieigenvector matrix $V=[v_1|\dots|v_k]$ to compute $c(i)=\|[v_1(i),\dots,v_k(i)]\|_2$. Complex-conjugate handling (via real and imaginary components) and special cases like zero eigenvalues and directed cycles are systematically addressed, with road networks illustrating a practical weighting scheme $a_{ij}=\frac{1}{1+t_{ij}}$, $t_{ij}=\sqrt{\frac{2d}{a}}$. Results on defined community networks (e.g., school contact graphs) show that local centrality at multiple eigengaps mirrors class-level centralities while revealing cross-community roles; undefined communities in road networks yield high-centrality hubs corresponding to locally and globally important nodes. A PageRank-based non-linear rescaling provides a principled way to mitigate localisation, linking deterministic multiscale centrality with stochastic flow measures. Overall, the approach offers a principled, multiscale centrality framework that complements Laplacian-based methods and provides actionable insights for networks with or without clear community delineations, with accompanying code and data available publicly.

Abstract

Eigenvector centrality is an established measure of global connectivity, from which the importance and influence of nodes can be inferred. We introduce a local eigenvector centrality that incorporates both local and global connectivity. This new measure references prominent eigengaps and combines their associated eigenspectrum, via the Euclidean norm, to detect centrality that reflects the influence of prominent community structures. In contact networks, with clearly defined community structures, local eigenvector centrality is shown to identify similar but distinct distributions to eigenvector centrality applied on each community in isolation and PageRank. Discrepancies between the two eigenvector measures highlight nodes and communities that do not conform to their defined local structures, e.g. nodes with more connections outside of their defined community than within it. While reference to PageRank's centrality assessment enables a mitigation strategy for localisation effects inherent in eigenvector-based measures. In networks without clearly defined communities, such as city road networks, local eigenvector centrality is shown to identify both locally prominent and globally connected hubs.

Figures (8)

• Figure 1: A four community graph example, where the complex conjugate pair occur at $v_2$ and $v_3$, with the real and imaginary values of $v_2$ displayed. (a) Eigengap analysis, (b) first eigenvector $v_1$, (c) absolute real component of $v_2$, (d) absolute imaginary component of $v_2$, (e) absolute $v_4$ values, and (f) local eigenvector centrality.
• Figure 2: Eigengaps for a 10-node directed cycle
• Figure 3: Centrality analysis of the Primary School contact network. (a) Eigengap plot showing prominent eigengaps at $i=1$,$5$, and $10$. (b) Eigenvector centrality distribution capturing global centrality. (c) Local eigenvector centrality for $i=5$. (d) Local eigenvector centrality for $i=10$. In panels (b–d), the circle sizes are proportional to the corresponding centrality values, highlighting the relative importance of each node within the network.
• Figure 4: Comparison of each class eigenvector centrality with local eigenvector centrality, i=10, for the Primary School network where circle sizes are proportional to the corresponding centrality values, (b) class eigenvector centrality and PageRank difference with respect to sorted local eigenvector centrality values, and (c) boxplot of the class eigenvector centrality and PageRank differences.
• Figure 5: Centrality analysis of a High School contact network. (a) Visualistion of class membership. (b) Eigengap plot showing prominent eigengap at $i=5$. (c) Comparison of each class eigenvector centrality with local eigenvector centrality for $i=5$. Circle sizes are proportional to the corresponding centrality values, illustrating the relative centrality of nodes according to these measures.