On the Stability of Graph Convolutional Neural Networks under Edge Rewiring

Abstract

Graph neural networks are experiencing a surge of popularity within the machine learning community due to their ability to adapt to non-Euclidean domains and instil inductive biases. Despite this, their stability, i.e., their robustness to small perturbations in the input, is not yet well understood. Although there exists some results showing the stability of graph neural networks, most take the form of an upper bound on the magnitude of change due to a perturbation in the graph topology. However, these existing bounds tend to be expressed in terms of uninterpretable variables, limiting our understanding of the model robustness properties. In this work, we develop an interpretable upper bound elucidating that graph neural networks are stable to rewiring between high degree nodes. This bound and further research in bounds of similar type provide further understanding of the stability properties of graph neural networks.

On The Stability of Polynomial Spectral Graph Filters

“…A similar bound has been recently derived to bound the change in feature representations of certain graph neural network architectures Kenlay et al (2020a).…”

“…It was recently proved that polynomial filters are linearly stable with respect to the shifted normalised Laplacian matrix L − I n Kenlay et al (2020b). A simpler proof with a tighter bound (smaller stability constant) was given to show linear stability with respect to the augmented adjacency matrix D−1/2 à D−1/2 where à = A + I n and D = D + I n Kenlay et al (2020a). In addition, the following more general result holds.…”

“…The work most related to ours is by Kenlay et al (2020a) as it provides structurally interpretable bound. The authors show that under degree preserving edge rewiring the change of spectral graph filters applied to the augmented adjacency matrix depends on the locality of edge rewiring and the degree of the endpoints of the edges being rewired.…”

“…Degree preserving rewiring is a type of edge rewiring such that the perturbation does not change the original degree distribution Kenlay et al (2020a). Given two edges u ∼ v and u ∼ v such that u ∼ v , u ∼ u , v ∼ v and v ∼ u , the double edge rewiring operation deletes the two edges and introduces the edges u ∼ u and v ∼ v (see Fig.…”

“…This limitation hampers the design of strategies that could defend efficiently against potential adversaries. A notable exception in the literature is the recent work of Kenlay et al (2020a); however, this work only considers degree preserving edge rewiring which is a stringent assumption that does not cover many perturbations observed in practical scenarios.…”