Giant Components in Biased Graph Processes

A random graph process, $\Gorg[1](n)$, 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, $\Gorg[K](n)$, which 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 \cite{SpencerWormald} 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 $\Gorg$, and in particular, we show that tc(K) strictly decreases from 3/2 to 0 as K increases from 0 to ∞, and that tc(K)=43K√(1+o(1)). Numerical approximations of the differential equations agree both with computer simulations of the process $\Gorg(n)$ and with the analytical results.