In this paper we introduce some new algebraic and geometric perspectives on networked space communications. Our main contribution is a novel definition of a time-varying graph (TVG), defined in terms of a matrix with values in subsets of the real line P(R). We leverage semi-ring properties of P(R) to model multi-hop communication in a TVG using matrix multiplication and a truncated Kleene star. This leads to novel statistics on the communication capacity of TVGs called lifetime curves, which we generate for large samples of randomly chosen STARLINK satellites, whose connectivity is modeled over day-long simulations. Determining when a large subsample of STARLINK is temporally strongly connected is further analyzed using novel metrics introduced here that are inspired by topological data analysis (TDA). To better model networking scenarios between the Earth and Mars, we introduce various semi-rings capable of modeling propagation delay as well as protocols common to Delay Tolerant Networking (DTN), such as store-and-forward. Finally, we illustrate the applicability of zigzag persistence for featurizing different space networks and demonstrate the efficacy of K-Nearest Neighbors (KNN) classification for distinguishing Earth-Mars and Earth-Moon satellite systems using time-varying topology alone.

翻译：本文提出了空间通信网络的一些新代数与几何视角。主要贡献在于一种新的时变图（TVG）定义，该定义基于实直线子集 P(R) 中取值的矩阵。我们利用 P(R) 的半环性质，通过矩阵乘法与截断Kleene星来模拟 TVG 中的多跳通信。由此导出 TVG 通信容量的新型统计数据——生命周期曲线，并针对大量随机选取的STARLINK卫星样本（其连通性经全天时长模拟建模）生成了该曲线。进一步，借助受拓扑数据分析（TDA）启发的新度量，分析了STARLINK大子样本何时在时间上强连通。为更好地模拟地球与火星间的组网场景，我们引入多种能够模拟传播延迟以及延迟容忍网络（DTN）常见协议（如存储转发）的半环。最后，我们展示了之字形持续性在特征化不同空间网络中的适用性，并论证了仅利用时变拓扑即可区分地火与地月卫星系统的K近邻（KNN）分类效能。