  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.
