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的代数和几何观点。主要贡献在于以实轴子集P(R)取值的矩阵形式，给出时变图(TVG)的新定义。我们利用P(R)的半环性质，通过矩阵乘法和截断Kleene星操作对TVG中的多跳通信进行建模。由此导出了关于TVG通信容量的新型统计量——生存期曲线，并基于长达整天的模拟连接性，对随机选取的大量STARLINK卫星样本生成了该曲线。通过借鉴拓扑数据分析(TDA)引入的新度量，进一步分析了STARLINK大子样本在时间上强连通的条件。为更好地建模地球与火星间的组网场景，我们引入了多种能够处理传播时延以及延迟容忍网络(DTN)常见协议（如存储转发）的半环结构。最后，演示了之字形持久性在空间网络特征化中的适用性，并证明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)分类方法的有效性。