On the distances within cliques in a soft random geometric graph

TL;DR

This paper investigates the structure of distances within cliques in a soft random geometric graph on a torus, revealing how the maximum intra-clique distances scale and their limiting distribution.

Contribution

It provides the first detailed analysis of maximal distances in cliques of soft random geometric graphs, including their asymptotic behavior and distributional limits.

Findings

Maximal clique distances scale with clique size.

Asymptotically, large-distance cliques contain only one remote point.

Maximal clique distances converge to a Fréchet distribution.

Abstract

We study the distances of edges within cliques in a soft random geometric graph on a torus, where the vertices are points of a homogeneous Poisson point process, and far-away points are less likely to be connected than nearby points. We obtain the scaling of the maximal distance between any two points within a clique of size $k$. Moreover, we show that asymptotically in all cliques with large distances, there is only one remote point and all other points are nearby. Furthermore, we prove that a re-scaled version of the maximal $k$-clique distance converges in distribution to a Fr\'echet distribution. Thereby, we describe the order of magnitude according to which the largest distance between two points in a clique decreases with the clique size.