Learning Representations over Temporal Simplicial Complexes

Temporal higher-order networks can be used to describe time-evolving complex systems in various domains, including biological, social, and collaboration networks, where groups of entities interact over time. Learning representations for these networks requires jointly modeling temporal evolution and higher-order topology while efficiently processing a continuous stream of interaction. Existing approaches typically rely on either clique-expanded graphs or hypergraphs: graph-based models offer computational efficiency but often fail to capture higher-order dependencies, whereas hypergraph-based models represent such dependencies explicitly at the expense of increased computational complexity and limited theoretical expressivity. To address these challenges, we introduce the Temporal Simplicial Complex (TSC), a formalism for representing time-evolving higher-order interactions as an event-driven sequence of simplicial complexes. We define TSC isomorphism and develop the Temporal Simplicial Weisfeiler-Lehman (TSWL) test to characterize the expressive power of temporal simplicial representation learning. Building on this framework, we propose the Temporal Simplicial State Space Model ($\text{TS³M}$), an online representation learning architecture that uses state space models to efficiently propagate information over the TSC. We establish an expressiveness bound for $\text{TS³M}$ in terms of its state dimension, showing that its expressive power increases with the state dimension and reaches that of the TSWL test in the infinite-dimensional limit. Numerical experiments on dynamic higher-order interaction prediction across standard benchmark datasets show that $\text{TS³M}$ achieves competitive predictive performance while delivering substantially faster training and inference compared to existing approaches.