HoloNets: Spectral Convolutions do extend to Directed Graphs
This work addresses a foundational problem in graph learning by enabling spectral methods on directed graphs, which is incremental but impactful for applications requiring directed data.
The authors tackled the limitation of spectral convolutional networks to undirected graphs by extending them to directed graphs using complex analysis and spectral theory, achieving new state-of-the-art results for heterophilic node classification on many datasets with stability to topological perturbations.
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.