Asymptotic analysis of k-hop connectivity in the 1D unit disk random graph model

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 a probability given by a function of the mutual distance between the cars, or between the cars and the RSU. We derive exact expressions 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 beh…

Connectivity of 1d random geometric graphs

A 1d random geometric graph (1d RGG) is built by joining a random sample of $n$ points from an interval of the real line with probability $p$. We count the number of $k$-hop paths between two vertices of the graph in the case where the space is the 1d interval $[0,1]$. We show how the $k$-hop path count between two vertices at Euclidean distance $|x-y|$ is in bijection with the volume enclosed by a uniformly random $d$-dimensional lattice path joining the corners of a $(k-1)$-dimensional hyperrectangular lattice. We are able to provide the probability generating function and distribution of th…

Connectivity and Centrality in Dense Random Geometric Graphs

Due to shorter range communication becoming more prevalent with the development of multiple-input, multiple-output antennas (MIMO) and millimeter wave communications, multi-hop, intra-cell communication is anticipated to play a major role in 5G. This is developed in this thesis. Our analysis involves a stochastic spatial network model called a random geometric graph, which we use to model a network of interconnected devices communicating wirelessly without any separate, pre-established infrastructure. [The remaining abstract is available in the front matter of this thesis.]

软随机几何图的连通性

Connectivity of soft random geometric graphs

考虑单位正方形中$n$均匀随机点上的图，如果点间距离至多为$r$，则每对点由概率为$p$的边连接。我们发现，作为$n\to\infty$的概率完全连通的是由没有孤立的顶点，本身由泊松近似的孤立顶点的数量，均匀地在所有的选择$p,r$。我们确定的渐近概率的连通性为所有的$(p_n,r_n)$受$r_n=O(n^{-\varepsilon})$，一些$\varepsilon >0$。我们将第一个结果推广到更高的维度和更大的一类连接概率函数。