Inference on random networks
Yann Issartel · HAL (Le Centre pour la Communication Scientifique Directe) · 2020
This thesis lies at the intersection of the theories of non-parametric statistics and statistical learning. Its goal is to provide an understanding of statistical problems in latent space random graphs. Latent space models have emerged as useful probabilistic tools for modeling large networks in various fields such as biology, marketing or social sciences.We first define an identifiable index of the dimension of the latent space and then a consistent estimator of this index. More generally, such identifiable and interpretable quantities alleviate the absence of identifiability of the latent space itself.We then introduce the pair-matching problem. From a non-observed graph, a strategy sequentially queries pairs of nodes and observes the presence/absence of edges. Its goal is to discover as many edges as possible with a fixed budget of queries. For this bandit type problem, we study optimal regrets in stochastic block models and random geometric graphs.Finally, we are interested in estimating the positions of the nodes in the latent space, in the particular situation where the space is a circle in the Euclidean plane.For each of the three problems, we obtain procedures that achieve the statistical optimal performance, as well as efficient procedures with theoretical guarantees. These algorithms are analysed from a non-asymptotic viewpoint, relying in particular on concentration inequalities.