The goal of many neuroimaging studies is to better understand how the functional connectivity structure of the brain changes with a given phenotype such as age. Functional connectivity can be characterised as a network, with nodes corresponding to brain regions and edges corresponding to statistical dependencies between the respective regional time series of activity. A typical neuroimaging dataset will thus consist of one or more networks for each individual in the study. Most statistical network models, however, were originally proposed to describe a single underlying relational structure such as friendships between individuals or hyperlinks between web pages. As a result, the development of these models has largely been restricted to the single network case. While one could in principle fit a single network model to each individual separately, it is not always straightforward to combine these individual results into a single group result.

In the first half of the thesis, we propose a multilevel framework for populations of networks based on exponential random graph models. By pooling information across the individual networks, this framework provides a principled approach to characterise the relational structure for an entire population. We use the framework to assess group-level variations in functional connectivity, providing a method for the inference of differences in the topological structure between groups of networks. Our motivation stems from the Cam-CAN project, a neuroimaging study on healthy ageing. Using this dataset, we illustrate how our method can be used to detect differences in functional connectivity between a group of young individuals and a group of old individuals.

In the second half of the thesis, we shift our focus to dynamic functional connectivity (dFC). Recent studies have found that using static measures may average over informative fluctuations in functional connectivity. Several methods have been developed to measure dFC in functional magnetic resonance imaging (fMRI) data. However, spurious group differences in measured dFC may be caused by other sources of heterogeneity between people. We use a generic simulation framework for fMRI data to investigate the effect of such heterogeneity on estimates of dFC and find that, despite no differences in true dFC, individual differences in measured dFC can result from other (non-dynamic) features of the data. We then add a natural and novel extension to our multilevel framework by inserting time windows as an intermediate level between time points and subjects. Using magnetoencephalography data from the Cam-CAN study, we apply our method to detect differences in time-varying connectivity between a young group and an old group.