# Algebraic and Geometric Models for Space Networking

arXiv: https://arxiv.org/abs/2304.01150 · HTML: https://mlsanity.com/paper/2304.01150/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 47 references that are in the map.
- First posted on arXiv 3 Apr 2023.

## Builds on (most cited first, 11 of 11)

- [Temporal Network Analysis Using Zigzag Persistence](https://mlsanity.com/paper/2205.11338/) (2022)
- [Algebraic Stability of Zigzag Persistence Modules](https://mlsanity.com/paper/1604.00655/) (2016)
- [Fast Computation of Zigzag Persistence](https://mlsanity.com/paper/2204.11080/) (2022)
- [Extracting Persistent Clusters in Dynamic Data via M"obius inversion](https://mlsanity.com/paper/1712.04064/) (2017)
- [Intrinsic Interleaving Distance for Merge Trees](https://mlsanity.com/paper/1908.00063/) (2019)
- [Computing Zigzag Persistence on Graphs in Near-Linear Time](https://mlsanity.com/paper/2103.07353/) (2021)
- [Parametric shortest-path algorithms via tropical geometry](https://mlsanity.com/paper/1904.01082/) (2019)
- [Categorified Reeb Graphs](https://mlsanity.com/paper/1501.04147/) (2015)
- [How to Compose Shortest Paths](https://mlsanity.com/paper/2205.15306/) (2022)
- [Topological Analysis of Temporal Hypergraphs](https://mlsanity.com/paper/2302.02857/) (2023)
- [Revisiting Graph Persistence for Updates and Efficiency](https://mlsanity.com/paper/2302.12796/) (2023)

## Abstract (arXiv)

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.