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Depth-Adaptive Graph Neural Networks via Learnable Bakry-'Emery Curvature

Asela Hevapathige, Ahad N. Zehmakan, Qing Wang

TL;DR

The paper tackles the challenge of leveraging geometric graph properties to enhance GNNs by introducing a learnable Bakry-Émery curvature estimator and a depth-adaptive mechanism that assigns per-vertex message-passing depth based on curvature. By connecting curvature to diffusion dynamics, the authors establish a theoretical link to feature distinctiveness and provide a scalable learning framework that jointly optimizes curvature estimation and task performance with a curvature-guided loss. Empirically, the approach yields consistent improvements across vertex and graph tasks on a wide range of datasets, while reducing oversmoothing and improving stability. The work offers a principled, diffusion-aware means to allocate computation where it benefits learning most, potentially impacting a broad set of graph-based applications.

Abstract

Graph Neural Networks (GNNs) have demonstrated strong representation learning capabilities for graph-based tasks. Recent advances on GNNs leverage geometric properties, such as curvature, to enhance its representation capabilities by modeling complex connectivity patterns and information flow within graphs. However, most existing approaches focus solely on discrete graph topology, overlooking diffusion dynamics and task-specific dependencies essential for effective learning. To address this, we propose integrating Bakry-Émery curvature, which captures both structural and task-driven aspects of information propagation. We develop an efficient, learnable approximation strategy, making curvature computation scalable for large graphs. Furthermore, we introduce an adaptive depth mechanism that dynamically adjusts message-passing layers per vertex based on its curvature, ensuring efficient propagation. Our theoretical analysis establishes a link between curvature and feature distinctiveness, showing that high-curvature vertices require fewer layers, while low-curvature ones benefit from deeper propagation. Extensive experiments on benchmark datasets validate the effectiveness of our approach, showing consistent performance improvements across diverse graph learning tasks.

Depth-Adaptive Graph Neural Networks via Learnable Bakry-'Emery Curvature

TL;DR

The paper tackles the challenge of leveraging geometric graph properties to enhance GNNs by introducing a learnable Bakry-Émery curvature estimator and a depth-adaptive mechanism that assigns per-vertex message-passing depth based on curvature. By connecting curvature to diffusion dynamics, the authors establish a theoretical link to feature distinctiveness and provide a scalable learning framework that jointly optimizes curvature estimation and task performance with a curvature-guided loss. Empirically, the approach yields consistent improvements across vertex and graph tasks on a wide range of datasets, while reducing oversmoothing and improving stability. The work offers a principled, diffusion-aware means to allocate computation where it benefits learning most, potentially impacting a broad set of graph-based applications.

Abstract

Graph Neural Networks (GNNs) have demonstrated strong representation learning capabilities for graph-based tasks. Recent advances on GNNs leverage geometric properties, such as curvature, to enhance its representation capabilities by modeling complex connectivity patterns and information flow within graphs. However, most existing approaches focus solely on discrete graph topology, overlooking diffusion dynamics and task-specific dependencies essential for effective learning. To address this, we propose integrating Bakry-Émery curvature, which captures both structural and task-driven aspects of information propagation. We develop an efficient, learnable approximation strategy, making curvature computation scalable for large graphs. Furthermore, we introduce an adaptive depth mechanism that dynamically adjusts message-passing layers per vertex based on its curvature, ensuring efficient propagation. Our theoretical analysis establishes a link between curvature and feature distinctiveness, showing that high-curvature vertices require fewer layers, while low-curvature ones benefit from deeper propagation. Extensive experiments on benchmark datasets validate the effectiveness of our approach, showing consistent performance improvements across diverse graph learning tasks.

Paper Structure

This paper contains 42 sections, 4 theorems, 28 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Curvature-based GNNs
  4. GNNs with Adaptive Architectures
  5. Background
  6. Graph Neural Networks
  7. Bakry-Émery Curvature Formulation
  8. Curvature and Information Propagation
  9. Depth-Adaptive GNNs
  10. Adaptive Layer Depth Mechansim
  11. Learning to Estimate Curvature
  12. GNN Training Loss Function
  13. Theoretical Analysis
  14. Complexity Analysis
  15. Feature Distinctiveness and Curvature
  16. ...and 27 more sections

Key Result

theorem 1

Let $G = (V, E, w)$ be an undirected, weighted graph with bounded degree and Laplacian $L$. If the local curvature-dimension inequality holds for every $f \in \ell^2(V)$ at vertex $x$, then for any $\epsilon \in (0,1)$, the local mixing time satisfies

Figures (8)

  • Figure 1: High-level overview of the proposed model: Vertices are clustered based on learned curvature, where message-passing depth is adaptively adjusted according to curvature. Both the curvature functions and the GNN are jointly optimized via a loss function.
  • Figure 2:
  • Figure 3:
  • Figure 5:
  • Figure 6:
  • ...and 3 more figures

Theorems & Definitions (4)

  • theorem 1: Mixing Time Bound
  • theorem 2: Feature Decay Bound
  • theorem 2: Mixing Time Bound
  • theorem 2: Feature Decay Bound