Graph Convolutional Neural Networks Sensitivity under Probabilistic Error Model

TL;DR

This paper analyzes how sensitive Graph Convolutional Neural Networks are to probabilistic errors in graph data, providing bounds and demonstrating stability under certain conditions, with validation through experiments.

Contribution

It introduces a framework linking GSO errors to GCNN output differences, establishing stability conditions and analyzing multilayer networks with theoretical bounds and practical examples.

Findings

GSO errors have a linear impact on GCNN output differences.

Single-layer GCNNs are stable under bounded GSO errors.

Multilayer GCNNs' output differences follow a recursive linear relationship.

Abstract

Graph Neural Networks (GNNs), particularly Graph Convolutional Neural Networks (GCNNs), have emerged as pivotal instruments in machine learning and signal processing for processing graph-structured data. This paper proposes an analysis framework to investigate the sensitivity of GCNNs to probabilistic graph perturbations, directly impacting the graph shift operator (GSO). Our study establishes tight expected GSO error bounds, which are explicitly linked to the error model parameters, and reveals a linear relationship between GSO perturbations and the resulting output differences at each layer of GCNNs. This linearity demonstrates that a single-layer GCNN maintains stability under graph edge perturbations, provided that the GSO errors remain bounded, regardless of the perturbation scale. For multilayer GCNNs, the dependency of system's output difference on GSO perturbations is shown to be a recursion of linearity. Finally, we exemplify the framework with the Graph Isomorphism Network (GIN) and Simple Graph Convolution Network (SGCN). Experiments validate our theoretical derivations and the effectiveness of our approach.