Freeman, N.P., Crane, E. and Tóth, B. (2015) Cluster growth in the dynamical Erdős-Rényi process with forest fires. Electronic Journal of Probability, 20 (101). pp. 1-33. ISSN 1083-6489
Abstract
We investigate the growth of clusters within the forest fire model of Ráth and Tóth [EJP, vol 14, paper no 45]. The model is a continuous-time Markov process, similar to the dynamical Erdős-Rényi random graph but with the addition of so-called fires. A vertex may catch fire at any moment and, when it does so, causes all edges within its connected cluster to burn, meaning that they instantaneously disappear. Each burned edge may later reappear.
We give a precise description of the process CtCt of the size of the cluster of a tagged vertex, in the limit as the number of vertices in the model tends to infinity. We show that CtCt is an explosive branching process with a time-inhomogeneous offspring distribution and instantaneous return to 1 on each explosion. Additionally, we show that the characteristic curves used to analyse the Smoluchowski-type coagulation equations associated to the model have a probabilistic interpretation in terms of the process CtCt.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | © 2015 The Authors. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
Keywords: | Erdos-Rényi random graph; forest fire; self-organized criticality; Smoluchowski coagulation equation. |
Dates: |
|
Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 22 Apr 2016 08:41 |
Last Modified: | 22 Apr 2016 08:41 |
Published Version: | http://dx.doi.org/10.1214/EJP.v20-4035 |
Status: | Published |
Publisher: | Institute of Mathematical Statistics |
Refereed: | Yes |
Identification Number: | 10.1214/EJP.v20-4035 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:98793 |