Learning Time-Varying Graph Signals via Koopman

Abstract--A wide variety of real-world data, such as sea measurements, e.g., temperatures collected by distributed sensors and multiple unmanned aerial vehicles (UA V) trajectories, can be naturally represented as graphs, often exhibiting non-Euclidean structures. These graph representations may evolve over time, forming time-varying graphs. Effectively modeling and analyzing such dynamic graph data is critical for tasks like predicting graph evolution and reconstructing missing graph data. In this paper, we propose a framework based on the Koopman autoencoder (KAE) to handle time-varying graph data. Specifically, we assume the existence of a hidden non-linear dynamical system, where the state vector corresponds to the graph embedding of the time-varying graph signals. T o capture the evolving graph structures, the graph data is first converted into a vector time series through graph embedding, representing the structural information in a finite-dimensional latent space. In this latent space, the KAE is applied to learn the underlying non-linear dynamics governing the temporal evolution of graph features, enabling both prediction and reconstruction tasks. A. Motivation Graphs are fundamental data structures for modeling the structure and interactions within complex systems [1] across a variety of domains, including, but not limited to, social networks [2], biological systems [3], transportation networks [4], and communication systems [5]. These data structures provide a versatile framework for representing relationships and dependencies, enabling insights into the organization and behavior of complex systems. In many real-world applications, the underlying graph data is not static; instead they evolve over time. Time-varying graphs [6] are a type of graph data characterized by temporal variations in their components or overall configuration. Unlike the commonly studied static graph structures, analyzing time-varying graph data introduces additional challenges. While the reconstruction of graph signals is necessary for recovering missing information, which is common in real-world sensor networks or data transmission scenarios, prediction, on the other hand, enables forecasting the future states of the systems and thus supports planning, decision-making, and control in dynamical environments. S. Krishnan and J. Choi are with the School of Electrical and Mechanical Engineering, The University of Adelaide, Australia (Emails:{jinho.choi,sivaram.krishan}@adelaide.edu.au), and J. Park is with the Information Systems Technology and Design Pillar, Singapore University of Technology and Design, Singapore (Email: jihong park@sutd.edu.sg).