Title: Understanding Signal Propagation in GNNs Via Observables
Primary area: learning on graphs and other geometries & topologies
Keywords: graph neural networks, spectral methods, unitary shift operator, Schrödinger equation, observables, oversmoothing, oversquashing

Abstract: Graph Neural Networks (GNNs) perform computations on graphs by routing the signal information between regions of the graph 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 at this place, 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 between graph regions.

TL;DR: We propose a theory for modeling the capacity of GNNs to rout signals between regions of the graph. We propose Schrödinger GNN, which has superior routing capabilities than previous methods