Are randomly grown graphs really random?

Duncan S. Callaway; John E. Hopcroft; Jon M. Kleinberg; M. E. J. Newman; Steven H. Strogatz

doi:10.1103/physreve.64.041902

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arXiv · cond-mat/0104546

Are randomly grown graphs really random?

Duncan S. Callaway·John E. Hopcroft·Jon M. Kleinberg·M. E. J. Newman·Steven H. Strogatz

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Abstract

We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability delta, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time steps. In the limit of large t, the resulting graph displays surprisingly rich characteristics. In particular, a giant component emerges in an infinite-order phase transition at delta = 1/8. At the transition, the average 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 diverges 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 fundamentally different from their static random graph counterparts.

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Duncan S. Callaway, John E. Hopcroft, Jon M. Kleinberg, M. E. J. Newman, Steven H. Strogatz. 2001-06-14. Are randomly grown graphs really random?. https://doi.org/10.1103/physreve.64.041902

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