Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems

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.

Automated summary

We analyze the stability of large, linear dynamical systems with inhomogeneous growth rates that interact through a fully connected random matrix. Our findings show that in the absence of correlations between the coupling strengths, systems with interactions are always less stable than systems without interactions. However, interactions that are antagonistic, characterized by negative correlations, can stabilize linear dynamical systems. Particularly, 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 using an exact theory for the spectral properties of fully connected random matrices with diagonal disorder.