  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.
