Network analysis under sparse condition

Abstract

In this thesis, we propose two general parametric models for network analysis including the situations for both directed and undirected graphs. For directed graph models, parametric pairwise comparison models play a crucial role in the statistical modeling of networks. Among many models in the field, the Bradley-Terry (BT) model is arguably one of the most popular ones owing to its simplicity and well-studied structures. In the BT model, observation of any two subjects under comparison is assumed binary and the corresponding link function depends on the relative score of the subjects. Restriction to the binary outcome makes the model far from accurate in situations where a non-binary result is preferable. To address this issue, a general framework allowing pairwise comparison data to be both discrete and continuous is proposed. Under such a framework, uniform consistency of the maximum likelihood estimator (MLE) for the latent scores is established whenever the mutual comparison rate is close to its optimal lower bound. The near-optimal sparsity condition ensures that MLE is applicable in large-scale networks where the relative number of comparison data is scarce. Among undirected graph models, the β-model plays a fundamental role and is widely applied to analyze network data. It assumes that the edge probability is linked with the sum of the strength parameters of the two vertices through a logistics function. Because of the univariate nature of the link function, this formulation, despite its popularity, can be restrictive for practical applications, even with a straightforward extension of the link function. For example, it is possible that vertices with similar strength parameters are more likely to be connected, in which case the edge probability depends on the distance of the strength parameters. Such common cases are not included in the β-model or its immediate extensions. We propose a bivariate gamma model that links the edge probability with the two strength parameters by some symmetric bivariate function. The proposed model is more flexible than the β-model and its existing variants. It is also applicable to mirror various undirected networks. Asymptotic theory is established to justify the consistency and asymptotic normality of the moment estimators. Numerical studies present evidence in support of the theory and an example involving real data further illustrates the applications.

Date of Award	2020
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology