Deviation Estimates for Extremal Relay Random Geometric Graphs
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)~\(G\) with adjacency distance~\(rn\) as relays. We call the resulting graph as a 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~\(G\) with independent, exponentially distributed weights and obtain bounds for the maximum possible weight~\(Wn\) of a relay RGG with a given length~\(Ln.\)
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