Analysis of annotated networks via the sparse β-model

Exponential random graph models have emerged as attractive models for modelling network data. However, mathematical theory so far is mostly limited to the case of dense networks, that is, networks in which the number of observed edges scales quadratically in the number of nodes. Real networks however, are almost exclusively sparse, i.e. the number of observed edges scales much more slowly. The Sparse \beta-model is a novel random network model designed to model sparse networks. It can be interpreted as an interpolation of the celebrated, yet simple, Erdős–Rényi model and the flexible \beta-model. In my talk I will show you how to consistently estimate the parameters in the sparse \beta-model when additional node covariates are observed. We will see that we obtain different rates of convergence for the local and global parameters of the model and discuss the difficulties emerging from the sparsity assumptions. I will also show you my most recent result, which is a central limit theorem for the parameter associated with the covariates.