Routeing properties in a Gibbsian model for highly dense multihop networks

TL;DR

This paper analyzes a probabilistic Gibbsian model for routing in dense multihop ad-hoc networks, revealing how trajectories behave under different regimes and how interference influences route selection.

Contribution

It provides a detailed qualitative analysis of typical routing scenarios and trajectory properties in high-density networks, extending previous work with new asymptotic insights.

Findings

Trajectories tend to be straight lines in large areas and high interference regimes.

The typical hop length diverges logarithmically with distance in large-scale limits.

Local and global repulsive effects influence trajectory patterns in dense subareas.

Abstract

We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution, which favours trajectories with low interference, measured in terms of signal-to-interference ratio. This model was introduced in our earlier paper [KT18], where we expressed, in the limit of a high density of users, the typical distribution of the family of trajectories in terms of a law of large numbers. In the present work, we derive its qualitative properties. We analytically identify the emerging typical scenarios in three extreme regimes. We analyse the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory approaches a straight line quickly, in regime (1) with equal hop lengths. Interestingly, in regime (1), the typical length of a hop diverges logarithmically in the distance of the transmitter to the base station. We further analyse (3) local and global repulsive effects of a densely populated subarea on the trajectories.