  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.
