One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade (Braunstein et al. in Phys Rev Lett 91(16) (3,4), distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n (τ −3)/(τ −1) . In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of Lévy processes "without replacement" (Aldous and Limic Electron in J Probab 3(3):59, 1998), yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent τ ∈ (3, 4) where edges are rescaled by n −(τ −3)/(τ −1) yielding the first rigorous proof of the above conjecture. in this case are compact "tree-like" random fractals with a dense collection of hubs (infinite degree vertices), a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal (τ − 2)/(τ − 3) a.s., in stark contrast to the Erdős-Rényi scaling limit whose Minkowski dimension is 2 a.s. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent τ ∈ (3, 4). Our proof makes crucial use of inhomogeneous continuum random trees (ICRT), which have previously arisen in the study of the entrance boundary of the additive coalescent. We show that tilted versions of the same objects using the associated mass measure, describe connectivity properties of the multiplicative coalescent. Since convergence of height processes of corresponding approximating p-trees is not known, we use general methodology in Athreya et al. (2014) and develop novel techniques relying on first showing convergence in the Gromov-weak topology and then extending this to Gromov-Hausdorff-Prokhorov convergence by proving a global lower mass-bound.:Keywords Multiplicative coalescent · p-trees · Inhomogeneous continuum random trees · Critical random graphs · Gromov-Hausdorff distance · Gromov-weak topology Mathematics Subject Classification Primary 60C05 · 05C80List of notation and terminology (S t Critical exponentsSpace of counting measures on R + equipped with the vague topologyMinimal number of open balls with radius δ required to cover a metric space M For a tree t ∈ T * I,(k+ ) , the functional defined in (4.24) ht(t)Height of a tree t with edge lengths incorporated into the distanceRoot-to-vertex weights and measures in T θ (∞) defined in (2.7) and (2.8). See ...

We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erd\H{o}s-R\'enyi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.Comment: 33 pages. Minor improvement

The last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models to gain insight into real-world systems. In this study we analyze the maximal components at criticality of one famous class of such models, the rank-one inhomogeneous random graph model [44], [17, Section 16.4]. Viewing these components as measured random metric spaces, under finite moment assumptions for the weight distribution, we show that the components in the critical scaling window with distances scaled by n −1/3 converge in the Gromov-Haussdorf-Prokhorov metric to rescaled versions of the limit objects identified for the Erdős-Rényi random graph components at criticality in [3]. A key step is the construction of connected components of the random graph through an appropriate tilt of a fundamental class of random trees called p-trees [8,22]. This is the first step in rigorously understanding the scaling limits of objects such as the minimal spanning tree and other strong disorder models from statistical physics [19] for such graph models. By asymptotic equivalence [34], the same results are true for the Chung-Lu model [24][25][26] and the Britton-Deijfen-Martin-Löf model [20]. A crucial ingredient of the proof of independent interest are tail bounds for the height of p-trees. The techniques developed in this paper form the main technical bedrock for the general program developed in [11] for proving universality of the continuum scaling limits in the critical regime for a wide array of other random graph models including the configuration model and inhomogeneous random graphs with general kernels [17].

Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter t (usually related to edge density) and a (model dependent) critical time t c which specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erdős-Rényi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like n 2/3 ) and (b) the structure of components (rescaled by n −1/3 ) converge to random fractals related to the continuum random tree. Till date, (a) has been proven for a number of models using different techniques while (b) has been proven for only two models, the classical Erdős-Rényi random graph and the rank-1 inhomogeneous random graph. The aim of this paper is to develop a general program for proving such results. The program requires three main ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent (ii) scaling exponents of susceptibility functions are the same as the Erdős-Rényi random graph and (iii) macroscopic averaging of expected distances between random points in the same component in the barely subcritical regime. We show that these apply to a number of fundamental random graph models including the configuration model, inhomogeneous random graphs modulated via a finite kernel and bounded size rules. Thus these models all belong to the domain of attraction of the classical Erdős-Rényi random graph. As a by product we also get results for component sizes at criticality for a general class of inhomogeneous random graphs.

We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such results were derived by Bhamidi, van der Hofstad, Sen (2018) [15]. We develop general principles under which the identical scaling limits as [15] can be obtained. Of independent interest, we derive refined asymptotics for various susceptibility functions and the maximal diameter in the barely subcritical regime.