A limit theorem for small cliques in inhomogeneous random graphs

Abstract

The theory of graphons comes with a natural sampling procedure, which results in an inhomogeneous variant of the Erdős-Rényi random graph, called W -random graphs. We prove, via the method of moments, a limit theorem for the number of r-cliques in such random graphs. We show that, whereas in the case of dense Erdős-Rényi random graphs the fluctuations are normal of order n r−1 , the fluctuations in the setting of W -random graphs may be of order n 0, r−1 , or n r−0.5 .Furthermore, when the fluctuations are of order n r−0.5 they are normal, while when the fluctuations are of order n r−1 they exhibit either normal or a particular type of chi-square behavior whose parameters relate to spectral properties of W . These results can also be deduced from a general setting, based on the projection method. In addition to providing alternative proofs, our approach makes direct links to the theory of graphons.

Higher-order fluctuations in dense random graph models

“…(1.7) (the values of the constants c 1 to c 17 can be found in the Appendix); these results are again consistent with Hladký et al (2019) and Chatterjee and Bhattacharya (2021). Note also that all these quantities are again uncorrelated and themselves sums of uncorrelated random variables, and they are scaled to be of order 1.…”

“…These convergence results were also obtained by Hladký et al (2019) and Chatterjee and Bhattacharya (2021). Note that even in the third case, the contribution of W in (1.5) does not vanish if α = β, although the fluctuation only contributes to a scale that is smaller than the dominating fluctuation.…”

“…There have been some recent efforts to understand the subgraphs counts in the context of graphons and dense graph limit theory. Hladký, Pelekis and Šileikis (2019) analysed the limiting distributions of r-clique counts of a random graph obtained through sampling from a graphon (which is our model G(n, κ) below), and Chatterjee and Bhattacharya (2021) generalised the results to arbitrary subgraphs. Maugis (2020) analysed localised versions, where the counts are not global, but only over one specific vertex.…”

Prominent examples of flip processes

“…We define the rescaling core of ( k ) k≥2 , which we denote by  • , to be the set of elements in the interior of  (with respect to the topology induced by the cut norm) at which the maps W  → Ψ t (W), for t ≥ 0, are continuous (with respect to the topology induced by the cut norm). In particular, the rescaling core of ( k ) k≥2 is a set of graphons W such that Ψ t (W) is a good approximation for Φ t∕(k) 2  k ( W) when k is sufficiently large and W is sufficiently close to W. (Let us note that in principle, we might investigate rescaled trajectories outside of the rescaling core as well, and this indeed seems to be doable using the main result from [13]. However, the favourable continuity property is lost in this case, and in particular such rescaled trajectories would have no connection to flip processes on finite graphs, as provided by Theorem 1.1.)…”

“…In particular, the rescaling core of $$$ {\left({\mathcal{R}}_k\right)}_{k\ge 2} $$$ is a set of graphons $$$ W $$$ such that $$$ {\Psi}^t(W) $$$ is a good approximation for $$$ {\Phi}_{{\mathcal{R}}_k}^{t/{(k)}_2}\left(\tilde{W}\right) $$$ when $$$ k $$$ is sufficiently large and $$$ \tilde{W} $$$ is sufficiently close to $$$ W $$$. (Let us note that in principle, we might investigate rescaled trajectories outside of the rescaling core as well, and this indeed seems to be doable using the main result from [13]. However, the favourable continuity property is lost in this case, and in particular such rescaled trajectories would have no connection to flip processes on finite graphs, as provided by Theorem 1.1.…”