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

Bias–Variance and Breadth–Depth Tradeoffs in Respondent-Driven Sampling

Respondent-driven sampling (RDS) is a link-tracing network sampling strategy for collecting data from hard-to-reach populations, such as injection drug users or individuals at high risk of being infected with HIV. The mechanism is to find initial participants (seeds), and give each of them a fixed number of coupons allowing them to recruit people they know from the population of interest, with a mutual financial incentive. The new participants are again given coupons and the process repeats. Currently, the standard RDS estimator used in practice is known as the Volz–Heckathorn (VH) estimator. It relies on strong assumptions about the underlying social network and the RDS process. Via simulation, we study the relative performance of the plain mean and VH estimators when assumptions of the latter are not satisfied, under different network types (including homophily and rich-get-richer networks), participant referral patterns, and varying number of coupons. The analysis demonstrates that the plain mean outperforms the VH estimator in many but not all of the simulated settings, including homophily networks. Also, we highlight the implications of multiple recruitment and varying referral patterns on the depth of RDS process. We develop interactive visualizations of the findings and RDS process to further build insight into the various factors contributing to the performance of current RDS estimation techniques.Statistic

Monte Carlo Maximum Likelihood for Exponential Random Graph Models: From Snowballs to Umbrella Densities

Performing maximum-likelihood estimation for parameters in an exponential random graph model is challenging because of the unknown normalizing constant. Geyer and Thompson (1992) provide a Monte Carlo algorithm that uses samples from a distribution with known parameters to approximate the full likelihood, which is then maximized to estimate the MLE. We refine the approximation to use sample draws collected from differently parameterized distributions, increasing the effective sample size and improving the accuracy of the MLE estimate. Substantially lower estimation variance is demonstrated in simulated and actual network data. We also propose a new method for finding a starting point: scaling the MLE parameters of a small graph subsampled from the original graph. Through simulation with the triad model, this starting point produces convergence in many cases where the standard starting point (based on pseudolikelihood) fails to give convergence, though the reverse is also true.Statistic

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 Erdős-Rényi model, such as dependent edges and non-binomial degrees. In this paper, we use a 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 surprisingly efficient sequential importance sampling. Applications are given, including simulating a biological network and estimating the number of graphs with a given degree sequence. 1. Introduction. Rando