Journal
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
Volume 102, Issue 29, Pages 10052-10057Publisher
NATL ACAD SCIENCES
DOI: 10.1073/pnas.0409296102
Keywords
random network; Boolean dynamics; cellular automata; associative memory
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A general class of dynamic models on scale-free networks is studied by analytical methods and computer simulations. Each network consists of N vertices and is characterized by its degree distribution, P(k), which represents the probability that a randomly chosen vertex is connected to k nearest neighbors. Each vertex can attain two internal states described by binary variables or Ising-like spins that evolve in time according to local majority rules. Scale-free networks, for which the degree distribution has a power law tail P(k) similar to k(-gamma), are shown to exhibit qualitatively different dynamic behavior for gamma < 5/2 and gamma > 5/2, shedding light on the empirical observation that many real-world networks are scale-free with 2 < gamma < 5/2. For 2 < gamma < 5/2, strongly disordered patterns decay within a finite decay time even in the limit of infinite networks. For gamma > 5/2, on the other hand, this decay time diverges as In(N) with the network size N. An analogous distinction is found for a variety of more complex models including Hopfield models for associative memory networks. In the latter case, the storage capacity is found, within mean field theory, to be independent of N in the limit of large N for gamma < 5/2 but to grow as N with alpha = (5 - 2 gamma)/(gamma - 1) for 2 < gamma < 5/2.
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