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Are Randomly Grown Graphs Really Random?

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

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.

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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    Cited by:

    1. 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 (2007-2020), Kiel Institute for the World Economy (IfW Kiel), vol. 3, pages 1-13.
    2. 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.
    3. 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.
    4. 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.
    5. Sabek, M. & Pigorsch, U., 2023. "Local assortativity in weighted and directed complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 630(C).
    6. Meysam Alizadeh & Claudio Cioffi-Revilla & Andrew Crooks, 2017. "Generating and analyzing spatial social networks," Computational and Mathematical Organization Theory, Springer, vol. 23(3), pages 362-390, September.
    7. Yang Zhang & Ying-Ju Chen, 2020. "Optimal Nonlinear Pricing in Social Networks Under Asymmetric Network Information," Operations Research, INFORMS, vol. 68(3), pages 818-833, May.
    8. Weaver, Iain S., 2015. "Preferential attachment in randomly grown networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 439(C), pages 85-92.
    9. Chris Fields, 2015. "How small is the center of science? Short cross-disciplinary cycles in co-authorship graphs," Scientometrics, Springer;Akadémiai Kiadó, vol. 102(2), pages 1287-1306, February.
    10. Haizheng Zhang & Baojun Qiu & Kristinka Ivanova & C. Lee Giles & Henry C. Foley & John Yen, 2010. "Locality and attachedness‐based temporal social network growth dynamics analysis: A case study of evolving nanotechnology scientific collaboration networks," Journal of the American Society for Information Science and Technology, Association for Information Science & Technology, vol. 61(5), pages 964-977, May.
    11. Swami Iyer & Timothy Killingback, 2016. "Evolution of Cooperation in Social Dilemmas on Complex Networks," PLOS Computational Biology, Public Library of Science, vol. 12(2), pages 1-25, February.
    12. 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.
    13. Oh, S.M. & Son, S.-W. & Kahng, B., 2021. "Percolation transitions in growing networks under achlioptas processes: Analytic solutions," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    14. Zhou, Li-xin & Lin, Jie & Wang, Yu-qing & Li, Yan-feng & Miao, Run-sheng, 2018. "Critical phenomena of spreading dynamics on complex networks with diverse activity of nodes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 439-447.
    15. Inoue, Masaaki & Pham, Thong & Shimodaira, Hidetoshi, 2020. "Joint estimation of non-parametric transitivity and preferential attachment functions in scientific co-authorship networks," Journal of Informetrics, Elsevier, vol. 14(3).

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