A Geometry Study on the Capacity of Wireless Networks via Percolation

We study the effect of various geometries on the capacity of wireless networks via percolation, which was not considered much before. Percolation theory was first applied to derive an achievable rate 1/√n in [5] by constructing a highway system, in contrast with the previous result Θ(1/√n log n) in [6], where n is the number of the nodes. While a highway system that consists of both horizontal and vertical edge-disjoint paths exists in a square network, B. Liu et al. in [1] pointed out that the horizontal paths will disappear if the width of a strip network is increasing more slowly than log n. In this paper, first we take a deeper look at the percolation in a strip network. We discover that when a highway system exists, the capacity is restricted by the maximum length of the sides. Moreover, a sub-highway system is still in presence when the highway system disappears. Secondly, we consider the situations in a triangle network. Conditions that percolation highway exists in it, and the achievable rate for a triangle network are discussed. We find that corner effect can be a bottleneck of the capacity. By combining the achievable rate of the former networks, we attribute the variance between them to their symmetry discrepancies. Finally, we turn to the capacity of three dimensional (3D) networks via percolation. The whole study shows that geometric symmetry plays a significant role in the percolation and the capacity, thereby shedding a light on the network design and the scheduling.