Line-of-sight Cox percolation on Poisson-Delaunay triangulation

TL;DR

A percolation tool, inspired by enhancement techniques, is used for the first time in the context of communication networks to take advantage of percolation properties of device-to-device (D2D) networks in urban environments.

Abstract

In this work, percolation properties of device-to-device (D2D) networks in urban environments are investigated. The street system is modeled by a Poisson-Delaunay triangulation (PDT). Users are of two types: given either by a Cox process supported by the edges of the PDT or by a Bernoulli process on the vertices of the PDT (i.e. on streets and at crossroads). Percolation of the resulting connectivity graph G p,$λ$,r is interpreted as long-range connection in the D2D network. According to the parameters p, $λ$, r of the model, we state several percolation regimes in Theorem 1 (see also Fig. 3). This work completes and specifies results of Le Gall et al [23]. To do it, we take advantage of a percolation tool, inspired by enhancement techniques, used to our knowledge for the first time in the context of communication networks.

Figures (9)

• Figure 1: On top is depicted a user at crossroad $x \in \mathbf{X}_p$ (the black point) and its four neighbors in the Delaunay triangulation $\mathbf{T}$. The four connection ranges $\frac{r'}{2} \mathcal{E}_{x,1},\ldots,\frac{r'}{2} \mathcal{E}_{x,4}$ are represented with bold lines. On this example, $\frac{r'}{2} \mathcal{E}_{x,4}$ is larger than the distance $\|x - x'\|$, but the corresponding excess does not matter in our model: this is why the bold line is stopped at $x'$. Hence, $x'$-- if it is open --will be automatically connected to $x$, as well as all users on the street $[x,x']$. Below, a simulation of the connectivity graph (in blue) $\boldsymbol{\mathcal{G}}_{p,\lambda,r}$ in the box $[-60,60]\times [-30,30]$ with $r'=2r=1.8$, $\lambda=1$ and $p=0.7$.
• Figure 2: Between the two dotted lines is the non-trivial regime corresponding to a non-trivial critical intensity. In this regime, the intensity $\lambda$ of users on streets really matters to determine if percolation occurs or not. This phase diagram presents three uncertain regions marked by the roman symbols 1, 2 and 3.
• Figure 3: Here is the phase diagram corresponding to Theorem \ref{['thm:mainResult']}. W.r.t. the phase diagram of le2021continuum and recalled in Fig. \ref{['fig:DiagPhase']}, three blurred regions have been removed: the hypothetical region 1 does not exist, the hypothetical region 2-- if it exists --is reduced to the curve $\{ (r,p) : p = p_c(r) \}$ and $p_c(r)$ tends to $1/2$ as $r$ tends to $\infty$. Remark that both opposite regimes, namely the permanently subcritical and supercritical ones, become very close from each other as $r \to \infty$.
• Figure 4: To the left: for the site percolation model on crossroads of a Voronoi tesselation, it is possible that no horizontally crossing open paths nor vertically crossing closed ones occur. To the right: focus only on $x,y,z \in \mathbf{X}$, $z'$ will be added further. Initially, there is no Poisson points in the circumscribed circle $\mathcal{C}$ to $x,y,z$ so the triangle defined by these three vertices is present in the Delaunay triangulation $\mathbf{T}$. $x,y$ (black points) are open, i.e. in $\mathbf{X}_p$, while $z$ (white point) is closed. In $\boldsymbol{\mathcal{G}}_{p,0,r}$, $x$ and $y$ are connected since $\| x - y \| \leq r$. Now, let us add a black point (namely $z' \in \mathbf{X}_p$) inside $\mathcal{C}$ but at distance from $y$ larger than $r$. The Delaunay triangulation changes (new edges are red): the adding of $z'$ destroyes the edge $\{x,y\}$ and new triangles appear ($xz'z$ and $zz'y$). If $x$ and $y$ were linked in $\boldsymbol{\mathcal{G}}_{p,0,r}$ only through their common edge, they are no longer linked in $\boldsymbol{\mathcal{G}}_{p,0,r}$ after adding $z' \in \mathbf{X}_p$.
• Figure 5: The event $\mathcal{C}_{n,\lambda,r}(0)$, merely denoted by $\mathcal{C}_{n,\lambda,r}$, relative to $B_{3n}$ is depicted.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof