Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems

TL;DR

This paper demonstrates that antagonistic interactions with negative correlations can enhance the stability of large, heterogeneous linear dynamical systems, especially when interaction strength is moderate, challenging previous assumptions about system stability.

Contribution

It provides an exact spectral theory showing how antagonistic interactions can stabilize systems with inhomogeneous growth rates, contrasting with uncorrelated interactions.

Findings

Antagonistic interactions can stabilize large systems.

Systems without interactions are less stable than those with antagonistic interactions.

Stability depends on the strength and correlation of interactions.

Abstract

We analyse the stability of large, linear dynamical systems of variables that interact through a fully connected random matrix and have inhomogeneous growth rates. We show that in the absence of correlations between the coupling strengths, a system with interactions is always less stable than a system without interactions. Contrarily to the uncorrelated case, interactions that are antagonistic, i.e., characterised by negative correlations, can stabilise linear dynamical systems. In particular, when the strength of the interactions is not too strong, systems with antagonistic interactions are more stable than systems without interactions. These results are obtained with an exact theory for the spectral properties of fully connected random matrices with diagonal disorder.