Energy-Balanced Hyperspherical Graph Representation Learning via Structural Binding and Entropic Dispersion

Graph Representation Learning (GRL) can be fundamentally modeled as a physical process of seeking an energy equilibrium state for a node system on a latent manifold. However, existing Graph Neural Networks (GNNs) often suffer from uncontrolled energy dissipation during message passing, driving the system towards a state of Thermal Death--manifested as feature collapse or over-smoothing--due to the absence of explicit thermodynamic constraints. To address this, we propose HyperGRL, a thermodynamics-driven framework that embeds nodes on a unit hypersphere by minimizing a Helmholtz free energy objective composed of two competing potentials. First, we introduce Structural Binding Energy (via Neighbor-Mean Alignment), which functions as a local binding force to strengthen structural cohesion, encouraging structurally related nodes to form compact local clusters. Second, to counteract representation collapse, we impose a Mean-Field Repulsive Potential (via Sampling-Free Uniformity), which acts as a global entropic force to maximize representation dispersion without the need for negative sampling. Crucially, to govern the trade-off between local alignment and global uniformity, we devise an Adaptive Thermostat. This entropy-guided strategy dynamically regulates the system's "temperature" during training, guiding the representation towards a robust metastable state that balances local cohesion with global discriminability. Extensive experiments on node classification, node clustering, and link prediction show that HyperGRL consistently achieves strong performance across diverse benchmark datasets, yielding more discriminative and robust representations while alleviating over-smoothing.