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Are Randomly Grown Graphs Really Random?


  • D. S. Callaway
  • J. E. Hopcroft
  • J. M. Kleinberg
  • M. E. J. Newman
  • S. H. Strogatz


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.

Suggested Citation

  • D. S. Callaway & J. E. Hopcroft & J. M. Kleinberg & M. E. J. Newman & S. H. Strogatz, 2001. "Are Randomly Grown Graphs Really Random?," Working Papers 01-05-025, Santa Fe Institute.
  • Handle: RePEc:wop:safiwp:01-05-025

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    5. Brot, Hilla & Muchnik, Lev & Goldenberg, Jacob & Louzoun, Yoram, 2012. "Feedback between node and network dynamics can produce real-world network properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6645-6654.
    6. Konno, Tomohiko, 2009. "Network structure of Japanese firms. Scale-free, hierarchy, and degree correlation: analysis from 800,000 firms," Economics - The Open-Access, Open-Assessment E-Journal, Kiel Institute for the World Economy (IfW), vol. 3, pages 1-13.
    7. Small, Henry, 2009. "Critical thresholds for co-citation clusters and emergence of the giant component," Journal of Informetrics, Elsevier, vol. 3(4), pages 332-340.
    8. Reppas, Andreas I. & Spiliotis, Konstantinos & Siettos, Constantinos I., 2015. "Tuning the average path length of complex networks and its influence to the emergent dynamics of the majority-rule model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 109(C), pages 186-196.
    9. Morehead, Raymond & Noore, Afzel, 2007. "Novel hybrid mitigation strategy for improving the resiliency of hierarchical networks subjected to attacks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(2), pages 603-612.

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