Consistent estimation of high-dimensional random graph models with dependent edge variables
Stewart, Jonathan Roy
Doctor of Philosophy
An important question in statistical network analysis is how to construct models of random graphs with dependent edges without sacrificing computational scalability and statistical guarantees. This thesis advances models, methods, and theory for dependent network data by introducing a simple and flexible approach to specifying random graph models that allow edges to be dependent and dependence among edges to propagate throughout the random graph. As examples, we develop generalizations of β-models with dependent edges capturing brokerage in networks. On the statistical side, we obtain the first consistency results and convergence rates for maximum likelihood in high-dimensional settings where a single observation of a network with dependent random variables is available and the number of parameters increases with network size. The theoretical results developed here are general and make weak assumptions, requiring nothing more than strictly positive distributions with exponential-family parameterizations, and may be of independent interest. We showcase consistency results and convergence rates in the special case of generalized β-models with dependent edges and parameter vectors of increasing dimension, and demonstrate through simulations that the statistical error is low even when the network has no more than 1,000 nodes. The thesis concludes with two applications, one involving social networks and the other one involving human brain networks.
statistical network analysis; network data; exponential families; exponential-family random graph models; social network analysis