Dynamic Topology Identification and Filtering Techniques for Higher-Order Networks (CYCLONE)

Traditional signal processing techniques focus on regular domains like time-series data and images. However, much of the natural and social world’s data exists on irregular, non-Euclidean domains, such as brain networks or social group interactions. Historically, such irregular structures have been represented by graphs, which capture pair-wise relationships. While graph-based techniques such as the Fourier transform, filtering, and topology identification have advanced significantly, they are limited in representing higher-order interactions inherent in many real-world systems.

Higher-order networks, such as hypergraphs and simplicial complexes, address these limitations. Hypergraphs generalize the concept of edges to connect groups of nodes, while simplicial complexes introduce hierarchical structures to represent signals on nodes, edges, triangles, and beyond. These models enable principled analysis of multi-way relationships, which is essential for understanding complex systems like ecological networks or collaboration dynamics. However, a critical challenge remains: these topologies often need to be inferred from data, especially when they are dynamically evolving over time. Our research focuses on this emerging and underexplored area: dynamically learning higher-order topologies from data. Accurately describing the evolving topology of such networks is vital, as it directly impacts the performance of downstream tasks such as signal filtering, denoising, and prediction. Current techniques for higher-order topology inference are primarily static, with hierarchical methods assuming the availability of lower-order structures and non-hierarchical methods jointly estimating lower- and higher-order structures. While these methods have seen some success, they fall short in scenarios where the topology evolves over time—common in real-world processes such as disease spread, financial transactions, or neural activity.

The CYCLONE project proposes leveraging dynamic state estimation techniques like Kalman and particle filters to infer evolving higher-order topologies. In the second phase, we aim to build filtering techniques tailored to these inferred topologies, enabling signal processing on dynamic higher-order networks. Extending these techniques from graphs to higher-order networks is non-trivial and requires substantial methodological innovation.