HoloNets: Spectral Convolutions do extend to Directed Graphs

Christian Koke, Daniel Cremers

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Keywords: Spectral Graph Convolutions, Directed Graphs, Weighted Graphs, Complex Analysis, Spectral Graph Theory, Heterophily, Node Classification, Graph Regression pectral Graph Theory, Rigorous Proofs

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TL;DR: We extend spectral convolutions to directed graphs. Corresponding networks are shown to set SOTA on heterophilic node classification tasks and to be stable to topological perturbations.

Abstract:

Within the graph learning community, conventional wisdom dictates that spectral convolutional networks may only be deployed on undirected graphs: Only there could the existence of a well-defined graph Fourier transform be guaranteed, so that information may be translated between spatial- and spectral domains. Here we show this traditional reliance on the graph Fourier transform to be superfluous and -- making use of certain advanced tools from complex analysis and spectral theory -- extend spectral convolutions to directed graphs.
We provide a frequency-response interpretation of newly developed filters, investigate the influence of the basis used to express filters and discuss the interplay with characteristic operators on which networks are based. In order to thoroughly test the developed theory, we conduct experiments in real world settings, showcasing that directed spectral convolutional networks provide new state of the art results for heterophilic node classification on many datasets and -- as opposed to baselines -- may be rendered stable to resolution-scale varying topological perturbations.

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Primary Area: learning on graphs and other geometries & topologies

Submission Number: 3367