On the Connectivity of Two-Hop Networks in Finite Domains

In this paper, we study the direct (1-hop) and 2-hop connectivity, between two fixed terminal nodes, amongst N uniformly distributed relay nodes, confined to a finite convex geometry. We develop closed form expressions for average connection probability for special cases of circular and rectangular geometries and consider the case where communication paths are subject to Rayleigh fading. Our analysis shows that connectivity is defined by domain geometry, the mid-point location of the chord separating the terminal nodes and their spatial separation, yielding an interesting property of rotational invariance on the terminal node positions about their mid-point. We show that 2-hop connectivity yields appreciable connectivity gain, over the 1-hop case, with largest gains realised at the domain boundaries. We show that, under a condition of low relay node intensity, a simpler approximation can be employed to evaluate this connectivity metric in a finite domain. The analysis provides an insight into 2-hop connectivity in bounded wireless network domains and has application in many fields including wireless sensor networks, device-to-device communication in 5G networks, which are often modelled as having finite extent.