Coevolution Regression and Composite Likelihood Estimation for Social Networks

Abstract

We study how social networks and nodal attributes influence each other over time. A multiplicative coevolution regression (MCR) model is proposed for longitudinal network and nodal attribute data. The coevolution model is based on the following three principles: autocorrelation, homophily and contagion. For the Gaussian MCR model, the maximum likelihood estimates can be obtained using ordinary least squares. We also extend the Gaussian MCR so that it can include latent factors or model ordinal data. A Bayesian method using Markov Chain Monte Carlo (MCMC) is used to estimate the parameters and latent factors. We then focus on developing a scalable method to estimate the parameters in models of very large binary network datasets. Maximum likelihood estimates are generally impossible to obtain because the full likelihood involves an intractable high dimensional integral. Also, full-likelihood Bayesian estimation is impractical for very large datasets as the MCMC algorithm is very slow. We propose a triadic composite likelihood estimation method for exchangeable latent Gaussian network models, and extend it to q-node composite likelihood estimation for other exchangeable and non-exchangeable models. The maximum composite likelihood estimates are obtained by optimizing the composite likelihood using a stochastic gradient-based algorithm, where the gradients are approximated using Monte Carlo samples. For networks of moderate size, we show via simulations that composite likelihood estimation provides estimates as accurate as those provided by fully Bayesian estimation using MCMC. For very large datasets, fully Bayesian estimation is impractical, but composite likelihood estimation is feasible as its computational cost is essentially constant as a function of the network size.