This proposal is focused on the study of networks as statistical data objects. Networks are fundamental to our modern world: they appear throughout science and society, and continue to grow in size, complexity and importance. Whenever we observe entities and relationships between them, we effectively define a network of some sort. As structural objects composed of nodes and links, networks play a strong and well defined role across mathematics, science and engineering. As statistical objects made up of collections of measurements, however, network datasets require significant advances to be made in mathematical knowledge if we are to achieve fundamental understanding. The crux of the problem, and the essence of the approach to be undertaken in this research, lies in finding the right balance between complexity and parsimony. Currently, the network models that we understand fully from a mathematical viewpoint are too simple to accurately describe modern data. At the same time, models sufficiently rich to provide accurate descriptions are presently beyond our mathematical comprehension, meaning that we cannot use them to draw sound and repeatable conclusions from data. This fundamental lack of understanding slows scientific progress and affects every single economic, social or other policy decision that relies on the analysis of network data. The main objectives of this research are therefore twofold: first, to develop the new statistical theory needed to view and interpret networks properly as data objects; and second, to transform this theory into new statistical methods that will allow us to model and draw inferences from network data in the real world. These objectives reflect the fact that network modelling and inference is an area of significant international importance. It spans the many diverse fields and contexts where inferences must be drawn and substantiated based on measurements of entities and the relationships or interactions between them.
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