Forest Fire Model on Configuration Graphs with Random Node Degree Distribution

Abstract

We consider two types of configuration graphs with node degrees being i.i.d. random variables following either the power-law or the Poisson distribution. The distribution parameter is a random variable following the uniform distribution on a predefined interval. We consider a destructive process which can be interpreted as a fire spreading over the graph links, and could be used for modeling forest fires as well as banking system defaults. This process is often referred to as a “forest fire model”. The probability of fire transfer over a graph link either possesses a predefined value and is fixed for all the graph links or is a random variable following the standard uniform distribution. By computer simulation we study the robustness of such graphs from a viewpoint of node survival in the two cases of starting a fire propagation process: the “random ignition” and the “targeted lightning-up”. The results on finding the optimal interval of the node degree distribution parameter that would ensure maximum survival of trees in case of a fire are presented. A comparative analysis of various graph models in terms of their robustness to various fire propagation processes was performed. 2010 Mathematics Subject Classification: 05C80, 05C82, 62G35.