A Survey of Mathematical Structures for Lunar Networks

To sustain the current and increasing accessibility of space, a scalable communications infrastructure (i.e. the Solar System Internet, SSI) is necessary. The goal of this paper is to begin the discovery of the fundamental underlying mathematical structure of space networks to help the research community harness these structures for algorithm development and optimization. To ensure the applicability of the research, the approaches are considered through the lens of simulated scenarios inspired by the Artemis Back-to-the-Moon mission set for 2024. We note that any approach to an SSI must fit under the umbrella of Delay Tolerant Networking (DTN), due to celestial mobility, high link latencies, high variance in link latencies, disconnections, lack of end-to-end paths, and so on. These difficulties are exacerbated by the fact that the underlying structure of a space network is a time-evolving network and may experience multiple discontinuities in its topology. In this paper we propose several novel approaches to a mathematical foundation for Delay Tolerant Networking Theory that fall outside the traditional scope of temporal network theory. These techniques include methods from Topological Data Analysis, Dynamic Graph Analysis, Applied Algebraic Geometry, Probability Theory, and Game Theory. Some of these methods include tools adapted to the study of dynamic metric spaces, such as zigzag persistent homology and their higher parameter analogs. We find that several of these methods target desired engineering outcomes such as discovery and automatic sub-netting. While each approach is theoretical, they are also algorithmic in nature and offer immediate practical applications. The paper concludes with comparisons of the various methods along with suggestions for future work.