Counting subgraphs in bounded-size Achlioptas processes

TL;DR

This paper adapts a static subgraph counting approach from Erdős–Rényi analysis to the more complex Achlioptas processes, enabling new insights into component structures near criticality.

Contribution

It extends the static subgraph count method to Achlioptas processes, providing new asymptotic formulas and bounds for component sizes near the critical point.

Findings

Expected number of k-vertex tree components is approximately c_{k,t}n.

Derived sharp bounds on the largest non-giant component near criticality.

Obtained the limiting distribution of fluctuations of the largest component.

Abstract

Achlioptas processes such as the Bohman--Frieze process are much harder to analyse than the classical Erd\H{o}s--R\'enyi process, due to the dependence between edges added at different stages. This dependence means that most analysis so far is dynamic, often based on the differential equation method. In the Erd\H{o}s--R\'enyi case there is an alternative static approach, pioneered by Erd\H{o}s, R\'enyi and Bollob\'as, based on evaluating the expectation (and higher moments) of various subgraph counts, and using this to study the component structure. Here we show that this latter approach can be applied (with some complications) to the Bohman--Frieze process. For example, we are able to show that the expected number $\mu_{k,t,n}$ of $k$-vertex tree components after $tn$ steps satisfies (essentially) $\mu_{k,t,n}=c_{k,t}n(1+O(k/\sqrt{n}))$. Our method gives a very complicated formula for $c_{k,t}$, which seems to be unusable. However, since $c_{k,t}$ does not depend on $n$, we may use recent results obtained by the differential equation method and branching process analysis to find the asymptotics of $c_{k,t}$ as $k\to\infty$. The latter results also give a formula for $\mu_{k,t,n}$ of the form $c_{k,t}n$ plus an error term, with a much more usable description of $c_{k,t}$ but a much worse error term. We combine the best of both worlds to prove a number of new results about the process near criticality. In particular, we obtain extremely sharp bounds on the size of the largest non-giant component near criticality, including the limiting distribution of its fluctuations.