Collapse of Resilience Patterns in Generalized Lotka-Volterra Dynamics and Beyond

TL;DR

This paper critically examines the limitations of a recent theoretical framework for simplifying multi-dimensional systems, revealing new conditions affecting its validity and extending its applicability to systems with mixed interaction signs.

Contribution

It demonstrates that the existing framework's assumptions are not universally valid, introduces a new limiting condition, and extends the approach to systems with mixed positive and negative interactions.

Findings

The framework's validity depends on specific network properties.

A new condition limits the framework's applicability.

The approach can be extended to systems with mixed interaction signs.

Abstract

Recently, a theoretical framework aimed at separating the roles of dynamics and topology in multi-dimensional systems has been developed (Gao et al, \textit{Nature}, Vol 530:307 (2016)). The validity of their method is assumed to hold depending on two main hypothesis: $(i)$ The network determined by the the interaction between pairs of nodes has negligible degree correlations; $(ii)$ The node activities are uniform across nodes on both the drift and pair-wise interaction functions. Moreover, the authors consider only positive (mutualistic) interactions. Here we show the conditions proposed by Gao and collaborators are neither sufficient nor necessary to guarantee that their method works in general, and validity of their results are not independent of the model chosen within the class of dynamics they considered. Indeed we find that a new condition poses effective limitations to their framework and we provide quantitative predictions of the quality of the one dimensional collapse as a function of the properties of interaction networks and stable dynamics using results from random matrix theory. We also find that multi-dimensional reduction may work also for interaction matrix with a mixture of positive and negative signs, opening up application of the framework to food-webs, neuronal networks and social/economic interactions.