Cliques in high-dimensional random geometric graphs

Random geometric graphs have become now a popular object of research. Defined rather simply, these graphs describe real networks much better than classical Erdős–Rényi graphs due to their ability to produce tightly connected communities. The n vertices of a random geometric graph are points in d -dimensional Euclidean space, and two vertices are adjacent if they are close to each other. Many properties of these graphs have been revealed in the case when d is fixed. However, the case of growing dimension d is practically unexplored. This regime corresponds to a real-life situation when one has a data set of n observations with a significant number of features, a quite common case in data science today. In this paper, we study the clique structure of random geometric graphs when n→ ∞ n → ∞ , and d → ∞ d → ∞ , and average vertex degree grows significantly slower than n . We show that under these conditions, random geometric graphs do not contain cliques of size 4 a. s. if only d log 1 + ε n d ≫ log 1 + ϵ n . As for the cliques of size 3, we present new bounds on the expected number of triangles in the case log ² n d log ³ n log 2 n ≪ d ≪ log 3 n that improve previously known results. In addition, we provide new numerical results showing that the underlying geometry can be detected using the number of triangles even for small n .