Convergence of Achlioptas processes via differential equations with unique solutions

TL;DR

This paper proves that for a broad class of Achlioptas processes, the rescaled size of the giant component converges and is continuous, based on the uniqueness of solutions to associated differential equations.

Contribution

It establishes a general framework linking Achlioptas processes to differential equations, ensuring convergence and continuity of the giant component size.

Findings

Rescaled giant component size converges for rules like the product rule.

Continuity of the limit function is guaranteed under unique differential equation solutions.

Applicable to a wide class of Achlioptas-like processes.

Abstract

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. The evolution of the rescaled size of the largest component in such variations of the Erd\H{o}s--R\'enyi random graph process has recently received considerable attention, in particular for for Bollob\'as's `product rule'. In this paper we establish the following result for rules such as the product rule: the limit of the rescaled size of the `giant' component exists and is continuous provided that a certain system of differential equations has a unique solution. In fact, our result applies to a very large class of Achlioptas-like processes. Our proof relies on a general idea which relates the evolution of stochastic processes to an associated system of differential equations. Provided that the latter has a unique solution, our approach shows that certain discrete quantities converge (after appropriate rescaling) to this solution.