Cliques in high-dimensional random geometric graphs

Description

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 $n \to \infty$, and $d \to \infty$, 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 $d >> \log^{1+\epsilon}(n)$. As for the cliques of size 3, we present new bounds on the expected number of triangles in the case $\log^2(n) << d << \log^3(n)$ 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 $n$.

Abstract

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