Networks are fast becoming part of the modern statistical landscape. Yet we lack a full understanding of their large-sample properties in all but the simplest settings, hindering the development of models and estimation methods that admit theoretical performance guarantees. A network histogram is obtained by fitting a stochastic blockmodel to a single observation of a network dataset. Blocks of edges play the role of histogram bins, and community sizes that of histogram bandwidths or bin sizes. Just as standard histograms allow for varying bandwidths, different blockmodel estimates can all be considered valid representations of an underlying probability model, subject to bandwidth constraints. We show that under these constraints, the mean integrated square error of the network histogram tends to zero as the network grows large, and we provide methods for optimal bandwidth selection-thus making the blockmodel a universal representation. With this insight, we discuss the interpretation of network communities in light of the fact that many different community assignments can all give an equally valid representation of the network. To demonstrate the fidelity-versus-interpretability tradeoff inherent in considering different numbers and sizes of communities, we show an example of detecting and describing new network community microstructure in political weblog data.
This is joint work with Patrick Wolfe - UCL