THESIS
2020
xi, 73 pages : illustrations ; 30 cm
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...[
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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.
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