Critical Phenomena in a Small World

  • Matthew B. Hastings
  • Balázs Kozma
Part II Network Dynamics
Part of the Lecture Notes in Physics book series (LNP, volume 650)


We consider the behavior of various systems on a small-world network near a critical point. Our starting point is a different, nonrandom system with combined short- and long-range interactions. We analyze this model and find that the critical behavior is mean-field in general, with mean-field amplitudes that depend in an anomalous way on the strength of the long-range interaction. We then compare this model to the original small-world model, and derive a general criterion which determines when the two models have the same scaling behavior. The criterion can be applied to a variety of equilibrium statistical mechanics models as well as to various non-equilibrium processes. Finally, we apply these results to the specific case of the Edwards-Wilkinson equation. There, we find that the mean-field behavior is valid for d>2 dimensions, but that for d≤ 2 dimensions there is anomalous scaling.


Processing Element Zero Mode Critical Phenomenon Small World Correlation Volume 
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Authors and Affiliations

  • Matthew B. Hastings
    • 1
  • Balázs Kozma
    • 2
  1. 1.Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, 87545USA
  2. 2.Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180USA

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