Circulant Type Formulas for the Eigenvalues of Linear Network Maps

TL;DR

This paper introduces network multipliers, matrices that determine the eigenvalues of linear admissible maps for homogeneous networks, providing a systematic way to analyze network spectra through these small, linearly independent matrices.

Contribution

It defines network multipliers and shows they determine eigenvalues of admissible maps for a class of networks, simplifying spectral analysis.

Findings

Eigenvalues of admissible maps are given by network multipliers.

Network multipliers depend linearly on response function coefficients.

Eigenvalues are independent of node phase space dimension.

Abstract

Given an admissible map F for a homogeneous network N, it is known that the Jacobian DF(x) around a fully synchronous point x = (x0, ..., x0) is again an admissible map for N. Motivated by this, we study the spectra of linear admissible maps for homogeneous networks. In particular, we define so-called network multipliers. These are (relatively small) matrices that depend linearly on the coefficients of the response function, and whose eigenvalues together make up the spectrum of the corresponding admissible map. More precisely, given a network N, we define a finite set of network multipliers L1 to Lk and a class of networks C containing N. This class is furthermore closed under taking quotient networks, subnetworks and disjoint unions. We then show that the eigenvalues of an admissible map for any network in C are given by those of (a subset of) the network multipliers, with fixed multiplicities m1 to mk and independently of the given (finite dimensional) phase space of a node. The coefficients of all the network multipliers of C are furthermore linearly independent, which implies that one may find the multiplicities m1 to mk by simply expressing the trace of an admissible map as a linear combination of the traces of the multipliers. In particular, we will give examples of networks where the network multipliers need not be constructed, but can be determined by looking at small networks in C. We also show that network multipliers are multiplicative with respect to composition of linear maps.