Cliques in high-dimensional random geometric graphs

Author:

Avrachenkov Konstantin E.,Bobu Andrei V.ORCID

Abstract

AbstractRandom 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 $$n\rightarrow \infty$$ n , and $$d \rightarrow \infty$$ 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 $$d \gg \log ^{1 + \epsilon } n$$ 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 $$\log ^2 n \ll d \ll \log ^3 n$$ 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.

Publisher

Springer Science and Business Media LLC

Subject

Computational Mathematics,Computer Networks and Communications,Multidisciplinary

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Limit theory of sparse random geometric graphs in high dimensions;Stochastic Processes and their Applications;2023-09

2. From Delaunay triangulation to topological data analysis: generation of more realistic synthetic power grid networks;Journal of the Royal Statistical Society Series A: Statistics in Society;2023-05-22

3. Cliques in geometric inhomogeneous random graphs;Journal of Complex Networks;2021-12-20

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