Spectral properties of the hierarchical product of graphs

TL;DR

This paper analyzes the eigenvalue spectrum of the adjacency matrix of the hierarchical product of graphs, providing asymptotic results and applications to epidemic modeling.

Contribution

It introduces a detailed spectral analysis of the hierarchical graph product, including asymptotic eigenvalue limits and their implications for network dynamics.

Findings

Derived exact limit points for eigenvalues in small and large coupling regimes.

Provided asymptotic formulas for eigenvalues based on component graphs.

Applied theory to predict epidemic thresholds in network models.

Abstract

The hierarchical product of two graphs represents a natural way to build a larger graph out of two smaller graphs with less regular and therefore more heterogeneous structure than the Cartesian product. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. Introducing a coupling parameter describing the relative contribution of each of the two smaller graphs, we perform an asymptotic analysis for the full spectrum of eigenvalues of the adjacency matrix of the hierarchical product. Specifically, we derive the exact limit points for each eigenvalue in the limits of small and large coupling, as well as the leading-order relaxation to these values in terms of the eigenvalues and eigenvectors of the two smaller graphs. Given its central roll in the structural and dynamical properties of networks, we study in detail the Perron-Frobenius, or largest, eigenvalue. Finally, as an example application we use our theory to predict the epidemic threshold of the Susceptible-Infected-Susceptible model on a hierarchical product of two graphs.