Communities in the Kuramoto model: dynamics and detection via path signatures

The behavior of multivariate dynamical processes is often governed by underlying struc- tural connections that relate the components of the system. For example, brain activity, which is often measured via time series is determined by an underlying structural graph, where nodes represent neurons or brain regions and edges represent cortical connectivity. Existing methods for inferring structural connections from observed dynamics, such as correlation-based or spectral techniques, may fail to fully capture complex relationships in high-dimensional time series in an interpretable way. Here, we propose the use of path signatures—a mathematical framework that encodes geometric and temporal properties of continuous paths—to address this problem. Path signatures provide a reparametrization-invariant characterization of dynamical data and, in partic- ular, can be used to compute the lead matrix, which reveals lead-lag phenomena. We showcase our approach on time series from coupled oscillators in the Kuramoto model defined on a stochastic block model graph, termed the Kuramoto Stochastic Block Model (KSBM). Using mean-field theory and Gaussian approximations, we analytically derive reduced models of KSBM dynamics in differ- ent temporal regimes and theoretically characterize the lead matrix in these settings. Leveraging these insights, we propose a novel signature-based community detection algorithm, achieving exact recovery of structural communities from observed time series in multiple KSBM instances. We also explored the performance of our community detection on a stochastic variant of the KSBM as well as on real neuropixels of cortical recordings to demonstrate applicability on real-world data. Our results demonstrate that path signatures provide a novel perspective on analyzing complex neural data and other high-dimensional systems, explicitly exploiting temporal functional relationships to infer underlying structure.