Eigenvector localization in hypergraphs: pair-wise vs higher-order links

TL;DR

This paper investigates how pair-wise and higher-order links influence eigenvector localization in hypergraph Laplacians, revealing their distinct roles in different eigenvalue regimes and implications for dynamical processes.

Contribution

It introduces a parameter b3 to quantify the impact of pair-wise versus higher-order interactions on eigenvector localization in hypergraphs, providing new insights into their roles.

Findings

Higher-order interactions induce localization in larger eigenvalues for b3 < 1.

Pair-wise links cause localization in small eigenvalues for b3 > 1.

Results enhance understanding of diffusion and random walks in complex systems with higher-order interactions.

Abstract

Localization behaviours of Laplacian eigenvectors of complex networks provide understanding to various dynamical phenomena on the corresponding complex systems. We numerically investigate role of hyperedges in driving eigenvector localization of hypergraphs Laplacians. By defining a single parameter \gamma which measures the relative strengths of pair-wise and higher-order interactions, we analyze the impact of interactions on localization properties. For, \gamma < 1 there exists no impact of pairwise links on eigenvector localization while the higher-order interactions instigate localization in the larger eigenvalues. For \gamma > 1, pair-wise interactions cause localization of eigenvector corresponding to small eigenvalues, where as higherorder interactions, despite being much lesser than the pair-wise links, keep driving localization of the eigenvectors corresponding to larger eigenvalues. The results will be useful to understand dynamical phenomena such as diffusion, and random walks on a range of real-world complex systems having higher-order interactions.