Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deviation Estimates for Extremal Relay Random Geometric Graphs

Ghurumuruhan Ganesan

TL;DR

The paper addresses how to realize a deterministic topology Γ inside the unit square using relay connections from a random geometric graph G(r_n), defining relay RGGs via vertex-disjoint relay paths that connect Γ's endpoints. It develops rigorous conditions on the graph size e_0 and the scaling r_n under which relay RGGs exist and are near-optimal in total length, and it analyzes extremal weights by assigning i.i.d. exponential weights to G's edges, deriving deviation bounds and L^2 convergence for the maximum relay weight W_n. The main theoretical contributions include precise concentration bounds for the minimum relay length L(G,Γ), an edge-count (har) bound bounding the relay path edge count by l_{tot}/r_n, and sharp, high-probability estimates for W_n in terms of δ_n = e_0 L_n log n, along with conditions ensuring convergence. These results provide principled guarantees for relay-based connectivity and throughput in wireless networks with geometric randomness, informing design under uncertain relay availability.

Abstract

In this paper, we consider a deterministic graph~\(Γ\) drawn on the unit square with straight line segments as edges and connect vertices of~\(Γ\) using edges of a random geometric graph (RGG)~\(G\) with adjacency distance~\(r_n\) as relays. We call the resulting graph as a \emph{relay} RGG and determine sufficient conditions under such relay RGGs exist and are also near optimal, in terms of the graph parameters of~\(Γ.\) We then equip edges of~\(G\) with independent, exponentially distributed weights and obtain bounds for the maximum possible weight~\(W_n\) of a relay RGG with a given length~\(L_n.\)

Deviation Estimates for Extremal Relay Random Geometric Graphs

TL;DR

The paper addresses how to realize a deterministic topology Γ inside the unit square using relay connections from a random geometric graph G(r_n), defining relay RGGs via vertex-disjoint relay paths that connect Γ's endpoints. It develops rigorous conditions on the graph size e_0 and the scaling r_n under which relay RGGs exist and are near-optimal in total length, and it analyzes extremal weights by assigning i.i.d. exponential weights to G's edges, deriving deviation bounds and L^2 convergence for the maximum relay weight W_n. The main theoretical contributions include precise concentration bounds for the minimum relay length L(G,Γ), an edge-count (har) bound bounding the relay path edge count by l_{tot}/r_n, and sharp, high-probability estimates for W_n in terms of δ_n = e_0 L_n log n, along with conditions ensuring convergence. These results provide principled guarantees for relay-based connectivity and throughput in wireless networks with geometric randomness, informing design under uncertain relay availability.

Abstract

In this paper, we consider a deterministic graph~ drawn on the unit square with straight line segments as edges and connect vertices of~ using edges of a random geometric graph (RGG)~ with adjacency distance~ as relays. We call the resulting graph as a \emph{relay} RGG and determine sufficient conditions under such relay RGGs exist and are also near optimal, in terms of the graph parameters of~ We then equip edges of~ with independent, exponentially distributed weights and obtain bounds for the maximum possible weight~ of a relay RGG with a given length~

Paper Structure

This paper contains 2 sections, 3 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Relay Random Geometric Graphs

Key Result

theorem 1

Suppose $\Gamma$ has $e_0$ edges and where $0 < \alpha,\beta < 1$ are constants satisfying $\beta < \frac{1-\alpha}{2}$ so that $e_0$ is much smaller than $nr_n^2.$ for every $\epsilon > 0$ there is a constant $C >0$ such that Moreover, if $0 < \beta < \frac{1-\alpha}{4},$ then there is a constant $D > 0$ such that

Figures (2)

  • Figure 1: The circles $\{S_i\}_{1 \leq i \leq W-1},$ each of diameter $b = \frac{2\delta r}{L} ,$ with corresponding centres that are spaced $a = r(1-\delta)$ apart. The vertices $u$ and $v$ are represented by $A = (0,0)$ and $B = (Kr,0),$ respectively and the dark dots represent the vertices of $G_{loc}$ that form the path $\pi$ from $A$ to $B,$ denoted by the solid line segments.
  • Figure 2: The path $\tau$ as described in the proof below with $U_0 = A,U_{N+1} = B$ and $U_i$ denoted as $i$ for $1 \leq i \leq N.$

Theorems & Definitions (3)

  • theorem 1
  • lemma 1
  • theorem 2