Isospectral Compression and Other Useful Isospectral Transformations of Dynamical Networks

TL;DR

This paper advances the theory of isospectral network transformations, enabling reduction of large networks to smaller spectrally equivalent ones, with applications to parameterized and large-scale networks.

Contribution

It extends previous work by applying isospectral transformations to families of parameterized networks and networks of arbitrary size.

Findings

Networks can be reduced while preserving spectral properties.

Spectral equivalence defines network similarity.

Applicable to large and parameterized networks.

Abstract

It is common knowledge that a key dynamical characteristic of a network is its spectrum (the collection of all eigenvalues of the network's weighted adjacency matrix). In \cite{BW10} we demonstrated that it is possible to reduce a network, considered as a graph, to a smaller network with fewer vertices and edges while preserving the spectrum (or spectral information) of the original network. This procedure allows for the introduction of new equivalence relations between networks, where two networks are spectrally equivalent if they can be reduced to the same network. Additionally, using this theory it is possible to establish whether a network, modeled as a dynamical system, has a globally attracting fixed point (is strongly synchronizing). In this paper we further develop this theory of isospectral network transformations and demonstrate that our procedures are applicable to families of parameterized networks and networks of arbitrary size.