Cliques in geometric inhomogeneous random graphs

Abstract

Many real-world networks were found to be highly clustered and contain a large amount of small cliques. We here investigate the number of cliques of any size $k$ contained in a geometric inhomogeneous random graph: a scale-free network model containing geometry. The interplay between scale-freeness and geometry ensures that connections are likely to form between either high-degree vertices, or between close by vertices. At the same time, it is rare for a vertex to have a high degree, and most vertices are not close to one another. This trade-off makes cliques more likely to appear between specific vertices. In this article, we formalize this trade-off and prove that there exists a typical type of clique in terms of the degrees and the positions of the vertices that span the clique. Moreover, we show that the asymptotic number of cliques as well as the typical clique type undergoes a phase transition, in which only $k$ and the degree-exponent $\tau$ are involved. Interestingly, this phase transition shows that for small values of $\tau$, the underlying geometry of the model is irrelevant: the number of cliques scales the same as in a non-geometric network model.

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

“…3−β , as was previously observed for hyperbolic random graphs [7] and for GIRGs of constant dimensionality [33]. The latter work explains this behavior by showing that for k < 2 3−β , the number of cliques is strongly dominated by "geometric" cliques forming among vertices whose distance is of order n −1/d regardless of their weight.…”

“…The behavior in the first column is the same as in hyperbolic random graphs [7], and the behavior in the third column is the same as in the IRG model [14]. Results marked with * were previously known for constant k [33].…”

“…In addition to investigating how the number and size of cliques depends on d, it remains to analyze among which vertices k-cliques form dominantly. For constant d and β ∈ (2, 3) this was previously done by Michielan and Stegehuis who noted that cliques of size k > 2 3−β are dominantly formed among vertices of weight in the order of √ n like in the IRG model, whereas cliques of size k < 2 3−β dominantly appear among vertices within distance in the order of n −1/d [33]. This characterizes the geometric and non-geometric nature of cliques of size larger and smaller than 2 3−β , respectively.…”

“…Geometric inhomogeneous random graphs are an extension of IRGs, where in addition to the weight, each vertex v is also equipped with a position x v in some geometric space and the probability for edges to form depends on their weights and the distance in the underlying space [27]. While, in its raw form, the GIRG framework is rather general, we align our paper with existing analysis on GIRGs [6,28,33] and consider the d-dimensional torus T d equipped with L ∞ -norm as the geometric ground space. More precisely, in what we call the standard GIRG model, the positions x of the vertices are drawn independently and uniformly at random from T d , according to the standard Lebesgue measure.…”

“…However, it was not known how the size and the number of cliques depends on d if we use the torus as our ground space, and how the clique structure in high-dimensions behaves for inhomogeneous weights. We give explicit bounds on the expected number of cliques of a given size k, which we afterwards turn into bounds on the clique number ω(G), i.e., the size of the largest clique in the graph G. While the expected number of cliques in the GIRG model was previously studied by Michielan and Stegehuis [33] when the power-law exponent of the degree distribution satisfies β ∈ (2, 3), to the best of our knowledge, the clique number of GIRGs remains unstudied even in the case of constant (but arbitrary) dimensionality. We close this gap, reproduce the existing results, and extend them to the case β ≥ 3 and the case where d can grow as a function of the number of vertices n in the graph.…”