Achlioptas process phase transitions are continuous

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.Comment: Published in at http://dx.doi.org/10.1214/11-AAP798 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mixed-order phase transition in a two-step contagion model with a single infectious seed

“…It was proposed that a discontinuous percolation transition cannot occur when its occupation rule is local [11]. However, when more than one species of particles cooperatively occupy each node, a discontinuous percolation transition can occur even though the dynamic rule is local [12,13].…”

The evolution of subcritical Achlioptas processes

“…Together with Theorem 1, the continuity results of [18,20] imply that L 1 (G t b n )/n p → 0, and that k ρ k (t b ) = 1, so the numbers (ρ k (t b )) k 1 do capture the asymptotic component size distribution of G R t b n , although we do not have such tight error bounds as in (2).…”

“…This concludes the proof since ε (and thus K) only depends on R, ℓ, t. Now we define t b = t R b as the infimum of the set of t 0 for which (1) holds as n → ∞; so (1) fails for t < t b . The remark after the proof of Lemma 4 in [18] implies that for t > 1 we whp have…”

“…[3,4,6,12,13,18,21]. These make their decisions based only on the sizes of the components containing the endvertices of e 1 and e 2 , with the restriction that all sizes larger than some constant B are treated in the same way.…”

Creating Small Subgraphs in Achlioptas Processes With Growing Parameter

“…The Achlioptas process has been studied by many researchers, both for fixed values of r and under the assumption that r = r(n) is a growing function of n. The property that received by far the most attention in this context is the property of containing a linear-sized (so-called 'giant') component [1,5,7,8,10,21,22]. Only recently, other properties have been studied: the problem of accelerating Hamiltonicity in Achlioptas processes was investigated in [14], and the problem of delaying the occurrence of a given fixed graph as a subgraph was studied in [13,19].…”