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

2025-01-29

AI Summary

In localized networks—such as core-periphery structures—the reciprocal of the nonbacktracking matrix’s principal eigenvalue severely underestimates the percolation threshold. To address this, we propose using the reciprocal of the subdominant real eigenvalue (associated with a delocalized eigenvector) as a refined threshold estimator. This approach breaks the conventional reliance solely on the principal eigenvalue: we quantify eigenvector localization via the inverse participation ratio (IPR), and rigorously characterize the regime of validity within theoretical models. We systematically validate the estimator across multiple classes of synthetic core-periphery random graphs and over a dozen large-scale real-world networks. Results show that the new estimator reduces mean absolute error by 30–65% compared to classical methods, delivering significantly improved accuracy and robustness. Our work establishes a more reliable spectral framework for percolation analysis in localized network topologies.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Game Theory and Economic Paradigms: Imperfect Information
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Networks
• Percolation Threshold
• Spectral Radius

Innovation

• Nonbacktracking Centrality
• Sub-leading Eigenvalues
• Percolation Threshold Estimation