Performance Evaluation of Distance-Hop Proportionality on Geometric Graph Models of Dense Sensor Networks

Wireless sensor networks can often be viewed in terms of a uniform deployment of a large number of nodes on a region in Euclidean space, e.g., the unit square. After deployment, the nodes self-organise into a mesh topology. In a dense, homogeneous deployment, a frequently used approximation is to take the hop distance between nodes to be proportional to the Euclidean distance between them. In this paper, we analyse the performance of this approximation. We show that nodes with a certain hop distance from a fixed anchor node lie within a certain annulus with probability approaching unity as the number of nodes n → ∞.

We take a uniform, i.i.d. deployment of n nodes on a unit square, and consider the geometric graph on these nodes with radius r(n) = c√1n n/n. We show that, for a given hop distance h of a node from a fixed anchor on the unit square, the Euclidean distance lies within [(1-ε)(h-1)r(n), hr(n)], for ε > 0, with probability approaching unity as n → ∞. This result shows that it is more likely to expect a node, with hop distance h from the anchor, to lie within this annulus centred at the anchor location, and of width roughly r(n), which decreases as n increases. We show that if the radius r of the geometric graph is fixed, the convergence of the probability is exponentially fast. Similar results hold for a randomised lattice deployment. We provide simulation results that illustrate the theory, and serve to show how large n needs to be for the asymptotics to be useful.