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Hybrid graphs as a framework for the small-world effect.

Katharina A Lehmann1, Hendrik D Post, Michael Kaufmann

  • 1Wilhelm-Schickard-Institut für Informatik, Universität Tübingen.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 29, 2006
PubMed
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This study formalizes the small-world effect in hybrid graphs, demonstrating how combining local and sparse random networks significantly reduces graph diameter. This research offers a flexible model for real-world network analysis.

Area of Science:

  • Network Science
  • Graph Theory
  • Complex Systems

Background:

  • The small-world effect describes how networks with local structure and sparse random connections have surprisingly short path lengths.
  • Existing models, like Watts and Strogatz, capture some small-world phenomena but not all hybrid graph structures.

Purpose of the Study:

  • To formalize and generalize the small-world effect in a broad family of hybrid graphs.
  • To provide a flexible framework for modeling real-world networks exhibiting small-world properties.

Main Methods:

  • Formalization of a generalized family of hybrid graphs combining local and random components.
  • Development of a novel proof technique to establish upper bounds on graph diameter.
  • Analysis of diameter scaling with varying numbers of random edges.

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Main Results:

  • Demonstrated that a large family of hybrid graphs exhibits the small-world effect with a diameter of O(ln n).
  • Showcased that this generalized family encompasses classic small-world models.
  • Provided detailed upper bounds for the hybrid graph diameter based on random edge density.

Conclusions:

  • The proposed flexible family of hybrid graphs effectively demonstrates and generalizes the small-world effect.
  • The findings offer a tunable model applicable to diverse real-world systems and network structures.