New results on the stability of large antagonistic systems on complex networks
: a random matrix approach

Student thesis: Doctoral ThesisDoctor of Philosophy

Abstract

We study the local stability of dynamical systems on complex networks with a random matrix approach. In this approach the Jacobian of the linearised dynamics at the fixed points is described by a sparse random matrix. Inspired by the ecological food webs, we consider degrees of freedom that have pairwise correlated interactions that can be predator-prey, competitive, or mutualistic. Matrices where all interactions are of the predator-prey type define the antagonistic ensemble, while matrices with all the three kinds of interactions define the mixture ensemble. We develop an exact theory to eval-uate the spectral properties of infinitely large sparse random matrices with pairwise correlated interactions, and use this theory to infer the system stability. This theory shows that the kind of the interactions plays a major role for the stability of sparse systems. In particular, we find that the leading eigenvalue of infinitely large antagonistic matrices is finite, while the leading eigenvalue of mixture ones is not. This implies that the fixed points of infinitely large antagonistic matrices can be stable, while mixture ones are unstable. In addition, it emerges that degree fluctuations in the network topology typically provide further stabilising effects for infinitely large antagonistic systems. Finally, we find a peculiar behaviour in the spectra of antagonistic matrices at small values of the mean degree, which we denominate the reentrance effect. The leading eigenvalue is imaginary in this case and consequently, as illustrated, the dynamical recovery to a fixed point of the corresponding antagonistic system is typically oscillatory. The reen-trance effect characterises a continuous phase transition from a region where the recovery is oscillatory for small enough mean degrees to a phase with a monotonic response for larger connectivities, as instead the leading eigenvalue is real.
Date of Award1 Aug 2021
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorIzaak Neri (Supervisor) & Chiara Cammarota (Supervisor)

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