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Related Experiment Videos

Are randomly grown graphs really random?

D S Callaway1, J E Hopcroft, J M Kleinberg

  • 1Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York 14853-1503, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 3, 2001
PubMed
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A new model of growing networks shows a giant component emerges at a phase transition point. This differs from static random graphs, highlighting unique properties of network growth.

Area of Science:

  • Network Science
  • Statistical Physics
  • Graph Theory

Background:

  • Understanding network structure is crucial in various scientific fields.
  • Growing networks exhibit distinct properties compared to static random graphs.
  • Phase transitions in complex systems reveal fundamental changes in behavior.

Purpose of the Study:

  • To analyze a minimal model of a growing network.
  • To investigate the emergence of a giant component in growing networks.
  • To compare the phase transition behavior of growing and static random graphs.

Main Methods:

  • A minimal growing network model was analyzed.
  • Vertices were added sequentially, with edges formed randomly with probability delta.
  • The study considered the limit of large time steps (t).

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Main Results:

  • A giant component emerges in the growing network at delta=1/8 via an infinite-order phase transition.
  • At this transition, the average component size jumps discontinuously but remains finite.
  • Static random graphs with similar degree distributions show a second-order phase transition at delta=1/4, with diverging average component size.

Conclusions:

  • Growing networks exhibit fundamentally different behavior than static random graphs.
  • Positive correlation between degrees of connected vertices in grown graphs drives these differences.
  • The age of vertices influences their degree and connectivity in grown networks.