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.

翻译：在图学习领域，传统观念认为谱卷积网络仅能应用于无向图：唯有在此类图上才能保证存在良定义的图傅里叶变换，从而在空间域与谱域之间传递信息。本文证明这种对图傅里叶变换的传统依赖是多余的——借助复分析与谱理论的先进工具——将谱卷积扩展至有向图。我们为新开发的滤波器提供频率响应解释，研究用于表达滤波器的基函数的影响，并讨论其与网络基础特征算子间的相互作用。为全面检验所提出的理论，我们在真实场景下开展实验，证明有向谱卷积网络在多个数据集上的异配节点分类任务中达到新的最优结果，并且——与基线方法不同——能够对分辨率尺度变化的拓扑扰动保持鲁棒性。