Random Graphs' Robustness in Random Environment

Abstract

We consider configuration graphs the vertex degrees of which are independent and  follow the power-law distribution. Random graphs dynamics takes place in a random  environment with the parameter of vertex degree distribution following  uniform distributions on finite fixed intervals. As the number of vertices tends  to infinity the limit distributions of the maximum vertex degree and the number  of vertices with a given degree were obtained. By computer simulations we study  the robustness of those graphs from the viewpoints of link saving and node survival  in the two cases of the destruction process: the ``targeted attack'' and the  ``random breakdown''. We obtained and compared the results under the conditions that  the vertex degree distribution was averaged with respect to the distribution of the  power-law parameter or that the values of the parameter were drawn from the uniform  distribution separately for each vertex.