On Attractors of Isospectral Compressions of Networks

TL;DR

This paper explores the behavior of isospectral network compressions, showing they converge to attractors with uniform characteristics, and introduces a new approach for analyzing and comparing network structures based on spectral properties.

Contribution

It introduces a dynamical systems perspective on isospectral network compressions and reveals convergence properties and characteristic-dependent attractors, advancing network analysis methods.

Findings

Orbits of isospectral compressions converge to attractors with uniform node or edge characteristics.

Different characteristics can lead to different or same attractors for the same network.

Networks can be spectrally equivalent under one characteristic but not under another.

Abstract

In the recently developed theory of isospectral transformations of networks isospectral compressions are performed with respect to some chosen characteristic (attribute) of nodes (or edges) of networks. Each isospectral compression (when a certain characteristic is fixed) defines a dynamical system on the space of all networks. It is shown that any orbit of such dynamical system which starts at any finite network (as the initial point of this orbit) converges to an attractor. Such attractor is a smaller network where a chosen characteristic has the same value for all nodes (or edges). We demonstrate that isospectral contractions of one and the same network defined by different characteristics of nodes (or edges) may converge to the same as well as to different attractors. It is also shown that spectrally equivalent with respect to some characteristic networks could be non-spectrally equivalent for another characteristic of nodes (edges). These results suggest a new constructive approach to analysis of networks structures and to comparison of topologies of different networks.