A local eigenvector centrality

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.