Modelling the structure of complex networks

A complex network is a systems in which a discrete set of units interact in a quantifiable manner. Representing systems as complex networks have become increasingly popular in a variety of scientific fields including biology, social sciences and economics. Parallel to this development complex networks has been independently studied as mathematical objects in their own right. As such, there has been both an increased demand for statistical methods for complex networks as well as a quickly growing mathematical literature on the subject.

In this dissertation we explore aspects of modelling complex networks from a probabilistic perspective. The first two chapter will be focused on the justification of the use of probabilistic methods for inference problems; we will look at the justification of probabilistic methods from the perspective of consistency and as a general method of updating beliefs. The next chapters will treat some of the various symmetries, representer theorems and probabilistic structures often deployed in the modelling complex networks, the construction of sampling methods and various network models.

The introductory chapters will serve to provide context for the included written work on the topics of (i) updating beliefs (ii) construction of samplers for partition-based problems (iii) applying non-parametric methods for modelling stationary and temporal network data.