Connectivity Thresholds for Bounded Size Rules

TL;DR

This paper analyzes how different Achlioptas processes, especially bounded size rules, influence the connectivity threshold in evolving random graphs, providing detailed distributional insights and identifying processes that optimize the connectivity transition.

Contribution

It offers a detailed analysis of bounded size rules' impact on graph connectivity and identifies processes that accelerate the connectivity transition.

Findings

Distribution of rounds until connectivity is established

Dynamics of component merging during graph evolution

Identification of processes that maximize the speed of connectivity

Abstract

In an Achlioptas process, starting with a graph that has n vertices and no edge, in each round $d \geq 1$ edges are drawn uniformly at random, and using some rule exactly one of them is chosen and added to the evolving graph. For the class of Achlioptas processes we investigate how much impact the rule has on one of the most basic properties of a graph: connectivity. Our main results are twofold. First, we study the prominent class of bounded size rules, which select the edge to add according to the component sizes of its vertices, treating all sizes larger than some constant equally. For such rules we provide a fine analysis that exposes the limiting distribution of the number of rounds until the graph gets connected, and we give a detailed picture of the dynamics of the formation of the single component from smaller components. Second, our results allow us to study the connectivity transition of all Achlioptas processes, in the sense that we identify a process that accelerates it as much as possible.