Betweenness centrality in dense random geometric networks

Random geometric networks consist of 1) a set of nodes embedded randomly in a bounded domain $\mathcal{V} \subseteq \mathbb{R}^d$ and 2) links formed probabilistically according to a function of mutual Euclidean separation. We quantify how often all paths in the network characterisable as topologically `shortest' contain a given node (betweenness centrality), deriving an expression in terms of a known integral whenever 1) the network boundary is the perimeter of a disk and 2) the network is extremely dense. Our method shows how similar formulas can be obtained for any convex geometry. Numerical corroboration is provided, as well as a discussion of our formula's potential use for cluster head election and boundary detection in densely deployed wireless ad hoc networks.Comment: 6 pages, 3 figure

Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model

We compute the expected number of cars that have at least one two-hop path to a fixed roadside unit in a one-dimensional vehicular ad hoc network in which other cars can be used as relays to reach a roadside unit when they do not have a reliable direct link. The pairwise channels between cars experience Rayleigh fading in the random connection model, and so exist, with probability function of the mutual distance between the cars, or between the cars and the roadside unit. We derive exact equivalents for this expected number of cars when the car density $\rho$ tends to zero and to infinity, and determine its behaviour using an infinite oscillating power series in $\rho$, which is accurate for all regimes. We also corroborate those findings to a realistic situation, using snapshots of actual traffic data. Finally, a normal approximation is discussed for the probability mass function of the number of cars with a two-hop connection to the origin. The probability mass function appears to be well fitted by a Gaussian approximation with mean equal to the expected number of cars with two hops to the origin.Comment: 21 pages, 7 figure

Counting geodesic paths in 1-D VANETs

In the IEEE 802.11p standard addressing vehicular communications, Basic Safety Messages (BSMs) can be bundled together and relayed as to increase the effective communication range of transmitting vehicles. This process forms a vehicular ad hoc network (VANET) for the dissemination of safety information. The number of "shortest multihop paths" (or geodesics) connecting two network nodes is an important statistic which can be used to enhance throughput, validate threat events, protect against collusion attacks, infer location information, and also limit redundant broadcasts thus reducing interference. To this end, we analytically calculate for the first time the mean and variance of the number of geodesics in 1D VANETs.Comment: 11 pages, 5 figure

Meta distribution of SIR in the Internet of Things modelled as a Euclidean matching

The Poisson bipolar model considers user-base station pairs distributed at random on a flat domain, similar to matchsticks scattered onto a table. Though this is a simple and tractable setting in which to study dense networks, it doesn't properly characterise the stochastic geometry of user-base station interactions in some dense deployment scenarios, which may involve short and long range links, with some paired very nearby optimally, and others sub-optimally due to local crowding. Since the users will pair one-to-one with base stations, we can consider using the popular bipartite Euclidean matching (BEM) from spatial combinatorics, and study the corresponding (meta) distribution of the signal-to-interference-ratio (SIR). This provides detailed information about the proportion of links in the network meeting a target reliability constraint. We can then observe via comparison the impact of taking into account the variable/correlated short-range distances between the transmitter-receiver pairs on the communication statistics. We illustrate and quantify how the widely-accepted bipolar model fails to capture the network-wide reliability of communication in a typical ultra-dense setting based on a binomial point process. We also show how assuming a Gamma distribution for link distances may be a simple improvement on the bipolar model. Overall, BEMs provide good grounds for understanding more sophisticated pairing features in ultra-dense networks

Two-Hop connectivity to the roadside in a VANET under the Random Connection Model

In this paper, we compute the expected number of vehicles with at least one two-hop path to a fixed roadside unit (RSU) in a multi-hop, one-dimensional vehicular ad hoc network (VANET) where other cars can act as relays. The pairwise channels experience Rayleigh fading in the random connection model, and so exist, with probability function of the mutual distance between the cars, or between the cars and the RSU. We derive exact equivalents for the expected number of cars with a two-hop connection to the RSU when the car density ρ tends to zero and infinity, and determine its behaviour using an infinite oscillating power series in ρ, which is accurate for all regimes. We also corroborate those findings to a realistic situation, using snapshots of actual traffic data. Finally, a normal approximation is discussed for the probability mass function of the number of cars with a two-hop connection to the RSU