Random Dot Product Graphs model a network by assigning each vertex a latent vector and taking edge probabilities to be inner products of these latent positions. This flexible framework includes many familiar network models, including many stochastic block models, as special cases, while also allowing for more general latent structures. A fundamental practical question is how to determine the unknown latent dimension from a single observed adjacency matrix.
In this talk, I will introduce a model-selection criterion designed to be consistent: as the number of observed vertices tends to infinity, it selects the correct latent dimension with probability tending to one. I will present consistency results in the dense regime and discuss the additional conditions required when the graph becomes sparse.
Finally, I will discuss the finite-sample implementation of the criterion, the choice of a data-driven penalty, and comparisons with commonly used alternatives such as eigengap and scree-plot methods.
