Energy-Driven Higher-Order Graph State Evolution for Brain Network Analysis

Characterizing group-wise interactions among brain regions is important for understanding the behaviors of brain disorders. While conventional graph learning methods propagate information through pairwise edges, higher-order graph models introduce group-wise structure but commonly use it as an aggregation pattern under predefined propagation rules. Physics-inspired graph models instead derive state evolution from energy-based dynamics, but existing formulations are built from node-wise or pairwise interactions on fixed graphs. In this regard, we introduce HOPE, a higher-order graph learning framework that discovers instance-specific latent factors with sparse soft memberships and converts their group-wise interactions into factor-wise energies governing node-state evolution. HOPE alternates factor inference with structure-aware gradient-flow updates, allowing higher-order interaction structures and node states to co-evolve, while each fixed-factor state evolution follows dissipative continuous-time dynamics. Experiments on diverse brain networks show that HOPE consistently improves disease prediction across neurodegenerative and neurodevelopmental disorders. Moreover, the learned factors and energies identify disease-relevant ROI groups, revealing structured higher-order interactions associated with heterogeneous brain disorders.