We introduce the monoidal Rips filtration, a filtered simplicial set for weighted directed graphs and other lattice-valued networks. Our construction generalizes the Vietoris-Rips filtration for metric spaces by replacing the maximum operator, determining the filtration values, with a more general monoidal product. We establish interleaving guarantees for the monoidal Rips persistent homology, capturing existing stability results for real-valued networks. When the lattice is a product of totally ordered sets, we are in the setting of multiparameter persistence. Here, the interleaving distance is bounded in terms of a generalized network distance. We use this to prove a novel stability result for the sublevel Rips bifiltration. Our experimental results show that our method performs better than Flagser in a graph regression task, and that combining different monoidal products in point cloud classification can improve performance.

翻译：本文引入幺半群Rips滤过，这是一种针对加权有向图及其他格值网络的滤过单纯集。该构造通过将决定滤过值的最大算子替换为更一般的幺半群积，推广了度量空间的Vietoris-Rips滤过。我们为幺半群Rips持续同调建立了交错保证，从而囊括了实值网络现有的稳定性结果。当格为全序集的乘积时，该框架对应于多参数持续性理论。在此情形下，交错距离可由广义网络距离界定。基于此，我们证明了子水平Rips双滤过的一个新颖稳定性定理。实验结果表明，在图回归任务中，本方法性能优于Flagser；在点云分类任务中，组合不同幺半群积可提升模型性能。