Abstract :

A simple yet sufficiently expressive and interpretable model for random graphs is the Random Dot Product Graph (RDPG), which also comes with an inference method with statistical guarantees known as Adjacency Spectral Embedding (ASE). In this talk, we present this model alongside generalisations that allow us to overcome some of its best-known limitations, in all cases seeking to preserve its fundamental properties of interpretability and guarantees. We frame ASE as an optimisation problem, which enables the definition of inference methods that handle missing data or dynamic networks, where the number of nodes may, for example, increase or decrease over time. We also propose new inference methods for directed graphs, and analyse the corresponding optimisation landscape,  proving that it is benign. Furthermore, we introduce a nonparametric latent position model for random weighted graphs by relating the inner product of the nodal latent position vectors to the moment generating function of the edge weight distribution, proving statistical guarantees such as consistency and asymptotic normality. These spectral embeddings can also be used to generate new graphs that statistically resemble the structure and edge weight distribution of a given real graph. Finally, we discuss some further limitations and new directions,  including heterophilious graphs and hyperbolic geometry.