Isospectral Graph Transformations, Spectral Equivalence, and Global Stability of Dynamical Networks

TL;DR

This paper introduces a spectral-preserving graph transformation technique that simplifies networks while maintaining key spectral properties, enabling improved analysis of global stability in dynamical systems.

Contribution

It presents a novel isospectral transformation method for weighted graphs, facilitating network reduction and expansion while preserving spectral characteristics for stability analysis.

Findings

Allows reduction or expansion of networks without losing spectral information

Provides stronger results on global stability and synchronization of dynamical networks

Establishes new equivalence relations among weighted graphs based on spectral properties

Abstract

In this paper we present a general procedure that allows for the reduction or expansion of any network (considered as a weighted graph). This procedure maintains the spectrum of the network's adjacency matrix up to a set of eigenvalues known beforehand from its graph structure. This procedure can be used to establish new equivalence relations on the class of all weighted graphs (networks) where two graphs are equivalent if they can be reduced to the same graph. Additionally, dynamical networks (or any finite dimensional, discrete time dynamical system) can be analyzed using isospectral transformations. By so doing we obtain stronger results regarding the global stability (strong synchronization) of dynamical networks when compared to other standard methods.