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Center for Nonlinear Studies |
Recently there has been considerable interest in large, stochastic networks, whether man-made (the Internet, networks of flight connections) or arising on their own in some natural context (networks of predator-prey, association between chemicals in the metabolic process, etc.). It is found that many of these nets have a degree (number of links emanating from each node) whose distribution falls off in power-law fashion, or a scale-free fashion. The scale-free distribution has been shown to give rise to exotic properties of the nets, such as increased resilience to random dilution (removal of nodes). The question arises whether there is anything magic about the scale-free architecture when it comes to diffusion, one of the most elementary models of transport. We study diffusion (random walks) on recursive scale-free graphs, and contrast the results to similar studies in other media. This allows us to identify ways in which diffusion is special in scale-free nets. Most notably, scale-free nets yield faster transit times between pre-existing nodes, when the network grows in size; and walks starting from the most connected nodes are recurrent, despite the network's infinite dimension.