A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees

Random graphs with given degrees are a natural next step in complexity beyond the Erdős–Rényi model, yet the degree constraint greatly complicates simulation and estimation. We use an extension of a combinatorial characterization due to Erdős 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 for surprisingly efficient sequential importance sampling. The resulting probabilities are easily computed on the fly, allowing the user to reweight estimators appropriately, in contrast to some ad hoc approaches that generate graphs with the desired degrees but with completely unknown probabilities. Applications are given, including simulating an ecological network and estimating the number of graphs with a given degree sequence.