Are randomly grown graphs really random? art. no. 041902

We analyze a minimal model of a growing network. At each time step, a new v ertex is added; then, with probability delta, two vertices are chosen unifo rmly at random and joined by an undirected edge. This process is repeated f or t time steps. In the limit of large t, the resulting graph displays surp risingly rich characteristics. In particular, a giant component emerges in an infinite-order phase transition at delta =1/8. At the transition, the av erage component size jumps discontinuously but remains finite. In contrast, a static random graph with the same degree distribution exhibits a second- order phase transition at delta =1/4, and the average component size diverg es there. These dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph-older vertices tend to have higher degree, and to link with other high-degree vertices, merely by virtue of their age. We conclude that grown graphs, however randomly they are constructed, are fundamentall y different from their static random graph counterparts.