Abstract
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 nrightarrow 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.
Highlights
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
Since p is the normalized surface area of a spherical cap of angle arccos tp,d, we learn from convex geometry that: 1 √ 6tp,d d
Summary
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
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