Inference of the size of nonlinear network systems from perceptible dynamics

Network dynamical systems are ubiquitous in science and engineering. The most basic property of a network dynamical system is its size, which, for scalar dynamics, corresponds to the number of nodes. For linear network systems, recent studies have developed reliable tools for inferring the size of the system from perceptible dynamics (measurements of one or some of the network nodes) across multiple experiments. Here, we extend these tools to nonlinear network systems by putting forward a model-agnostic approach that combines clustering techniques, the use of detection matrices, and spectral analysis. The theoretical premise of the algorithm is that, under mild assumptions, the variation between the dynamics of some nodes across multiple measurements can be used to bound the variation between the dynamics of all nodes across the same measurements. By applying clustering techniques on perceptible dynamics, we identify nearby measurements, about which the variational dynamics are approximately linear and the use of the detection matrix is valid. From the spectrum of the detection matrix, we infer its rank, which corresponds to the size of the nonlinear network system. We demonstrate our approach via numerical experiments on different nonlinear network systems, including different types of hypergraphs. Whether nonlinearity comes from individual dynamics of the nodes or the interactions among them, it is rarely a feature that one can dismiss. Our work paves the way to infer the size of a nonlinear network system when governing equations are unknown and only limited data are accessible.