Design Your Own Universe: A Physics-Informed Agnostic Method for Enhancing Graph Neural Networks

TL;DR

This work addresses the persistent challenges of over-smoothing, over-squashing, and heterophily in graph neural networks by introducing a physics-inspired, model-agnostic framework that augments graphs with Collapsing Nodes (CNs) connected to labeled data with signed edges. CNs act as gravitational sources that induce attractive forces among same-label nodes and repulsive forces across different labels, enabling both diffusion and rewiring that improve information flow and label separation. The authors provide theoretical analyses showing OSM can be avoided and OSQ mitigated, aided by the appearance of negative Laplacian eigenvalues that enable both smoothing and sharpening dynamics, along with curvature-based arguments linking CN-induced structure to OSQ trade-offs. Empirical results across homophilic, heterophilic, and long-range graph benchmarks demonstrate significant performance gains for CN-enhanced GNNs (UYGCN, UYGAT) over strong baselines, validating the approach as a broadly applicable enhancement that scales to large graphs and deep architectures.

Abstract

Physics-informed Graph Neural Networks have achieved remarkable performance in learning through graph-structured data by mitigating common GNN challenges such as over-smoothing, over-squashing, and heterophily adaption. Despite these advancements, the development of a simple yet effective paradigm that appropriately integrates previous methods for handling all these challenges is still underway. In this paper, we draw an analogy between the propagation of GNNs and particle systems in physics, proposing a model-agnostic enhancement framework. This framework enriches the graph structure by introducing additional nodes and rewiring connections with both positive and negative weights, guided by node labeling information. We theoretically verify that GNNs enhanced through our approach can effectively circumvent the over-smoothing issue and exhibit robustness against over-squashing. Moreover, we conduct a spectral analysis on the rewired graph to demonstrate that the corresponding GNNs can fit both homophilic and heterophilic graphs. Empirical validations on benchmarks for homophilic, heterophilic graphs, and long-term graph datasets show that GNNs enhanced by our method significantly outperform their original counterparts.

Figures (3)

• Figure 1: Model performance under the layer increases.
• Figure 2: Learning Accuracy of UYGCN with different number of CNs.
• Figure 3: Learning Accuracy (F1 score) of UYGCN, UYGAT and their original models, GCN and GAT on long-term graph benchmarks.

Theorems & Definitions (11)

• Definition 1: Homophily and Heterophily
• Definition 2: Collapsing Nodes
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• proof
• Proposition 3
• proof : Proof of Proposition \ref{['UYGNN_OSQ']}