The Relay Random Tree: A Stochastic Geometry Approach of Multihop Relay in an Urban Visibility Setting

TL;DR

This paper extends a stochastic geometry model for urban LoS connections with RIS relays, analyzing multi-hop links and relay load, and classifies the resulting network within geometric random graph theory.

Contribution

It introduces arbitrary building height distributions and multi-hop connections into the model, linking it to Eternal Family Trees classification.

Findings

Model accommodates arbitrary building height distributions.

Multi-hop connections are analyzed within the model.

Network classification aligns with Eternal Family Trees theory.

Abstract

In a recent work (Lee, Baccelli $'25$), a one dimensional stochastic geometry model was introduced to study Line of Sight (LoS) connections using Reconfigurable Intelligent Surfaces (RIS), in the context of non terrestrial networks. In this model, signal can be propagated in a urban environment, with buildings acting as obstacles with RIS (which, for the scope of this present article can essentially be thought of as relays) on their rooftops, relaying the connection. The present paper extends this model by both allowing arbitrary distributions for the buildings heights, and considering multi-hop connections. Those generalities also lead to considering structural problems linked to the total load of a relay. Furthermore, studying this Line of Sight connection geometry at the light of geometric random graph theory, we show that it constitutes a computationally well understood example that highlights the different classes of the Eternal Family Trees (EFTs) classification.