A Stochastic Block Model for Multilevel Networks: Application to the Sociology of Organisations
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This work is motivated by the analysis of multilevel networks. We define a multilevel network as the junction of two interaction networks, one level representing the interactions between individuals and the other one the interactions between organisations. The levels are linked by an affiliation relationship, each individual belonging to a unique organisation. We design a Stochastic block model (SBM) suited to multilevel networks. SBM is a latent variable model for networks, where the connections between nodes depend on a latent clustering (blocks), thus modeling some connection heterogeneity. We prove the identifiability of our model. The parameters of the model are estimated with a variational EM algorithm. An Integrated Completed Likelihood criterion is developed not only to select the number of blocks but also to detect whether the two levels (individuals and organisations) are dependent or not. In a comprehensive simulation study, we exhibit the benefit of considering our approach, illustrate the robustness of our parameter estimation and highlight the reliability of our model selection criterion. Our approach is applied on a sociological dataset collected during a television programmers trade fare. The inter-organisational level is the economic networks between companies and the inter-individual level is the informal network between their representatives.
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