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The small-world phenomenon: a model, explanations, characterizations, and examples

Author

Listed:
  • Leo Egghe

    (Hasselt University)

  • Ronald Rousseau

    (KU Leuven
    University of Antwerp)

Abstract

We introduce and define three types of small worlds: small worlds based on the diameter of the network (SWD), those based on the average geodesic distance between nodes (SWA), and those based on the median geodesic distance (SWMd). These types of networks are defined as limiting properties of sequences of sets. We show the exact relation between these three types, namely that each SWD network is also an SWA network and that each SWA network is also an SWMd network. Yet, having the small-world property is a phenomenon that can easily occur in the sense that most networks are small-world networks in one of the three ways. We introduce sequences of distance frequencies, so-called alpha-sequences, and prove a relation between the majorization property between alpha-sequences and small-world properties.

Suggested Citation

  • Leo Egghe & Ronald Rousseau, 2024. "The small-world phenomenon: a model, explanations, characterizations, and examples," Scientometrics, Springer;Akadémiai Kiadó, vol. 129(9), pages 5599-5614, September.
    • Handle: RePEc:spr:scient:v:129:y:2024:i:9:d:10.1007_s11192-024-05119-8
    DOI: 10.1007/s11192-024-05119-8
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    References listed on IDEAS

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    2. Shiu-Wan Hung & An-Pang Wang, 2010. "Examining the small world phenomenon in the patent citation network: a case study of the radio frequency identification (RFID) network," Scientometrics, Springer;Akadémiai Kiadó, vol. 82(1), pages 121-134, January.
    3. Réka Albert & Hawoong Jeong & Albert-László Barabási, 1999. "Diameter of the World-Wide Web," Nature, Nature, vol. 401(6749), pages 130-131, September.
    4. Jon M. Kleinberg, 2000. "Navigation in a small world," Nature, Nature, vol. 406(6798), pages 845-845, August.
    5. Joel A. C. Baum & Andrew V. Shipilov & Tim J. Rowley, 2003. "Where do small worlds come from?," Industrial and Corporate Change, Oxford University Press and the Associazione ICC, vol. 12(4), pages 697-725, August.
    6. Matthew O. Jackson & Brian W. Rogers, 2007. "Meeting Strangers and Friends of Friends: How Random Are Social Networks?," American Economic Review, American Economic Association, vol. 97(3), pages 890-915, June.
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