Avoiding a giant component

Abstract

Let e1, e′1; e2, e′2;…;ei, e′i;⋅⋅⋅ be a sequence of ordered pairs of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e. we sample with replacement). This sequence is used to form a graph by choosing at stage i, i=1,…, one edge from ei,e′i to be an edge in the graph, where the choice at stage i is based only on the observation of the edges that have appeared by stage i. We show that these choices can be made so that whp the size of the largest component of the graph formed at stage 0.535n is polylogarithmic in n. This resolves a question of Achlioptas. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 75–85, 2001

The evolution of subcritical Achlioptas processes

“…[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.…”

Social climber attachment in forming networks produces a phase transition in a measure of connectivity

“…We demon-strate this application by showing that networks formed under SC attachment exhibit enhanced spreading properties with respect to the susceptible-infected-susceptible (SIS) model [4,5], a contagion model with many applications including the dissemination of information, sometimes referred to as "gossip based" communication or epidemic routing [17]. We note that the development this potential application may be facilitated by the fact that SC attachment may be combined with arbitrary percolation processes, such as Erdös-Rényi (ER) percolation [18] and Achlioptas processes [19][20][21], to independently control cluster aggregation (determined by the percolation process) and connectivity within clusters (determined by SC attachment).…”

Creating Small Subgraphs in Achlioptas Processes With Growing Parameter

“…This process was first considered in [5], and is known in the literature as the Achlioptas process with parameter r. It is the most prominent example of a random graph process that is not completely random but involves some limited amount of choice (by some 'player' or 'online algorithm'). Other graph processes of that type are the Ramsey process [3,4,6,11,16,17], in which random edges appear one by one and have to be colored with one of r available colors, and the Balanced Ramsey process [12,14,15,20], in which at each step r random edges appear and have be colored using each of the r available colors exactly once (this can be seen as a combination of the previous two processes).…”

“…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].…”