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 Fréchet 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.

References in corpus (14)

• Network robustness and fragility: Percolation on random graphs
• Resilience of the Internet to random breakdowns
• Hyperbolic Geometry of Complex Networks
• Central limit theorems for $U$-statistics of Poisson point processes
• Connectivity of soft random geometric graphs
• Number of cliques in random scale-free network ensembles
• Counting cliques and cycles in scale-free inhomogeneous random graphs
• Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry
• Small worlds and clustering in spatial networks
• Connectivity of Soft Random Geometric Graphs Over Annuli
• Detecting hyperbolic geometry in networks: why triangles are not enough
• Statistical mechanics of random geometric graphs: Geometry-induced first order phase transition
• Cliques in geometric inhomogeneous random graph
• The longest edge of the one-dimensional soft random geometric graph with boundaries