The low-rank hypothesis of complex systems

Abstract

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.

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