Percolation and localisation: Sub-leading eigenvalues of the nonbacktracking matrix

TL;DR

This paper explores how analyzing sub-leading eigenvalues of the nonbacktracking matrix improves percolation threshold estimates in networks with localized eigenvectors, especially in core-periphery structures.

Contribution

It introduces a spectral method focusing on sub-leading eigenvalues to better estimate percolation thresholds in localized networks, surpassing traditional leading eigenvalue approaches.

Findings

Sub-leading eigenvalues provide more accurate threshold estimates in localized networks.

Theoretical and experimental validation on model and real-world networks.

Approach improves understanding of percolation in core-periphery structures.

Abstract

The spectrum of the nonbacktracking matrix associated to a network is known to contain fundamental information regarding percolation properties of the network. Indeed, the inverse of its leading eigenvalue is often used as an estimate for the percolation threshold. However, for many networks with nonbacktracking centrality localised on a few nodes, such as networks with a core-periphery structure, this spectral approach badly underestimates the threshold. In this work, we study networks that exhibit this localisation effect by looking beyond the leading eigenvalue and searching deeper into the spectrum of the nonbacktracking matrix. We identify that, when localisation is present, the threshold often more closely aligns with the inverse of one of the sub-leading real eigenvalues: the largest real eigenvalue with a "delocalised" corresponding eigenvector. We investigate a core-periphery network model and determine, both theoretically and experimentally, a regime of parameters for which our approach closely approximates the threshold, while the estimate derived using the leading eigenvalue does not. We further present experimental results on large scale real-world networks that showcase the usefulness of our approach.