In recent years, graph signal processing has emerged as a powerful framework at the intersection of signal processing and graph theory, providing tools for the analysis of signals defined on nodes while accounting for their relationships represented by edges. These tools have been successfully applied to various settings, including statistical hypothesis testing. In particular, non-parametric approaches based on surrogate generation have been proposed for signals on undirected graphs. However, they are yet to be extended to directed graphs. In this work, we first revisit the notion of stationary graph signals on directed graphs. Specifically, and through the eigendecomposition of the graph shift operator, we define directed graph wide-sense stationary signals. Then, we propose a new framework to generate surrogate graph signals that preserve covariance structure under stationarity assumptions. Null distributions of the test metric can then be constructed from these surrogates and serve as a reference for the empirical data. Finally, we provide guiding examples and an application on real data, in which we compare the performance of our framework with existing techniques for undirected graphs or based on naive permutation, demonstrating feasibility and superiority of the proposed approach.

翻译：近年来，图信号处理已成为信号处理与图论交叉领域中的一个强大框架，它提供了分析节点上定义信号、同时考虑边所表示关系的方法。这些工具已成功应用于多种场景，包括统计假设检验。特别是，针对无向图上的信号，已有基于代理生成的非参数方法被提出。然而，这些方法尚未扩展到有向图。本研究首先重新审视了有向图上平稳图信号的概念。具体而言，通过图移位算子的特征分解，我们定义了有向图广义平稳信号。然后，我们提出了一种新框架，用于生成在平稳性假设下保持协方差结构的代理图信号。据此可构建检验统计量的零分布，作为经验数据的参考。最后，我们提供了指导性示例及真实数据上的应用，将本框架与现有面向无向图或基于朴素置换的方法进行性能比较，论证了所提方法的可行性与优越性。