The generalised random dot product graph

This paper introduces a latent position network model, called the generalised random dot product graph, comprising as special cases the stochastic blockmodel, mixed membership stochastic blockmodel, and random dot product graph. In this model, nodes are represented as random vectors on $\mathbb{R}^d$, and the probability of an edge between nodes $i$ and $j$ is given by the bilinear form $X_i^T I_{p,q} X_j$, where $I_{p,q} = \mathrm{diag}(1,\ldots, 1, -1, \ldots, -1)$ with $p$ ones and $q$ minus ones, where $p+q=d$. As we show, this provides the only possible representation of nodes in $\mathbb{R}^d$ such that mixed membership is encoded as the corresponding convex combination of latent positions. The positions are identifiable only up to transformation in the indefinite orthogonal group $O(p,q)$, and we discuss some consequences for typical follow-on inference tasks, such as clustering and prediction.