# Optimal Paths on the Space-Time SINR Random Graph

## TL;DR

This paper investigates optimal packet transmission paths in a space-time SINR wireless network model, revealing conditions under which the minimum delay between nodes is finite or infinite, based on network infrastructure.

## Contribution

It introduces a novel analysis of space-time SINR graphs, establishing conditions for finite or infinite delay constants in wireless networks with random node placement and transmission policies.

## Key findings

- Time constant is infinite in pure Poisson networks under natural conditions.
- Adding a small periodic infrastructure makes the time constant finite.
- Results inform the design of more reliable wireless network routing.

## Abstract

We analyze a class of Signal-to-Interference-and-Noise-Ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on the SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. This paper studies optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both "positive" and "negative" results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to infinity. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite.