Collapse of resilience patterns in generalized Lotka-Volterra dynamics and beyond

Recently, a theoretical framework aimed at separating the roles of dynamics and topology in multidimensional systems has been developed [Gao et al., Nature (London) 530, 307 (2016)10.1038/nature16948]. The validity of their method is assumed to hold depending on two main hypotheses: (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 the pairwise interaction functions. Moreover, the authors consider only positive (mutualistic) interactions. Here we show the conditions proposed by Gao and collaborators [Nature (London) 530, 307 (2016)10.1038/nature16948] 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 multidimensional reduction may work also for an interaction matrix with a mixture of positive and negative signs, opening up an application of the framework to food webs, neuronal networks, and social and economic interactions.