Understanding Oversquashing in GNNs through the Lens of Effective Resistance

TL;DR

This paper analyzes oversquashing in GNNs using effective resistance, proposing a new bound and an algorithm to add edges that reduce oversquashing and improve GNN performance.

Contribution

It introduces the use of total effective resistance as a bound for oversquashing and develops a graph rewiring algorithm to mitigate this issue.

Findings

Total effective resistance bounds oversquashing in GNNs.

Edge addition based on effective resistance improves GNN performance.

Rewiring strategies effectively reduce oversquashing in empirical tests.

Abstract

Message passing graph neural networks (GNNs) are a popular learning architectures for graph-structured data. However, one problem GNNs experience is oversquashing, where a GNN has difficulty sending information between distant nodes. Understanding and mitigating oversquashing has recently received significant attention from the research community. In this paper, we continue this line of work by analyzing oversquashing through the lens of the effective resistance between nodes in the input graph. Effective resistance intuitively captures the ``strength'' of connection between two nodes by paths in the graph, and has a rich literature spanning many areas of graph theory. We propose to use total effective resistance as a bound of the total amount of oversquashing in a graph and provide theoretical justification for its use. We further develop an algorithm to identify edges to be added to an input graph to minimize the total effective resistance, thereby alleviating oversquashing. We provide empirical evidence of the effectiveness of our total effective resistance based rewiring strategies for improving the performance of GNNs.