Sharp Bounds for Poly-GNNs and the Effect of Graph Noise

TL;DR

This paper provides sharp bounds for the classification capabilities of poly-GNNs under a stochastic block model, showing that increasing depth does not improve class separation and that graph noise can negate the benefits of deeper networks.

Contribution

It offers a theoretical analysis of poly-GNNs, demonstrating that network depth does not enhance class separation and quantifies the impact of graph noise on GNN performance.

Findings

Deeper poly-GNNs do not outperform shallow ones in class separation.

Graph noise can dominate signal, reducing the benefit of network depth.

Differences exist between even and odd-layered GNNs in noise propagation.

Abstract

We investigate the classification performance of graph neural networks with graph-polynomial features, poly-GNNs, on the problem of semi-supervised node classification. We analyze poly-GNNs under a general contextual stochastic block model (CSBM) by providing a sharp characterization of the rate of separation between classes in their output node representations. A question of interest is whether this rate depends on the depth of the network $k$, i.e., whether deeper networks can achieve a faster separation? We provide a negative answer to this question: for a sufficiently large graph, a depth $k > 1$ poly-GNN exhibits the same rate of separation as a depth $k=1$ counterpart. Our analysis highlights and quantifies the impact of ``graph noise'' in deep GNNs and shows how noise in the graph structure can dominate other sources of signal in the graph, negating any benefit further aggregation provides. Our analysis also reveals subtle differences between even and odd-layered GNNs in how the feature noise propagates.