From Spectra to Localized Networks: A Reverse Engineering Approach

TL;DR

This paper develops an analytical scheme to construct highly localized networks based on spectral properties, revealing sensitivity of eigenvector localization to edge rewiring and eigenvalue crossing phenomena, with implications for understanding various dynamical processes.

Contribution

The paper introduces a novel analytical method to design localized networks and explores the spectral sensitivity and eigenvalue crossing phenomena related to eigenvector localization.

Findings

Localization of the principal eigenvector is sensitive to single edge rewiring.

Eigenvalue crossing phenomena are observed due to edge modifications.

The approach aids in understanding steady-state behaviors in dynamical processes.

Abstract

Understanding the localization properties of eigenvectors of complex networks is important to get insight into various structural and dynamical properties of the corresponding systems. Here, we analytically develop a scheme to construct a highly localized network for a given set of networks parameters that is the number of nodes and the number of interactions. We find that the localization behavior of the principal eigenvector (PEV) of such a network is sensitive against a single edge rewiring. We find evidences for eigenvalue crossing phenomena as a consequence of the single edge rewiring, in turn providing an origin to the sensitive behavior of the PEV localization. These insights were then used to analytically construct the highly localized network for a given set of networks parameters. The analysis provides fundamental insight into relationships between the structural and the spectral properties of networks for PEV localized networks. Further, we substantiate the existence of the eigenvalue crossing phenomenon by considering a linear-dynamical process, namely the ribonucleic acid (RNA) neutral network population dynamical model. The analysis presented here on model networks aids in understanding the steady-state behavior of a broad range of linear-dynamical processes, from epidemic spreading to biochemical dynamics associated with the adjacency matrices.