[Paper Review] Interpretable Stability Bounds for Spectral Graph Filters

Henry Kenlay, Dorina Thanou+1 more|arXiv (Cornell University)|Feb 18, 2021

Advanced Graph Neural Networks32 references4 citations

TL;DR

This paper introduces interpretable stability bounds for spectral graph filters under topological perturbations, expressing the change in filter output in terms of endpoint degrees and spatial proximity of deleted/added edges. The key contribution is a theoretically grounded, structurally interpretable bound showing that filters are stable when perturbations target high-degree nodes and are spatially distributed, validated through extensive synthetic experiments on diverse graph models.

ABSTRACT

Graph-structured data arise in a variety of real-world context ranging from sensor and transportation to biological and social networks. As a ubiquitous tool to process graph-structured data, spectral graph filters have been used to solve common tasks such as denoising and anomaly detection, as well as design deep learning architectures such as graph neural networks. Despite being an important tool, there is a lack of theoretical understanding of the stability properties of spectral graph filters, which are important for designing robust machine learning models. In this paper, we study filter stability and provide a novel and interpretable upper bound on the change of filter output, where the bound is expressed in terms of the endpoint degrees of the deleted and newly added edges, as well as the spatial proximity of those edges. This upper bound allows us to reason, in terms of structural properties of the graph, when a spectral graph filter will be stable. We further perform extensive experiments to verify intuition that can be gained from the bound.

Motivation & Objective

• To address the lack of theoretical understanding of spectral graph filter stability under topological perturbations.
• To develop a stability bound that is interpretable in terms of graph structural properties, not just perturbation magnitude.
• To identify sufficient conditions under which spectral graph filters remain robust to edge deletions and additions.
• To validate theoretical insights through extensive experiments on diverse graph models and perturbation strategies.
• To guide the design of robust graph-based machine learning models, especially graph neural networks.

Proposed method

• Derive an upper bound on the change in spectral graph filter output after edge deletions and additions.
• Express the bound in terms of the degrees of the endpoints of perturbed edges and their spatial proximity.
• Use a chain of inequalities involving the graph shift operator and spectral norms to establish the bound.
• Analyze the bound's looseness across different graph models (ER, BA, WS, etc.) and perturbation strategies (random, adversarial, robust).
• Design and evaluate three perturbation strategies: random rewiring, adversarial PGD, and a robust strategy informed by the bound.
• Conduct experiments on synthetic graphs with varying degree distributions to assess the impact of structural properties on filter stability.

Experimental results

Research questions

• RQ1 Under what structural conditions is a spectral graph filter stable when edges are deleted or added?
• RQ2 How do the degrees of the endpoints of perturbed edges influence filter stability?
• RQ3 To what extent does the spatial concentration of perturbations affect filter output change?
• RQ4 How do different graph models (e.g., scale-free vs. regular) affect the stability of spectral graph filters?
• RQ5 Can the theoretical bound guide the design of robust perturbation strategies that minimize filter output change?

Key findings

• The proposed stability bound is interpretable and explicitly depends on the degrees of the endpoints of deleted and added edges.
• Filters are more stable when perturbations target high-degree nodes, as confirmed by lower output changes in experiments.
• Spatial distribution of perturbations significantly affects stability: concentrated changes lead to larger output variations.
• The bound is tightest for rewiring strategies and performs reasonably well on adversarial (PGD) and robust strategies, especially on graphs with high degree variance.
• Experiments show that PGD attacks primarily target low-degree nodes, while the robust strategy preferentially modifies edges connected to high-degree hubs.
• On K-regular graphs, the robust strategy distributes edge modifications more evenly across the graph, aligning with the theoretical insight on spatial distribution.