Abstract
We consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs. arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), the asymptotic homogeneous case as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or the power-law case as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).
Highlights
Motivation and model Random graphs have generated a large amount of literature
From the point of view of precise asymptotics, one of the most important papers is certainly the contribution of Aldous [3], who introduced a stochastic process point of view and paved the way towards the study of scaling limits of critical random graphs. He obtained the asymptotics for the sequence of sizes of the connected components of G(n, p) in the so-called critical window where the phase transition occurs
The tree corresponding to the clients of the continuous analogue of the Markovian queue that is driven by X is the Lévy tree yielded by X, which is defined through its contour process as introduced by Le Gall & Le Jan [31]
Summary
Motivation and model Random graphs have generated a large amount of literature. This is even the case for one single model: the Erdos–Rényi graph G(n, p) (graph with n vertices connected pairwise in an i.i.d. way with probability p ∈ [0, 1]). The tree corresponding to the clients of the continuous analogue of the Markovian queue that is driven by X is the Lévy tree yielded by X , which is defined through its contour process as introduced by Le Gall & Le Jan [31]. Since bn is an explorationtime factor, we require that bn E[Cn], where Cn stands for the number of clients who are served before the arrival of Client 1 (i.e. the client corresponding to the largest weight w1(n) ) in the wn-LIFO queue encoding Gwn. Let us denote by Dn the sum of the weights of the vertices explored before visiting Client 1.
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