Exact eigenvalue spectrum of a class of fractal scale-free networks

TL;DR

This paper analytically derives the complete eigenvalue spectrum of a class of fractal scale-free networks and uses it to understand their dynamical properties, such as random walk times and spanning tree counts.

Contribution

It provides a closed-form analytical solution for the eigenvalues and degeneracies of the transition matrix of these networks, a novel result in network spectral analysis.

Findings

Exact eigenvalues and degeneracies are obtained for the network class.

Eigenvalues are used to compute eigentime for random walks.

Spectrum analysis confirms the number of spanning trees.

Abstract

The eigenvalue spectrum of the transition matrix of a network encodes important information about its structural and dynamical properties. We study the transition matrix of a family of fractal scale-free networks and analytically determine all the eigenvalues and their degeneracies. We then use these eigenvalues to evaluate the closed-form solution to the eigentime for random walks on the networks under consideration. Through the connection between the spectrum of transition matrix and the number of spanning trees, we corroborate the obtained eigenvalues and their multiplicities.