Percolation and Connectivity on the Signal to Interference Ratio Graph

TL;DR

This paper investigates percolation and connectivity in a wireless network modeled by a Poisson point process, showing conditions under which large connected clusters form or do not form based on SIR thresholds and node density.

Contribution

It provides new theoretical results on percolation thresholds and connectivity conditions in SIR graphs considering path-loss and fading models.

Findings

Percolation occurs for small thresholds within certain density ranges.

High thresholds prevent percolation at high densities.

Frequency division guarantees connectivity in large networks.

Abstract

A wireless communication network is considered where any two nodes are connected if the signal-to-interference ratio (SIR) between them is greater than a threshold. Assuming that the nodes of the wireless network are distributed as a Poisson point process (PPP), percolation (unbounded connected cluster) on the resulting SIR graph is studied as a function of the density of the PPP. For both the path-loss as well as path-loss plus fading model of signal propagation, it is shown that for a small enough threshold, there exists a closed interval of densities for which percolation happens with non-zero probability. Conversely, for the path-loss model of signal propagation, it is shown that for a large enough threshold, there exists a closed interval of densities for which the probability of percolation is zero. Restricting all nodes to lie in an unit square, connectivity properties of the SIR graph are also studied. Assigning separate frequency bands or time-slots proportional to the logarithm of the number of nodes to different nodes for transmission/reception is sufficient to guarantee connectivity in the SIR graph.