Giant components in biased graph processes

Abstract

A random graph process, G1(n), is a sequence of graphs on n vertices which begins with the edgeless graph, and where at each step a single edge is added according to a uniform distribution on the missing edges. It is well known that in such a process a giant component (of linear size) typically emerges after (1 + o(1))n2 edges (a phenomenon known as “the double jump”), i.e., at time t = 1 when using a timescale of n/2 edges in each step. We consider a generalization of this process, GK(n), proposed by Itai Benjamini in order to model the spreading of an epidemic. This generalized process gives a weight of size 1 to missing edges between pairs of isolated vertices, and a weight of size K ∈ [0,∞) otherwise. This corresponds to a case where links are added between n initially isolated settlements, where the probability of a new link in each step is biased according to whether or not its two endpoint settlements are still isolated. Combining methods of [13] with analytical techniques, we describe the typical emerging time of a giant component in this process, tc(K), as the singularity point of a solution to a set of differential equations. We proceed to analyze these differential equations and obtain properties of GK , and in particular, we show that tc(K) strictly decreases from 32 to 0 as K increases from 0 to ∞, and that tc(K) = 4 √3K (1 + o(1)), where the o(1)-term tends to 0 as K → ∞. Numerical approximations of the differential equations agree both with computer simulations of the process GK(n) and with the analytical results.