On the distances within cliques in a soft random geometric graph

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.