Circulant type formulas for the eigenvalues of linear network maps

Lee DeVille, Eddie Nijholt

Research output: Contribution to journalArticlepeer-review

Abstract

Given an admissible map γf for a homogeneous network N, it is known that the Jacobian Dγf(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 (Λl)l=1k 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 (ml)l=1k 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 (ml)l=1k 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.

Original languageEnglish (US)
Pages (from-to)379-439
Number of pages61
JournalLinear Algebra and Its Applications
Volume610
DOIs
StateAccepted/In press - 2020

Keywords

  • Admissible map
  • Eigenvalues
  • Homogeneous network
  • Jacobian
  • Linear analysis
  • Networks

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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