The augmented multiplicative coalescent, bounded size rules and critical dynamics of random graphs

Abstract

Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of n vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed K ≥ 1, all components of size greater than K are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous's standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is 'nearly' Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erdős-Rényi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from [8], on the size of the largest component in the barely subcritical regime.

Critical window for the configuration model: finite third moment degrees

“…We use percolation on a super-critical configuration model to show the joint convergence of the scaled vectors of component sizes at multiple locations of the percolation scaling window. We also obtain the asymptotic distribution of the number of surplus edges in each component and show that the sequence of vectors consisting of the re-scaled component sizes and surplus converges to a suitable limit under a strong topology as discussed in [6]. These results give very strong evidence in favor of the structural similarity of the component sizes of CM n (d) and Erdős-Rényi random graphs at criticality.…”

A new encoding of coalescent processes: applications to the additive and multiplicative cases

“…In the multiplicative case, we also construct a version of the standard augmented multiplicative coalescent of Bhamidi et al [10] as a "decorated" process of γ × . For a connected graph, let the excess be the minimum number of edges that one must remove in order to obtain a tree.…”

Critical random forests

“…Aldous's result has been extended in multiple ways. The same Brownian scaling limit has been shown to arise in more general settings including configuration models and inhomogeneous random graphs, provided the tail of the degree distribution is sufficiently light [32,28,19,10,8,16]. In some such cases, finer scaling limits describing the metric structure of the large components, as well as their size, have been obtained, in terms of objects related to the Brownian continuum random tree [1,2,7,9].…”

“…Z N,p,R tN 2/3 +1 − Z N,p,R tN 2/3 Z N,p,R tN 2/3 = xN 1/3 → b R (t, x),uniformly for t ∈ [0, T ] and x in any compact interval in (0, ∞). We define the rescaled processZ N,λ,R from Z N,p,R analogously to(8). Then we haveZ N,p,R d → Z λ,R uniformly on [0, T ].From this, P sup n∈[0,T N 2/3 ] Z N,p,R n > RN 1/3 → 0, as R → ∞, and so as processes on [0, T ], the law ofZ N,p,R converges to the law of Z N,p as R → ∞, and the law of Z λ,R converges to the law of Z λ .…”