Network data are frequently collected in modern society and science. Stylized features of a typical network include network sparsity, degree heterogeneity and homophily. This talk introduces a framework with a class of sparse models that utilize parameters to explicitly account for these network features. In particular, degree heterogeneity is handled by node-specific parameters while homophily is captured by the use of covariates. To avoid over-parametrization due to the former, we differentially assign parameters to nodes that are important in certain sense. We start by discussing the sparse β model when no covariates are present, and proceed to discuss a generalized model to include covariates. Interestingly for the former we can use ℓ0 penalization to identify and estimate the heterogeneity parameters, while for the latter we resort to penalized logistic regression with an ℓ1 penalty, thus immediately connecting our methodology to the lasso literature. Along the way, we demonstrate the fallacy of what we call data-selective inference, a common practice in the literature to discard less well-connected nodes in order to fit a model, which can be of independent interest.