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.

随机几何图现在已经成为流行的研究对象。这些图的定义非常简单，因为它们产生紧密联系的社区的能力，它们比传统的Erdős-Rényi图更好地描述了真实的网络。随机几何图的n个顶点是d维欧几里得空间中的点，并且如果两个顶点彼此靠近，则两个顶点相邻。在固定d的情况下，这些图的许多特性已被揭示。但是，实际上还没有探索尺寸d增大的情况。当一个人的数据集为n时，这种状态对应于现实情况具有大量特征的观测，这在当今的数据科学中非常普遍。在本文中，我们研究了\（n \ rightarrow \ infty \）和\（d \ rightarrow \ infty \）时随机几何图的集团结构，平均顶点度的增长明显慢于n。我们表明在这些条件下，随机几何图不包含大小为4 a的集团。s。如果仅\（d \ gg \ log ^ {1 + \ epsilon} n \）。至于大小为3的集团，在\（\ log ^ 2 n \ ll d \ ll \ log ^ 3 n \）的情况下，我们给出了三角形的预期数目的新边界可以改善以前已知的结果。此外，我们提供了新的数值结果，表明即使使用很小的n，也可以使用三角形的数量来检测基本几何形状。