TY - JOUR AU - Avrachenkov, Konstantin E. AU - Bobu, Andrei V. PY - 2020 DA - 2020/11/23 TI - Cliques in high-dimensional random geometric graphs JO - Applied Network Science SP - 92 VL - 5 IS - 1 AB - 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 $$n\rightarrow \infty$$, and $$d \rightarrow \infty$$, 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 $$d \gg \log ^{1 + \epsilon } n$$. As for the cliques of size 3, we present new bounds on the expected number of triangles in the case $$\log ^2 n \ll d \ll \log ^3 n$$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. SN - 2364-8228 UR - https://doi.org/10.1007/s41109-020-00335-6 DO - 10.1007/s41109-020-00335-6 ID - Avrachenkov2020 ER -