The phase transition in the uniformly grown random graph has infinite order

• Authors: Béla Bollobás;Svante Janson;Oliver Riordan
• Venue: Random Structures & Algorithms - Proceedings of the Eleventh International Conference "Random Structures and Algorithms," August 9—13, 2003, Poznan, Poland
• Year: 2005

The aim of this paper is to study the emergence of the giant component in the uniformly grown random graph Gn(c), 0 c n] = {1, 2, …, n} in which each possible edge ij is present with probability c/ max{i, j}, independently of all other edges. Equivalently, we may start with the random graph Gn(1) with vertex set[n], where each vertex j is joined to each “earlier” vertex i j with probability 1/j, independently of all other choices. The graph Gn(c) is formed by the open bonds in the bond percolation on Gn(1) in which a bond is open with probability c. The model Gn(c) is the finite version of a model proposed by Dubins in 1984, and is also closely related to a random graph process defined by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz [Phys Rev E 64 (2001), 041902]. Results of Kalikow and Weiss [Israel J Math 62 (1988), 257–268] and Shepp [Israel J Math 67 (1989), 23–33] imply that the percolation threshold is at c = 1/4. The main result of this paper is that for c = 1/4 + ε, ε 0, the giant component in Gn(c) has order exp(-&THgr;(1/√ ε)) n. In particular, the phase transition in the bond percolation on Gn(1) has infinite order. Using nonrigorous methods, Dorogovtsev, Mendes, and Samukhin [Phys Rev E 64 (2001), 066110] showed that an even more precise result is likely to hold. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005Research supported by NSF Grant ITR 0225610 and DARPA Grant F33615-01-C-1900