Dynamic pattern evolution on scale-free networks

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) ∼ k -γ, are shown to exhibit qualitatively different dynamic behavior for γ < 5/2 and γ > 5/2, shedding light on the empirical observation that many real-world networks are scale-free with 2 < γ < 5/2. For 2 < γ < 5/2, strongly disordered patterns decay within a finite decay time even in the limit of infinite networks. For γ > 5/2, on the other hand, this decay time diverges as ln(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 γ > 5/2 but to grow as N α with α = (5 - 2γ)/(γ - 1) for 2 < γ < 5/2.