Communities in the Kuramoto Model: Dynamics and Detection via Path Signatures

2025-03-21

AI Summary

To address the limited interpretability of structural connectivity inference from high-dimensional dynamical time series—such as neural activity—the paper introduces a novel structure learning framework grounded in path signatures. It pioneers the application of path signatures to coupled oscillator systems, integrating mean-field theory and Gaussian approximation to derive reduced dynamics and a lead-lag relationship matrix for a multiscale stochastic Kuramoto block model (KSBM). Building on this, the authors design a signature-driven algorithm with provable exact community recovery guarantees. Evaluated on multiple KSBM benchmarks, the method substantially outperforms conventional correlation- and spectral-based approaches, achieving error-free community reconstruction. This work establishes a new paradigm for structural inference in complex systems—particularly neuroscientific data—uniquely combining geometric interpretability, temporal sensitivity, and rigorous theoretical guarantees.

Technology Category

• Machine Learning: Structured Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

The behavior of multivariate dynamical processes is often governed by underlying structural 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 particular, 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 different 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. 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.

Problem

• Infer structural connections from complex high-dimensional time series
• Characterize lead-lag phenomena in dynamical systems using path signatures
• Detect structural communities in Kuramoto model via signature-based algorithm

Innovation

• Path signatures encode geometric and temporal properties
• Lead matrix reveals lead-lag phenomena
• Signature-based algorithm recovers structural communities