  Complex systems are high-dimensional nonlinear dynamical systems with
intricate interactions among their constituents. To make interpretable
predictions about their large-scale behavior, it is typically assumed, without
a clear statement, that these dynamics can be reduced to a few number of
equations involving a low-rank matrix describing the network of interactions --
what we call the low-rank hypothesis. Our paper sheds light on this assumption
and questions its validity. By leveraging fundamental theorems on singular
value decomposition, we expose the hypothesis for various random graphs, either
by making explicit their low-rank formulation or by demonstrating the
exponential decrease of their singular values. Notably, we verify the
hypothesis experimentally for real networks by revealing the rapid decrease of
their singular values, which has major consequences on their effective ranks.
We then evaluate the impact of the low-rank hypothesis for general dynamical
systems on networks through an optimal dimension reduction. This allows us to
prove that recurrent neural networks can be exactly reduced, and to connect the
rapidly decreasing singular values of real networks to the dimension reduction
error of the nonlinear dynamics they support, be it microbial, neuronal or
epidemiological. Finally, we prove that higher-order interactions naturally
emerge from the dimension reduction, thus providing theoretical insights into
the origin of higher-order interactions in complex systems.
