The phase transition in bounded-size Achlioptas processes

TL;DR

The paper resolves the finite-size scaling of percolation for all bounded-size Achlioptas processes, showing Erdős–Rényi-like phase transitions with rule-dependent constants. It develops a robust methodology combining a two-round exposure, differential-equation and PDE analysis, and branching-process couplings to handle edge dependencies. The results establish analytic scaling laws: the largest component grows linearly just above criticality, small components follow Erdős–Rényi-like trees with exponential cutoffs, and susceptibility moments share Erdős–Rényi exponents across all bounded-size rules. This work places bounded-size Achlioptas processes into the same universality class as Erdős–Rényi, highlighting the robustness of the phase-transition picture beyond independence assumptions and informing understanding of dynamic random graphs with dependencies.

Abstract

Perhaps the best understood phase transition is that in the component structure of the uniform random graph process introduced by Erdős and Rényi around 1960. Since the model is so fundamental, it is very interesting to know which features of this phase transition are specific to the model, and which are `universal', at least within some larger class of processes (a `universality class'). Achlioptas process, a class of variants of the Erdős--Rényi process that are easy to define but difficult to analyze, have been extensively studied from this point of view. Here, settling a number of conjectures and open problems, we show that all `bounded-size' Achlioptas processes share (in a strong sense) all the key features of the Erdős--Rényi phase transition. We do not expect this to hold for Achlioptas processes in general.

Figures (7)

• Figure 1: Simulation of the fraction $L_1(G^{{\mathcal{R}}}_{n,tn})/n$ of vertices in the largest component after $tn$ steps in various Achlioptas processes for $n=10^{13}$, giving an approximation to the limiting curve $\rho^{{\mathcal{R}}}(t)$. The rules are Erdős--Rényi (ER), Bohman--Frieze (BF) and Bohman--Kravitz (BK), all of which are bounded-size, and the sum and product rules (SR and PR), which are not.
• Figure 2: The basic proof setup. We condition on the component structure of the graph after $i_0=(t_{\mathrm{c}}-\sigma)n$ steps, and then reveal information about steps $i_0, \ldots, i_1=(t_{\mathrm{c}}+\varepsilon)n$ via a two-round exposure. The crux will be that the distribution of the second exposure round is extremely well behaved (consisting of many independent random choices), which eventually allows us to analyze the component size distribution of the resulting graph $G_i$ in step $i_0 < i \leqslant i_1$ via branching process methods.
• Figure 3: Example of the 'partial graph' used for the second exposure round of the graph $G_i$ in step $i_0 \leqslant i \leqslant i_1$. After the first exposure round we have revealed a certain subgraph $H_i$ of $G_i$, together with the $V_S$--endvertices of all $V_S$--$V_L$ edges in $G_i\setminus H_i$ (the endvertices in $V_L$ are still uniformly random), and the number of $V_L$--$V_L$ edges of $G_i\setminus H_i$ (these edges are still uniform). We obtain the graph $G_i$ by (a) connecting the undetermined $V_S$--$V_L$ edges to random vertices in $V_L$, and (b) adding the correct number of random $V_L$--$V_L$ edges. The point of this description is that it involves many independent random choices, so it allows us to analyze the component size distribution of the resulting graph $G_i$ using branching process techniques.
• Figure 4: An example (the same as in Figure \ref{['fig:po2']}) of the auxiliary graph $H_i$, which is determined by the first exposure round. A component of type $(k,r)$ contains $k$ vertices from $V_S$ and has $r$ incident $V_S$--$V_L$ edges, whose so-far unrevealed $V_L$--vertices are represented by stubs (here depicted by the circle-shaped endpoints in $V_L$). We obtain $G_i$ by (i) connecting each stub to a randomly chosen vertex from $V_L$, and (ii) adding $Q_{0,2}(i)$ random $V_L$--$V_L$ edges to $H_i$.
• Figure 5: Example of the initial (marked) graph $H=H({\mathfrak S})$ and the induced subgraph $H_L=H[V_L]$. $H_L$ is a standard (unmarked) graph, consisting of $N_k({\mathfrak S})/k$ components of size $k$ for each $k>K$. In $H$, for each $k,r \geqslant 0$ there are also $Q_{k,r}$ (or $\mathrm{Po}(Q_{k,r})$ in the Poissonized model) $(k,r)$-components: $k$-vertex components with $r$ stubs.

Theorems & Definitions (152)

• Remark 1.1
• Theorem 1.2
• Remark 1.3: Notation
• Remark 1.4: Critical window
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Theorem 2.5: Linear growth of the scaling limit
• Remark 2.6