Achlioptas process phase transitions are continuous

TL;DR

This paper proves that Achlioptas processes, including the product rule, do not exhibit explosive percolation, and instead have continuous phase transitions, settling longstanding conjectures in the study of random graph processes.

Contribution

The authors provide a simple proof that Achlioptas processes do not have discontinuous phase transitions and establish the continuity of the giant component size in these processes.

Findings

Achlioptas processes do not exhibit explosive percolation.

The size of the giant component converges continuously.

The paper settles several conjectures about phase transition nature.

Abstract

It is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on $n$ vertices, suppose that at each step two pairs of vertices are chosen uniformly at random, but only one pair is joined, namely, one minimizing the product of the sizes of the components to be joined. Making explicit an earlier belief of Achlioptas and others, in 2009, Achlioptas, D'Souza and Spencer [Science 323 (2009) 1453-1455] conjectured that there exists a $\delta>0$ (in fact, $\delta\ge1/2$) such that with high probability the order of the largest component "jumps" from $o(n)$ to at least $\delta n$ in $o(n)$ steps of the process, a phenomenon known as "explosive percolation." We give a simple proof that this is not the case. Our result applies to all "Achlioptas processes," and more generally to any process where a fixed number of independent random vertices are chosen at each step, and (at least) one edge between these vertices is added to the current graph, according to any (online) rule. We also prove the existence and continuity of the limit of the rescaled size of the giant component in a class of such processes, settling a number of conjectures. Intriguing questions remain, however, especially for the product rule described above.