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A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees
Random graphs with a given degree sequence are a useful model capturing several features absent in the classical Erd˝os-R´enyi model, such as dependent edges and non-binomial degrees. In this paper, we
use a characterization due to Erd˝os and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly
efficient sequential importance sampling. Applications are given, including simulating a biological network and estimating the number of graphs with a given degree sequence.Statistic
Upper bounds for number of removed edges in the Erased Configuration Model
Models for generating simple graphs are important in the study of real-world
complex networks. A well established example of such a model is the erased
configuration model, where each node receives a number of half-edges that are
connected to half-edges of other nodes at random, and then self-loops are
removed and multiple edges are concatenated to make the graph simple. Although
asymptotic results for many properties of this model, such as the limiting
degree distribution, are known, the exact speed of convergence in terms of the
graph sizes remains an open question. We provide a first answer by analyzing
the size dependence of the average number of removed edges in the erased
configuration model. By combining known upper bounds with a Tauberian Theorem
we obtain upper bounds for the number of removed edges, in terms of the size of
the graph. Remarkably, when the degree distribution follows a power-law, we
observe three scaling regimes, depending on the power law exponent. Our results
provide a strong theoretical basis for evaluating finite-size effects in
networks
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