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.

介绍了一种新颖的基于时间变化图的方法来模拟星际通信，包括通信容量的统计，拓扑数据分析等。

空间网络的代数和几何模型

Algebraic and Geometric Models for Space Networking

Social connectivity is the key process that characterizes the structural
properties of social networks and in turn processes such as navigation,
influence or information diffusion. Since time, attention and cognition are
inelastic resources, humans should have a predefined strategy to manage their
social interactions over time. However, the limited observational length of
existing human interaction datasets, together with the bursty nature of dyadic
communications have hampered the observation of tie dynamics in social
networks. Here we develop a method for the detection of tie
activation/deactivation, and apply it to a large longitudinal, cross-sectional
communication dataset ($\approx$ 19 months, $\approx$ 20 million people).
Contrary to the perception of ever-growing connectivity, we observe that
individuals exhibit a finite communication capacity, which limits the number of
ties they can maintain active. In particular we find that men have an overall
higher communication capacity than women and that this capacity decreases
gradually for both sexes over the lifespan of individuals (16-70 years). We are
then able to separate communication capacity from communication activity,
revealing a diverse range of tie activation patterns, from stable to
exploratory. We find that, in simulation, individuals exhibiting exploratory
strategies display longer time to receive information spreading in the network
those individuals with stable strategies. Our principled method to determine
the communication capacity of an individual allows us to quantify how
strategies for human interaction shape the dynamical evolution of social
networks.

通过开发一种检测关系激活 / 失活的方法并将其应用于一个大型的纵向交流数据集，研究表明相比于个体不断增长的互联程度的看法，人们的交流能力是有限的，尤其是随着时间的流逝。男性的通信能力总体上高于女性，且男女通信能力都会随着年龄的增长而逐渐降低，同时该方法也揭示了从稳定到探索性的各种关系激活模式以及这些模式对社交网络的动态演化造成的影响。