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Theses

Maximum likelihood estimation in dynamic or spatial stochastic block models

Abstract : This thesis deals with maximum likelihood estimation in dynamic and spatial extensions of the stochastic block model (SBM), based respectively on hidden Markov chains and fields. First, we consider a dynamic version of the stochastic block model, suited for the observation of networks at multiple time steps. In this dynamic SBM, the nodes are partitioned into latent classes and the connection between two nodes is drawn from a Bernoulli distribution depending on the classes of these two nodes. The temporal evolution of the nodes memberships is modeled by a hidden Markov chain. We prove the consistency (as the numbers of nodes and time steps increase) of the maximum likelihood and variational estimators of the parameters and obtain upper bounds on their rates of convergence. We also explore the case where the number of time steps is fixed and the connectivity parameters are allowed to vary. Besides, we obtain some results regarding parameter identifiability. Second, we introduce a spatial version of the SBM, suited for the observation of networks at different spatial locations. As before, the nodes are partitioned into latent classes and the connection is drawn from a Bernoulli distribution depending on the classes of these two nodes. The spatial evolution of the nodes memberships is modeled through hidden Markov random fields. We first prove that the parameter is generically identifiable under certain conditions. For the estimation of the parameters, we propose an algorithm based on the simulated field Expectation-Maximisation (EM) algorithm, a variation of the EM algorithm relying on a mean field like approximation based on the simulation of latent configurations.
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Léa Longepierre. Maximum likelihood estimation in dynamic or spatial stochastic block models. Statistics [math.ST]. Sorbonne Université, 2020. English. ⟨NNT : 2020SORUS184⟩. ⟨tel-03343958⟩

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