Isospectral Reductions of Dynamical Networks

TL;DR

This paper introduces a flexible method for reducing or expanding dynamical networks while preserving their spectral properties, enabling easier analysis and visualization of complex network structures.

Contribution

It presents a novel, simple, and general procedure for isospectral reduction of dynamical networks with respect to various network characteristics.

Findings

Preserves the spectrum of the adjacency matrix during reduction.

Allows reduction with respect to centrality, betweenness, etc.

Establishes new equivalence classes of networks based on spectral properties.

Abstract

We present a general and flexible procedure which allows for the reduction (or expansion) of any dynamical network while preserving the spectrum of the network's adjacency matrix. Computationally, this process is simple and easily implemented for the analysis of any network. Moreover, it is possible to isospectrally reduce a network with respect to any network characteristic including centrality, betweenness, etc. This procedure also establishes new equivalence relations which partition all dynamical networks into spectrally equivalent classes. Here, we present general facts regarding isospectral network transformations which we then demonstrate in simple examples. Overall, our procedure introduces new possibilities for the analysis of networks in ways that are easily visualized.