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:在本文中,我们在网络化空间通信上引入了新的代数和几何视角。我们的主要贡献是用值在实线子集 P(R) 上的矩阵定义了一个时变图 (TVG) 的新颖概念。我们利用 P(R) 的半环性质,利用矩阵乘法和截断的 Kleene 星模型多跳通信。这导致了所谓的生命曲线的新颖通信容量统计,我们对大量随机选择的 STARLINK 卫星样品进行了模拟来生成这些统计。我们还引入了一些半环来更好地模拟地球和火星之间的网络场景,这些半环可以模拟传播延迟以及 Delay Tolerant Networking(DTN) 中常见的协议,如存储和转发。最后,我们演示了利用锯齿持久性来特征化不同的空间网络,以及仅基于时变拓扑来区分地球-火星和地球-月卫星系统的 K-近邻(KNN)分类效果。