Single-Snapshot Inference of Network Couplings from Universal Dynamics at Relative Equilibrium
Many real-world systems can be modelled as complex networks whose collective behaviour is governed by hidden interactions between nodes. Existing methods for inferring these interactions typically require controlled perturbations, time-resolved observations or multiple independent snapshots, all of which are often unavailable in practice. Here we show that class-based coupling strengths can be inferred from a single snapshot of node states when the system is observed close to a relative equilibrium. In this regime, all nodes share a common velocity, which can be absorbed into an effective class bias, transforming the inverse problem into a homogeneous linear system. The coefficients of this linear system are determined entirely by the observed local neighbourhoods and their coupling mechanism, enabling the application to arbitrary known coupling functions. We validate the approach on three different linear and nonlinear dynamical systems, recovering relative class-based couplings and, in special cases, absolute couplings. These results show that spatial heterogeneity can substitute for temporal sampling, enabling single-snapshot inference of hidden coupling strengths in networked dynamical systems.