Sparse β model for the directed networks

Many complex systems involve plenty of units that interact with each other. The study of networks has attracted increasing attention in a wide variety of areas. This paper concerns a sparse \beta model for directed networks (DS\betaM) by allowing self-loops, which can capture degree heterogeneity. Specifically, we have derived consistency and asymptotic normality by reparameterizing the global and local density parameters when the support is known. For the other case when the support is unknown, we propose the monotonicity lemma with the l_0 penalized likelihood approach to estimate the parameters. The simulation results help prove our theoretical properties and the real data analyses confirm the usefulness of our model.