Local-Global Geometric Insights for Graph Neural Networks via Entropic Curvature

2026-07-27 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This work introduces Entropic Curvature, a novel global, transport-based curvature for Graph Neural Networks (GNNs), extending the Lott–Sturm–Villani framework to graphs. Unlike local measures like Ollivier–Ricci or Forman curvature, it addresses GNN issues such as oversmoothing and oversquashing by quantifying long-distance information propagation. A tractable proxy, Weak Entropic Curvature (κw), is defined, computable in O(kiter dmax^2) per node. From this, the authors derive a Poincaré-type inequality controlling oversmoothing, a transport–entropy generalization bound, and an "expansion paradox" unifying oversmoothing and oversquashing. The theory yields three practical mechanisms: the E-Gate aggregator, ENT structural encoding, and Midpoint-Completion Rewiring (MCR). Benchmarks show E-Gate improves performance in 16 of 20 backbone-dataset pairs (p<0.05), ENT outperforms other encodings on 4 of 5 benchmarks, and MCR consistently enhances spectral gaps over ORC and FRC across six graph families. Generalization bounds are established for both positive and negative curvature regimes.

Key takeaway

For Machine Learning Engineers designing robust GNNs, consider integrating entropic curvature to mitigate oversmoothing and oversquashing. This framework provides a unified geometric understanding, allowing you to improve generalization and spectral properties. Implement E-Gate for adaptive aggregation or ENT for structural encoding. You can also use Midpoint-Completion Rewiring to enhance graph connectivity and reduce bottlenecks, especially in sparse networks.

Key insights

Entropic curvature offers a global, transport-theoretic framework to unify and address GNN oversmoothing and oversquashing.

Principles

• Graph curvature can unify oversmoothing and oversquashing.
• Positive entropic curvature controls information diffusion.
• Sparse, spectrally expanding, and positively curved graphs cannot coexist.

Method

Weak Entropic Curvature (κw) is computed node-wise by maximizing a local functional HL(z,S2(z)) over a probability simplex using SLSQP, operationalizing displacement convexity in 2-hop neighborhoods.

In practice

• Use E-Gate aggregator to modulate GNN message passing.
• Employ ENT as a globally-aware structural encoding.
• Apply Midpoint-Completion Rewiring to reduce bottlenecks.

Topics

• Graph Neural Networks
• Entropic Curvature
• Optimal Transport
• Graph Rewiring
• GNN Generalization
• Oversmoothing

Code references

• DSE-MSU/DeepRobust

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.