A scaling law for random walks on networks
Date
2014
Authors
Perkins, T.J.
Foxall, E.
Glass, L.
Edwards, Roderick
Journal Title
Journal ISSN
Volume Title
Publisher
Nature Communications
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.
Description
Keywords
biological physics, networks and systems biology, scaling laws
Citation
Perkins, T.J., Foxall, E., Glass, L., & Edwards, R. (2014). A scaling law for random walks on networks. Nature Communications, 5. https://doi.org/10.1038/ncomms6121