The generalised random dot product graph

Because they appear in virtually every facet of the digital world, there is considerable value in being able to make inference and predictions based on networks. In Statistics, such endeavours often start with a probability model, mapping unknown quantities of interest to the data, and, here, one is proposed which strikes a promising balance of generality and interpretability. Our focus is on the simplest case of modelling a graph, that is, a set of nodes and (undirected) edges. To start discussions, we consider first the benefits and drawbacks of a foundational model known as the stochastic blockmodel (Holland et al., 1983). In this model, the nodes of the graph can be grouped into k communities, such that the probability of two nodes forming an edge is dependent only on the two communities involved, and is given by a k k inter-community edge probability matrix B. Under basic exchangeability assumptions (Aldous, 1981; Hoover, 1979), the model can be regarded as providing a piecewise constant, or even histogram (Olhede and Wolfe, 2014), approximation to any random graph model satisfying basic exchangeability assumptions (Aldous, 1981; Hoover, 1979). Its generality yet simple interpretation make it a natural candidate for exploratory data analysis and the model is very popular in practice. However, one obvious issue is its discrete structure, in particular, the'hard' assignment of every node to a single community. We would often prefer to describe node behaviour in a more continuous way. In a seminal paper, Hoff et al. (2002) considered a number of latent position models where, in abstract terms, each node i is mapped to a point X