Barabási-Albert random graphs, scale-free distributions and bounds for approximation through Stein's method

• Main Author: Ford, Elizabeth
• Other Authors: Reinert, Gesine
• Published: University of Oxford 2009
• Subjects: 519; Probability theory and stochastic processes : random graph : scale-free : power law : Stein's method
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.510953

Barabási-Albert random graph models are a class of evolving random graphs that are frequently used to model social networks with scale-free degree distributions. It has been shown that Barabási-Albert random graph models have asymptotic scale-free degree distributions as the size of the graph tends to infinity. Real world networks, however, have finite size so it is important to know how close the degree distribution of a Barabási-Albert random graph of a given size is to its asymptotic distribution. Stein’s method is chosen as one main method for obtaining explicit bounds for the distance between distributions. We derive a new version of Stein’s method for a class of scale-free distributions and apply the method to a Barabási-Albert random graph. We compare the evolution of a sequence of Barabási-Albert random graphs with continuous time stochastic processes motivated by Yule’s model for evolution. Through a coupling of the models we bound the total variation distance between their degree distributions. Using these bounds, we extend degree distribution bounds that we find for specific models within the scheme to find bounds for every member of the scheme. We apply the Azuma-Hoeffding inequality and Chernoff bounds to find bounds between the degree sequences of the random graph models and the given scale-free distribution. These bounds prove that the degree sequences converge completely (and therefore also converge almost surely) to our scale-free distribution. We discuss the relationship between the random graph processes and the Chinese restaurant process. Aided by the construction of an inhomogeneous Markov chain, we apply our results for the degree distribution in a Barabási-Albert random graph to a particular statistic of the Chinese restaurant process. Finally, we explore how our methods can be adapted and extended to other evolving random graph processes. We study a Bernoulli evolving random graph process, for which we bound the distance between its degree distribution and a geometric distribution and we bound the distance between the number of triangles in the graph and a normal distribution.