Theory and algorithms for hop-count-based localization with random geometric graph models of dense sensor networks

Reviewer: Somayeh Taheri

This paper investigates the frequently used approximation of the Euclidean distance between nodes as proportional to the hop distance between them. This approximation is quite often used in range-free localization protocols to avoid the cost of actually measuring the distance between the nodes. Studying the relationship between Euclidean distance (ED) and hop-count distance (HD) between nodes is motivated by range-free localization approaches that are based on hop count. The authors consider the sensor network as "a large number of nodes in a region of Euclidean space" viewed as a random geometric graph. There are a number of anchor nodes that can determine their own location, for instance, using the global positioning system (GPS). After introducing the problem and the assumptions, the authors obtain high probability bounds on ED for the nodes that are located h hops away from a given anchor. For this purpose, they consider four different cases, which are discussed in section 4. In the first case, nodes are located randomly and uniformly in the area, and the radius of the geometric graph is chosen as r ( n ) = c √ ln ( n )/ n where n is the number of nodes in the unit area and c is a constant. Given these values, the network remains asymptotically connected. The radius of the geometric graph refers to the maximum distance between two nodes that can still be considered neighbors and therefore connected in the graph. This kind of radius scaling is called the critical radius, and is based on the lemma proved in the paper. It guarantees the connectivity of nodes to all anchors with a high probability. In this case, the authors prove that ED is bounded as (1- e ) ( h a -1) r ( n ) < ED < h a r ( n ), in which h a is the hop distance from anchor a , and e is given such that e >0. The probability of the bounds is also calculated as a function of e . The lesson learned from this is that, in a dense network (where n is very large), the bounds of node distances to anchors is met because their probability converges to 1. Also, one can see that the smaller the value is for e , the slower the rate will be for convergence of the probability. Based on this result, the precision of localization assuming these ED bounds will increase with n , as the probability converges more rapidly for high n values. In the second case, the radius of a geometric graph is fixed to r . As in sensor networks, it should be the radio range of the nodes, which is constant and does not decrease with n as in r ( n ). Therefore, we have (1- e ) ( h a -1) r < ED < h a r . Here, the calculated bound probability convergence shows an exponential convergence and therefore, unlike the previous case, the precision of localization remains fixed at r and does not increase with n . In the third case, instead of the previously used uniform deployment, the nodes are placed in the area using a randomized lattice deployment, in which the unit area is divided into n cells and exactly one node is located in each cell. The authors have shown that, with this setting for both geometric graphs of r ( n ) and fixed r , the same results obtained for uniform node placement in the previous sections are valid. Finally, in the fourth case, the number of nodes is considered to have a Poisson distribution with a mean of n . Here, because the union bound is not used, the expression for the probability is exact, but it converges with ab = n , following the exponential power law. After the discussion of cases, the authors verify the validation of the bounds obtained in the previous section using simulations. The simulations show that the obtained bounds get tighter with increases in the number of nodes, n , in the unit of area. This is because of the definition of r ( n ), which decreases with n and squeezes the range for ED. The simulations also show how the lower bound on the probability of the bounds being respected increases with n . In section 5, the paper deals with obtaining the probability density function (pdf) of the pairwise ED given HD, that is, determining the probability distribution of ED when HD is given. After reviewing the state of the art, the authors obtain the probability distribution of the ED of a node that is located a given number of hops away from an anchor using a greedy algorithm in which the central limit theorem is used to estimate the pdf. Naturally, the estimation is more valid for large n values, because the central limit theorem is exploited. With this value, they can determine the distribution of the location of the node located HD hops away from the considered anchor. After demonstrating the relationship between ED and HD, the results are used in section 6 to develop two hop-count-derived localization algorithms, hop-count-derived ED bounds-based localization (HCBL) and hop-count-derived ED distribution-based localization (HCDL). These algorithms use the hop count information of the nodes to estimate their locations. HCBL localizes a node by finding the intersection of the areas given by the upper and lower bounds of its ED from L anchors when HDs are known. HCDL measures the hop distance of a node to L anchors, and uses the distribution function of the location of such a node based on the HDs to estimate its location via the minimum mean square error method. The authors then show with simulations that these localization approaches outperform some well-known existing approaches in the literature. Section 7 aims to improve the accuracy of the hop-count-based localization algorithms using belief propagation (BP). In this context, belief propagation means using the hop count information of neighboring nodes to improve the accuracy of the estimation of a node's location. On average, simulation results show a noticeable improvement in localization accuracy when it is combined with BP. The authors' approach to the relationship between hop-count distance and Euclidean distance in wireless networks is novel. In addition, the two localization approaches that are based on this relationship perform better than existing approaches, according to the presented results. Online Computing Reviews Service