Topological descriptors have been increasingly utilized for capturing multiscale structural information in relational data. In this work, we consider various filtrations on the (box) product of graphs and the effect on their outputs on the topological descriptors - the Euler characteristic (EC) and persistent homology (PH). In particular, we establish a complete characterization of the expressive power of EC on general color-based filtrations. We also show that the PH descriptors of (virtual) graph products contain strictly more information than the computation on individual graphs, whereas EC does not. Additionally, we provide algorithms to compute the PH diagrams of the product of vertex- and edge-level filtrations on the graph product. We also substantiate our theoretical analysis with empirical investigations on runtime analysis, expressivity, and graph classification performance. Overall, this work paves way for powerful graph persistent descriptors via product filtrations. Code is available at https://github.com/Aalto-QuML/tda_graph_product.

翻译：拓扑描述符在捕捉关系数据的多尺度结构信息方面日益得到应用。本文研究了图（盒）乘积上各类滤过结构及其对拓扑描述符——欧拉示性数（EC）与持续同调（PH）——输出结果的影响。具体而言，我们完整刻画了EC在基于一般着色的滤过结构上的表达能力。同时，我们证明了（虚拟）图乘积的PH描述符所包含的信息严格多于对各独立图进行计算所得的信息，而EC则不具备此性质。此外，我们提出了计算图乘积上顶点级与边级滤过结构PH图的算法，并通过运行时间分析、表达能力验证和图分类性能测试等实证研究佐证了理论分析。总体而言，本研究为通过乘积滤过构建强大的图持续描述符开辟了道路。代码发布于 https://github.com/Aalto-QuML/tda_graph_product。