Analysis of networks via the sparse β-model

Abstract

Data in the form of networks are increasingly available in a variety of areas, yet statistical models allowing for parameter estimates with desirable statistical properties for sparse networks remain scarce. To address this, we propose the Sparse (Formula presented.) -Model ((Formula presented.)), a new network model that interpolates the celebrated Erdős–Rényi model and the (Formula presented.) -model that assigns one different parameter to each node. By a novel reparameterization of the (Formula presented.) -model to distinguish global and local parameters, our (Formula presented.) can drastically reduce the dimensionality of the (Formula presented.) -model by requiring some of the local parameters to be zero. We derive the asymptotic distribution of the maximum likelihood estimator of the (Formula presented.) when the support of the parameter vector is known. When the support is unknown, we formulate a penalized likelihood approach with the (Formula presented.) -penalty. Remarkably, we show via a monotonicity lemma that the seemingly combinatorial computational problem due to the (Formula presented.) -penalty can be overcome by assigning non-zero parameters to those nodes with the largest degrees. We further show that a (Formula presented.) -min condition guarantees our method to identify the true model and provide excess risk bounds for the estimated parameters. The estimation procedure enjoys good finite sample properties as shown by simulation studies. The usefulness of the (Formula presented.) is further illustrated via the analysis of a microfinance take-up example.