Critical behavior of two-choice rules: a class of Achlioptas processes

Abstract

Achlioptas processes are a class of dynamically grown random graphs where on each step several edges are chosen at random but only one is added. The sum rule, product rule, and bounded size rules have been extensively studied. Here we introduce a new collection of rules called two-choice rules. In these systems one first pick $m$ vertices at random from the graph and chooses a vertex $v$ according to some rule based on their cluster sizes. The procedure is then repeated with a second independent sample to pick a vertex $v'$ and we add an edge from $v$ and $v'$. These systems are tractable because the cluster size distribution satisfies an analog of the Smoluchowski equation. We study the critical exponents associated with the phase transitions in five of these models. In contrast to the situation for $d$-dimensional percolation we show that all of the critical exponents can be computed if we know $\beta$, the exponent associated with the size of the giant component. When $\beta=1$ all the critical exponents are the same as for the \ER graph.