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.
Highlights
Random graphs have been widely used for modeling complex networks such as the Internet, social, transport or telecommunication networks
Real data observations showed (Faloutsos, Faloutsos, and Faloutsos 1999; Durrett 2007) that their topology can be adequately represented by configuration graphs with vertex degrees being independent identically distributed (i.i.d.) random variables possessing natural values
In (Leri and Pavlov 2014, 2016) we studied the robustness of configuration graphs to targeted and random destruction influences
Summary
Random graphs have been widely used for modeling complex networks such as the Internet, social, transport or telecommunication networks (see, e.g., Durrett 2007). In (Reittu and Norros 2004) the authors showed that in modeling of huge networks it is more appropriate to use the following vertex degree distribution of the random variable ξ: P{ξ = k} = k−τ − (k + 1)−τ ,. In (Pavlov 2016) we considered configuration graphs where vertex degrees follow the distribution (1) under the condition that the parameter τ is a random variable uniformly distributed on the interval [a, b], where 0 < a < b < ∞. Comparison of the results showed their similarity It means that the study of the graphs’ behaviour in the considered random environment can be replaced by the study of the model with an averaged vertex degree distribution (2).
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