title: A scaling law for random walks on networks.
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.
authors.affiliation: Ottawa Hospital Research Institute, 501 Smyth Road, Ottawa, Ontario, Canada K1H 8L6.
authors.forename: Theodore J
authors.initials: TJ
authors.lastname: Perkins
authors.affiliation: Department of Mathematics and Statistics, University of Victoria, P. O. Box 1700 STN CSC, Victoria, British Columbia, Canada V8W 2Y2.
authors.forename: Eric
authors.initials: E
authors.lastname: Foxall
authors.affiliation: Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec, Canada H3G 1Y6.
authors.forename: Leon
authors.initials: L
authors.lastname: Glass
authors.affiliation: Department of Mathematics and Statistics, University of Victoria, P. O. Box 1700 STN CSC, Victoria, British Columbia, Canada V8W 2Y2.
authors.forename: Roderick
authors.initials: R
authors.lastname: Edwards