We introduce hierarchical neighbor graphs, a new architecture for connecting ad hoc wireless nodes distributed in a plane. The structure has the flavor of hierarchical clustering and requires only local knowledge and minimal computation at each node to be formed and repaired. Hence, it is a suitable interconnection model for an ad hoc wireless sensor network. The structure is able to use energy efficiently by reorganizing dynamically when the battery power of heavily utilized nodes degrades and is able to achieve throughput, energy efficiency and network lifetimes that compare favorably with the leading proposals for data collation in sensor networks such as LEACH (Heinzelman et. al., 2002). Additionally, hierarchical neighbor graphs have low power stretch i.e. the power required to connect nodes through the network is a small factor higher than the power required to connect them directly. Our structure also compares favorably to mathematical structures proposed for connecting points in a plane e.g. nearest-neighbor graphs (Ballister et. al., 2005), @math -graphs (Ruppert and Seidel, 1991), in that it has expected constant degree and does not require any significant computation or global information to be formed.
On the mathematical front, nearest neighbor models have been studied by H "aggstr "om and Meester @cite_18 who proved that there is a critical value @math dependent on the dimension of the space such that if each point is connected to its @math nearest neighbors for any @math , an infinite component exists almost surely. Restricting this model to a square box of area @math , Xue and Kumar @cite_7 showed that @math must be at least 0.074 @math for the graph to be connected. This lower bound was improved to 0.3043 @math by Ballister, Bollob 'as, Sarkar and Walters @cite_1 who also showed that the threshold for connectivity is at most @math as well as corresponding results for the directed version of the problem. The advantage our model enjoys over this model is that by selecting neighbors carefully from the set of proximate nodes, hierarchical neighbor graphs achieve constant degree in expectation. Additionally our hierarchical structure also ensures paths with fewer hops.
@cite_18 @cite_1 @cite_7
2015859644 2056986231
Consider a Poisson process X in Rd with density 1. We connect each point of X to its k nearest neighbors by undirected edges. The number k is the parameter in this model. We show that, for k = 1, no percolation occurs in any dimension, while, for k = 2, percolation occurs when the dimension is sufficiently large. We also show that if percolation occurs, then there is exactly one infinite cluster. Another percolation model is obtained by putting balls of radius zero around each point of X and let the radii grow linearly in time until they hit another ball. We show that this model exists and that there is no percolation in the limiting configuration. Finally we discuss some general properties of percolation models where balls placed at Poisson points are not allowed to overlap (but are allowed to be tangent). 0 1996 John Wiley & Sons, Inc. Let P be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n , k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n , k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n , k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n , k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k > 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.