Random geometric graph diameter in the unit ball

This is the author's accepted manuscript

On the relation between graph distance and Euclidean distance in random geometric graphs

Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=¿(vlogn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).Peer ReviewedPostprint (author's final draft

On the Data Gathering Capacity and Latency in Wireless

In this paper, we investigate the fundamental properties of data gathering in wirelesssensor networks, in terms of both transport capacity and latency. We consider a scenarioin which s(n) out of n total network nodes have to deliver data to a set of d(n) sink nodesat a constant rate f(n; s(n); d(n)). The goal is to characterize the maximum achievablerate, and the latency in data delivery. We present a simple data gathering scheme thatachieves asymptotically optimal data gathering capacity and latency with arbitrary net-work deployments when d(n) = 1, and for most scaling regimes of s(n) and d(n) whend(n) > 1 in case of square grid and random node deployments. Differently from mostprevious work, our results and the presented data gathering scheme do not sacrifice en-ergy efficiency to the need of maximizing capacity and minimizing latency. Finally, weconsider the effects of a simple form of data aggregation on data gathering performance,and show that capacity can be increased of a factor f(n) with respect to the case of nodata aggregation, where f(n) is the node density. To the best of our knowledge, theones presented in this paper are the first results showing that asymptotically optimal datagathering capacity and latency can be achieved in arbitrary networks in an energy efficientway

The Fundamental Limits of Broadcasting in Dense Wireless Mobile Networks

In this paper, we investigate the fundamental properties of broadcasting in {em mobile} wireless networks. In particular, we characterize broadcast capacity and latency of a mobile network, subject to the condition that the stationary node spatial distribution generated by the mobility model is uniform. We first study the intrinsic properties of broadcasting, and present the {sc RippleCast} broadcasting scheme that simultaneously achieves asymptotically optimal broadcast capacity and latency, subject to a weak upper bound on maximum node velocity and under the assumption of static broadcast source. We then extend {sc RippleCast} with the novel notion of center-casting, and prove that asymptotically optimal broadcast capacity and latency can be achieved also when the broadcast source is mobile. This study intendedly ignores the burden related to the selection of broadcast relay nodes within the mobile network, and shows that optimal broadcasting in mobile networks is, in principle, possible. We then investigate the broadcasting problem when the relay selection burden is taken into account, and present a combined distributed leader election and broadcasting scheme achieving a broadcast capacity and latency which is within a $Theta((log n)^{1+frac{2}{alpha}})$ factor from optimal, where $n$ is the number of mobile nodes and $alpha>2$ is the path loss exponent. However, this result holds only under the assumption that the upper bound on node velocity converges to zero (although with a very slow, poly-logarithmic rate) as $n$ grows to infinity

Reconstruction of random geometric graphs: breaking the Ω(r) distortion barrier

Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r = n α for any 0 < α < 1/2, with high probability reconstructs the vertex positions with a maximum error of O(n β ) where β = 1/2−(4/3)α, until α ≥ 3/8 where β = 0 and the error becomes O( √ log n). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in R 3 and to hypercubes in any constant fixed dimension.Josep Díaz: partially supported by PID-2020-112581GB-C21 (MOTION). Cristopher Moore: partially supported by National Science Foundation grant IIS-1838251.Peer ReviewedPostprint (published version