Graph Convolutional Neural Networks Sensitivity under Probabilistic Error Model

TL;DR

This paper proposes an analysis framework to investigate the sensitivity of GCNNs to probabilistic graph perturbations, directly impacting the graph shift operator (GSO), and reveals a linear relationship between GSO perturbations and the resulting output differences at each layer of GCNNs.

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.

Figures (8)

• Figure 1: Visual representation of the probabilistic graph error model applied to a random geometric graph. From left to right: (a) Original graph; (b) Graph after edge deletions ($\epsilon_1=0.3,\epsilon_2=0$); (c) Graph after edge additions ($\epsilon_1=0,\epsilon_2=0.1$); (d) Graph after both edge deletions and additions ($\epsilon_1=0.3,\epsilon_2=0.1$). Deleted edges are marked in red and added edges are marked in blue. The transformations effectively illustrate the impact of perturbations modeled by \ref{['eq:basic_ermodel']}.
• Figure 2: Comparative analysis of our bound in Theorem \ref{['thm:Case1Adj']}, the deterministic bound in Theorem 2 of Wang22-Eusipco, and the empirical GSO distance in $\ell_2$ norm.
• Figure 3: Theoretical (bound in Thm. \ref{['thm:Case1Adj']}) and empirical bounds ($\ell_1$ and $\ell_2$ norms) for the perturbed Cora graph with ${\mathbf S} = {\mathbf A}$. Left panel: varying $\epsilon_1$ with fixed $\epsilon_2=0$. Right panel: varying $\epsilon_2$ with fixed $\epsilon_1=0.5$.
• Figure 4: Theoretical (bound in Prop. \ref{['prop:Case2NorAdj']}) and empirical bounds ($\ell_1$ and $\ell_2$ norms) for the perturbed Cora graph with ${\mathbf S} = {\mathbf A}_\textrm{n}$, under identical $(\epsilon_1, \epsilon_2)$ settings as Fig. \ref{['fig:theo_bdcrr_case1']}.
• Figure 5: Comparison of Theorem \ref{['thm:gfdistance']} bounds (solid lines) and empirical GF distances (scatter points) with fixed $\epsilon_2=0.05$ and varying $\epsilon_1$ in $[0.1,0.2,0.3]$.

Theorems & Definitions (9)

• Remark 1: Why not use $\ell_2$ norm?
• proof
• proof
• proof
• proof
• proof : Proof of Lemma \ref{['lmm:EuNormalized']}
• proof : Proof of Theorem \ref{['thm:gfdistance']}
• proof : Proof of Theorem \ref{['thm:gcnsensitivity']}
• proof