Inferring Higher-Order Interactions from Dynamics via Hodge-based message passing

Higher-order networks represent group interactions that shape collective dynamics. The joint influence of pairwise and higher-order interactions on node evolution makes inferring these structures from observed node states challenging. In this work, we propose SCHOD (Simplicial Complex Inference with Hodge-based Dynamic Modeling), a framework for jointly inferring pairwise and 2-simplex interactions without prior knowledge of the underlying dynamical equations. The Hodge $1$-Laplacian captures lower- and higher-order relations among edges in a simplicial complex. SCHOD builds on this property by incorporating learnable structural probabilities into its lower and upper components, enabling edge-to-edge message passing for structure inference. To account for node-level variation in the contributions of pairwise and $2$-simplex interactions, an adaptive gating mechanism modulates their aggregated messages for state prediction. Experiments on synthetic networks demonstrate the effectiveness of SCHOD in recovering higher-order interactions across different network types.