Understanding over-squashing and bottlenecks on graphs via curvature

TL;DR

This paper analyzes the over-squashing problem in graph neural networks, linking it to graph bottlenecks and negative curvature, and proposes a curvature-based rewiring method to mitigate it.

Contribution

It introduces a new edge-based curvature measure, proves its relation to over-squashing, and develops a curvature-based graph rewiring technique to improve GNN performance.

Findings

Negatively curved edges cause over-squashing.

Curvature-based rewiring alleviates over-squashing.

The method improves information flow in GNNs.

Abstract

Most graph neural networks (GNNs) use the message passing paradigm, in which node features are propagated on the input graph. Recent works pointed to the distortion of information flowing from distant nodes as a factor limiting the efficiency of message passing for tasks relying on long-distance interactions. This phenomenon, referred to as 'over-squashing', has been heuristically attributed to graph bottlenecks where the number of $k$-hop neighbors grows rapidly with $k$. We provide a precise description of the over-squashing phenomenon in GNNs and analyze how it arises from bottlenecks in the graph. For this purpose, we introduce a new edge-based combinatorial curvature and prove that negatively curved edges are responsible for the over-squashing issue. We also propose and experimentally test a curvature-based graph rewiring method to alleviate the over-squashing.