UCLA

With a rise in the amount of network data comes increased need for flexible and interpretable network models. Exponential-family random graph models (ERGM) are widely used to analyze small- to medium-sized networks, but suffer from model degeneracy which detracts from their application. In Part I of this dissertation we address this problem by developing novel statistics for ERGM. We focus on the modeling of transitivity in networks as it is a key feature of many real-world networks, but most attempts to account for it within ERGM have induced model degeneracy. The statistics we propose combine the strategies of the transformed statistics proposed by Horvat et al. (2015) and the regularized statistics proposed by Fellows (2012b). They include statistics to capture transitivity, clustering, and a new class of moment statistics to improve goodness of fit. We characterize our newly introduced statistics along with those of Horvat et al. (2015) and Fellows (2012a) using recent theoretical developments regarding ERGM degeneracy. We also compare them theoretically and in practice to the geometrically weighted statistics of Snijders et al. (2006) that are currently the most commonly used to model transitivity in ERGM.

In Part II of this dissertation we develop models to rate and rank items based on network data, and demonstrate many advantageous properties of these models. The impetus for this work came from research on ranking statistics journals by Varin et al. (2016). They present a quasi-Stigler model that is a great improvement over the commonly used but statistically indefensible Impact Factor, especially in the quantification of ratings uncertainty. However, the quasi-Stigler model does not fully leverage the network structure of the data and underestimates uncertainty. In addition to applying latent space models to the network rating problem, we identify a fast computational method for fitting the models. We also develop a new latent network model that leverages the symmetric and asymmetric patterns in directed relational data. This model has many potential applications beyond item rating.