Percolation and connectivity on the signal to interference ratio graph

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 (formation of an unbounded connected cluster) on the resulting SIR graph is studied as a function of the density of the PPP. 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, it is shown that for a large enough threshold, there exists a closed interval of densities for which the probability of percolation is zero. Connectivity properties of the SIR graph are also studied by restricting all the nodes to lie in a bounded area. Assigning separate frequency bands or time-slots proportional to the logarithm of the number of nodes to different nodes for transmission/reception is shown to be necessary and sufficient for guaranteeing connectivity in the SIR graph.