Cliques in High-Dimensional Random Geometric Graphs

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.

Cliques in geometric inhomogeneous random graph

“…The required parameters for each sample of the GIRG model are: n, τ , d and the temperature 1/γ. Moreover, in the code we add an additional parameter c, corresponding to the constant factor in the weight distribution (1). This factor affects the edge probability in (3): the mean number of edges in the graph increases as c increases.…”

“…Results on the number of cliques in random graphs with underlying geometry are less well-studied, as the presence of geometry creates correlations between the presence of different edges, making it difficult to compute the probability that a given clique is present. Still, some results are known for high-dimensional geometric random graphs [1] and hyperbolic random graphs [4], showing that these types of random graphs typically contain a larger number of cliques than non-geometric models as long as the dimension of the underlying space is not too large. Particular attention has been given to the clique number: the largest clique in the network [4,12,7].…”

“…Equation (3) shows an interesting relation between the properties of the vertices and their connection probabilities: Two vertices with high weights are more likely to connect. However, vertices with high weights are rare due to (1). Moreover, two vertices are more likely to connect if their positions are close.…”

Cliques in high-dimensional random geometric graphs

“…This paper is an extended version of the Complex Networks conference paper Avrachenkov and Bobu (2019). Compared to the short paper, we present here absolutely new numerical results and more specifically discuss the application of the present work to real-life tasks of machine learning.…”

“…Compared to the short paper, we present here absolutely new numerical results and more specifically discuss the application of the present work to real-life tasks of machine learning. Also, we have improved Theorems 4 and 5 and give full proofs of the results mentioned in Avrachenkov and Bobu (2019).…”