Beyond Oversquashing: Understanding Signal Propagation in GNNs Via Observables

TL;DR

This paper introduces a quantum-inspired framework using observables to analyze and improve signal propagation in GNNs, addressing issues like oversquashing and oversmoothing.

Contribution

It models signal localization and flow in GNNs via observables, proving standard spectral GNNs are limited and proposing a Schrödinger GNN with enhanced routing capabilities.

Findings

Standard spectral GNNs have poor signal propagation.

Schrödinger GNNs demonstrate superior signal routing.

The quantum-inspired model offers new insights into GNN behavior.

Abstract

Graph Neural Networks (GNNs) perform computations on graphs by routing the signal between graph regions using a graph shift operator or a message passing scheme. Often, the propagation of the signal leads to a loss of information, where the signal tends to diffuse across the graph instead of being deliberately routed between regions of interest. Two notions that depict this phenomenon are oversmoothing and oversquashing. In this paper, we propose an alternative approach for modeling signal propagation, inspired by quantum mechanics, using the notion of observables. Specifically, we model the place in the graph where the signal lies, how much the signal is concentrated there, and how much of the signal is propagated towards a location of interest when applying a GNN. Using these new concepts, we prove that standard spectral GNNs have poor signal propagation capabilities. We then propose a new type of spectral GNN, termed Schr\"odinger GNN, which we show has a superior capacity to route the signal across the graph.