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Are Randomly Grown Graphs Really Random?

Listed author(s):
  • D. S. Callaway
  • J. E. Hopcroft
  • J. M. Kleinberg
  • M. E. J. Newman
  • S. H. Strogatz
Registered author(s):

    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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    Paper provided by Santa Fe Institute in its series Working Papers with number 01-05-025.

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    Date of creation: May 2001
    Handle: RePEc:wop:safiwp:01-05-025
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