[Paper Review] Bohman-Frieze processes at criticality and emergence of the giant component

Shankar Bhamidi, Amarjit Budhiraja+1 more|arXiv (Cornell University)|Jun 6, 2011

Stochastic processes and statistical mechanics34 references3 citations

TL;DR

This paper establishes that the Bohman-Frieze random graph process exhibits the same critical behavior as the classical Erdős-Rényi model: at criticality, component sizes properly rescaled and re-centered converge to the same limiting distribution as the standard multiplicative coalescent. The authors prove this via a novel coupling with near-critical multitype branching processes and functional analytic tools, confirming universality in the critical window despite delayed giant component emergence due to the preferential edge selection rule.

ABSTRACT

The evolution of the usual Erdős-Rényi random graph model on n vertices can be described as follows: At time 0 start with the empty graph, with n vertices and no edges. Now at each time k, choose 2 vertices uniformly at random and attach an edge between these two vertices. Let \bfG_n(k) be the graph obtained at step k. Refined analysis in random graph theory now shows that for fixed t\in \Rbold, when k(n) = n/2+ n^{2/3} t/2, the sizes of the components in \bfG_n(k(n)) scale like n^{2/3} and rescaled component sizes converge to the standard multiplicative coalescent at time $t$. The last decade has seen variants of this process introduced, under the name Achlioptas processes, to understand the effect of simple changes in the edge formation scheme on the emergence of the giant component. Stimulated by a question of Achlioptas, one of the simplest and most popular of such models is the Bohman Frieze (BF) model wherein at each stage $k$, 2 edges e_1(k)=(v_1,v_2) and e_2(k) = (v_3, v_4) are chosen uniformly at random. If at this time v_1, v_2 are both isolated then this edge is added, otherwise e_2 is added. Then \cite{bohman2001avoiding} (and further analysis in \cite{spencer2007birth}) show that once again there is a critical parameter, which is larger than 1, above and below which the asymptotic behavior is as in the Erdős-Rényi setting. While an intense study for this and related models seems to suggest that at criticality, this model should be in the same universality class as the original Erdős-Rényi process, a precise mathematical treatment of the dynamics in the critical window has to date escaped analysis. In this work we study the component structure of the BF model in the critical window and show that at criticality the sizes of components properly rescaled and re-centered converge to the standard multiplicative coalescent.

Motivation & Objective

• To rigorously analyze the component structure of the Bohman-Frieze random graph model in the critical window, where the giant component emerges.
• To determine whether the Bohman-Frieze process belongs to the same universality class as the classical Erdős-Rényi model at criticality.
• To establish the convergence of rescaled component sizes to the standard multiplicative coalescent, despite the delayed emergence of the giant component due to the edge-formation rule.
• To develop a coupling framework using near-critical multitype branching processes and functional analytic methods to analyze the dynamics in the critical regime.

Proposed method

• Construct two auxiliary processes, $\bar{\boldsymbol{C}}_n^{-}$ and $\bar{\boldsymbol{C}}_n^{+}$, that stochastically bound the Bohman-Frieze process $\bar{\boldsymbol{C}}_n^{\scriptscriptstyle BF}$ in the component size space $l^{2}_{\downarrow}$.
• Use the multiplicative coalescent as the limiting process for both bounding processes, leveraging known convergence results from Aldous and others.
• Apply functional convergence techniques in the space $\mathcal{D}((-∞,\infty):l^{2}_{\downarrow})$ to establish convergence of the Bohman-Frieze process to the same limit.
• Employ tightness criteria from Aldous for $\mathcal{D}$-space processes, particularly analyzing the difference in squared $l^2$-norms of the bounding processes.
• Use the Markov property and moment bounds of the multiplicative coalescent to verify tightness of the difference process $\mathcal{V}(\lambda) = \mathcal{U}_+(\lambda) - \mathcal{U}_-(\lambda)$.
• Leverage the partial order $\preceq$ on $l^{2}_{\downarrow}$ to compare component configurations and ensure monotonicity in the coupling.

Experimental results

Research questions

• RQ1 Does the Bohman-Frieze process exhibit the same critical behavior as the classical Erdős-Rényi model, specifically convergence to the multiplicative coalescent in the critical window?
• RQ2 How does the delayed giant component emergence in the Bohman-Frieze model affect the scaling limits of component sizes at criticality?
• RQ3 Can the dynamics of the Bohman-Frieze process be approximated by near-critical multitype branching processes in the critical regime?
• RQ4 Is the limiting distribution of rescaled component sizes in the Bohman-Frieze model identical to that of the Erdős-Rényi model, despite different edge-formation rules?

Key findings

• The rescaled component size process $\bar{\boldsymbol{C}}_n^{\scriptscriptstyle BF}(\lambda)$ converges in distribution to the standard multiplicative coalescent $\boldsymbol{X}(\lambda)$ as $n \to \infty$, for all $\lambda \in \mathbb{R}$.
• The convergence holds in the Skorokhod space $\mathcal{D}((-∞,\infty):l^{2}_{\downarrow})$, confirming functional convergence of the entire trajectory.
• The critical window scaling $k(n) = n/2 + n^{2/3}t/2$ leads to component sizes of order $n^{2/3}$, matching the Erdős-Rényi case.
• The limiting process is universal: despite the edge-formation rule favoring isolated vertices, the critical behavior matches that of the classical model.
• The tightness of the difference process $\mathcal{V}(\lambda)$ is established via moment bounds and the Markov property of the multiplicative coalescent.
• The coupling argument via $\bar{\boldsymbol{C}}_n^{-} \preceq \bar{\boldsymbol{C}}_n^{\scriptscriptstyle BF} \preceq \bar{\boldsymbol{C}}_n^{+}$ ensures convergence by squeezing the process between two known limits.