Limitless stability for Graph Convolutional Networks

• URL: http://arxiv.org/abs/2301.11443v3
• Date: Sat, 30 Sep 2023 15:24:11 GMT
• Title: Limitless stability for Graph Convolutional Networks
• Authors: Christian Koke
• Abstract summary: This work establishes rigorous, novel and widely applicable stability guarantees and transferability bounds for graph convolutional networks. It is showcased that graph convolutional networks are stable under graph-coarse-graining procedures precisely if the GSO is the graph Laplacian and filters are regular at infinity.
• Score: 8.1585306387285
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: This work establishes rigorous, novel and widely applicable stability guarantees and transferability bounds for graph convolutional networks -- without reference to any underlying limit object or statistical distribution. Crucially, utilized graph-shift operators (GSOs) are not necessarily assumed to be normal, allowing for the treatment of networks on both undirected- and for the first time also directed graphs. Stability to node-level perturbations is related to an 'adequate (spectral) covering' property of the filters in each layer. Stability to edge-level perturbations is related to Lipschitz constants and newly introduced semi-norms of filters. Results on stability to topological perturbations are obtained through recently developed mathematical-physics based tools. As an important and novel example, it is showcased that graph convolutional networks are stable under graph-coarse-graining procedures (replacing strongly-connected sub-graphs by single nodes) precisely if the GSO is the graph Laplacian and filters are regular at infinity. These new theoretical results are supported by corresponding numerical investigations.

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