Symmetric and Asymmetric Tendencies in Stable Complex Systems

TL;DR

This paper analyzes stability in complex systems using Jacobian eigenvalue bounds, revealing tendencies for asymmetrical mutualistic and competitive relationships and symmetrical trophic relationships, with implications for system stabilization.

Contribution

It introduces eigenvalue bounds for Jacobians to predict relationship tendencies and defines interdependence diversity, linking it to stability in complex systems.

Findings

Stable systems favor asymmetrical mutualistic and competitive relationships.

Trophic relationships tend to be symmetrical in stable systems.

Higher interdependence diversity destabilizes the system, especially in trophic relationships.

Abstract

A commonly used approach to study stability in a complex system is by analyzing the Jacobian matrix at an equilibrium point of a dynamical system. The equilibrium point is stable if all eigenvalues have negative real parts. Here, by obtaining eigenvalue bounds of the Jacobian, we show that stable complex systems will favor mutualistic and competitive relationships that are asymmetrical (non-reciprocative) and trophic relationships that are symmetrical (reciprocative). Additionally, we define a measure called the interdependence diversity that quantifies how distributed the dependencies are between the dynamical variables in the system. We find that increasing interdependence diversity has a destabilizing effect on the equilibrium point, and the effect is greater for trophic relationships than for mutualistic and competitive relationships. These predictions are consistent with empirical observations in ecology. More importantly, our findings suggest stabilization algorithms that can apply very generally to a variety of complex systems.