Understanding the Effect of GCN Convolutions in Regression Tasks

TL;DR

This paper provides a theoretical analysis of how GCN convolutions influence regression tasks, focusing on bias-variance trade-offs and the impact of graph topology, supported by synthetic experiments.

Contribution

It introduces a statistical framework for understanding GCN convolution effects in regression, highlighting the influence of neighborhood size and graph structure.

Findings

Bias-variance trade-off depends on neighborhood size.

Certain graph topologies reduce convolution effectiveness.

Theoretical insights are validated with synthetic data.

Abstract

Graph Convolutional Networks (GCNs) have become a pivotal method in machine learning for modeling functions over graphs. Despite their widespread success across various applications, their statistical properties (e.g., consistency, convergence rates) remain ill-characterized. To begin addressing this knowledge gap, we consider networks for which the graph structure implies that neighboring nodes exhibit similar signals and provide statistical theory for the impact of convolution operators. Focusing on estimators based solely on neighborhood aggregation, we examine how two common convolutions - the original GCN and GraphSAGE convolutions - affect the learning error as a function of the neighborhood topology and the number of convolutional layers. We explicitly characterize the bias-variance type trade-off incurred by GCNs as a function of the neighborhood size and identify specific graph topologies where convolution operators are less effective. Our theoretical findings are corroborated by synthetic experiments, and provide a start to a deeper quantitative understanding of convolutional effects in GCNs for offering rigorous guidelines for practitioners.