Cliques in High-Dimensional Random Geometric Graphs

Konstantin Avrachenkov,Andrei Bobu

doi:10.1007/978-3-030-36687-2_49

Abstract

Random geometric graphs are good examples of random graphs with a tendency to demonstrate community structure. Vertices of such a graph are represented by points in Euclid space $R^d$ , and edge appearance depends on the distance between the points. Random geometric graphs were extensively explored and many of their basic properties are revealed. However, in the case of growing dimension $d → ∞$ practically nothing is known; this regime corresponds to the case of data with many features, a case commonly appearing in practice. In this paper, we focus on the cliques of these graphs in the situation when average vertex degree grows significantly slower than the number of vertices n with $n → ∞$ and $d → ∞$. We show that under these conditions random geometric graphs do not contain cliques of size 4 a.s. As for the size 3, we will present new bounds on the expected number of triangles in the case $log^2(n) << d << log^3(n)$ that improve previously known results.

Highlights

In which the link presence is a random variable, appear in the 20 th century with the simplest model proposed by Erdős and Rényi
The edges between vertices appear independently with equal probability. This model fails to describe some important properties of many real networks, such as the community structure
Before we present our main result on the clique number, let us prove a lemma that will be useful later

Summary

Introduction

Graphs are the most natural way to model many real-world networks. In which the link presence is a random variable, appear in the 20 th century with the simplest model proposed by Erdős and Rényi (see Erdős and Rényi 1960; Bollobás 2001; Alon and Spencer 2004). In this model, the edges between vertices appear independently with equal probability. The edges between vertices appear independently with equal probability This model fails to describe some important properties of many real networks, such as the community structure. Perhaps random geometric graphs are the simplest natural model where the edge appearance depends only on the Euclidean distance between given

Objectives

Results

Conclusion