## Section: New Results

### Random graph theory

Participant : Matthieu Lerasle.

In collaboration with R. Chetrite and R. Diel, Matthieu Lerasle published an article on the number of potential winners in the Bradley-Terry model in random environments. They proposed the first mathematical study of the Bradley-Terry model where the values of players are i.i.d. realisations of some distribution. They proved that a Bradley-Terry tournament is fair (in the sense that the best player ends up with the largest number of victories) under a certain convexity condition on the tail distribution of the values. They also showed that this condition is sharp and provided sharp estimate of the number of potential winners when the condition fails.

He also collaborated with R. Diel and S. Le Corff on learning latent structures of large random graphs, investigating the possibility of estimating latent structure in sparsely observed random graphs. The main example was a Bradley-Terry tournament where each team has only played a few games. It is well known that individual values of the teams cannot be consistently estimated in this setting. They showed that their distribution on the other hand can be, and provide general tools for bounding the risk of the maximum likelihood estimator (submitted).