  The collection of data on populations of networks is becoming increasingly
common, where each data point can be seen as a realisation of a network-valued
random variable. A canonical example is that of brain networks: a typical
neuroimaging study collects one or more brain scans across multiple
individuals, each of which can be modelled as a network with nodes
corresponding to distinct brain regions and edges corresponding to structural
or functional connections between these regions. Most statistical network
models, however, were originally proposed to describe a single underlying
relational structure, although recent years have seen a drive to extend these
models to populations of networks. Here, we propose one such extension: 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. To perform inference, we devise a novel
exchange-within-Gibbs MCMC algorithm that generates samples from the
doubly-intractable posterior. To illustrate our framework, we use it to assess
group-level variations in networks derived from fMRI scans, enabling the
inference of age-related differences in the topological structure of the
brain's functional connectivity.
