Are Randomly Grown Graphs Really Random?
AbstractWe 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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Bibliographic InfoPaper provided by Santa Fe Institute in its series Working Papers with number 01-05-025.
Date of creation: May 2001
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This paper has been announced in the following NEP Reports:
- NEP-ENT-2001-07-17 (Entrepreneurship)
- NEP-EVO-2001-07-17 (Evolutionary Economics)
- NEP-NET-2001-07-17 (Network Economics)
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