A scaling law for random walks on networks

Theodore J. Perkins (), Eric Foxall, Leon Glass and Roderick Edwards
Additional contact information
Theodore J. Perkins: Ottawa Hospital Research Institute
Eric Foxall: University of Victoria
Leon Glass: McGill University
Roderick Edwards: University of Victoria

Nature Communications, 2014, vol. 5, issue 1, 1-7

Abstract: Abstract The dynamics of many natural and artificial systems are well described as random walks on a network: the stochastic behaviour of molecules, traffic patterns on the internet, fluctuations in stock prices and so on. The vast literature on random walks provides many tools for computing properties such as steady-state probabilities or expected hitting times. Previously, however, there has been no general theory describing the distribution of possible paths followed by a random walk. Here, we show that for any random walk on a finite network, there are precisely three mutually exclusive possibilities for the form of the path distribution: finite, stretched exponential and power law. The form of the distribution depends only on the structure of the network, while the stepping probabilities control the parameters of the distribution. We use our theory to explain path distributions in domains such as sports, music, nonlinear dynamics and stochastic chemical kinetics.

Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:nat:natcom:v:5:y:2014:i:1:d:10.1038_ncomms6121

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DOI: 10.1038/ncomms6121

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