  The $\beta$-model is a powerful tool for modeling network generation driven
by degree heterogeneity. Its simple yet expressive nature particularly
well-suits large and sparse networks, where many network models become
infeasible due to computational challenge and observation scarcity. However,
existing estimation algorithms for $\beta$-model do not scale up; and
theoretical understandings remain limited to dense networks. This paper brings
several significant improvements to the method and theory of $\beta$-model to
address urgent needs of practical applications. Our contributions include: 1.
method: we propose a new $\ell_2$ penalized MLE scheme; we design a novel fast
algorithm that can comfortably handle sparse networks of millions of nodes,
much faster and more memory-parsimonious than all existing algorithms; 2.
theory: we present new error bounds on $\beta$-models under much weaker
assumptions than best known results in literature; we also establish new
lower-bounds and new asymptotic normality results; under proper parameter
sparsity assumptions, we show the first local rate-optimality result in
$\ell_2$ norm; distinct from existing literature, our results cover both small
and large regularization scenarios and reveal their distinct asymptotic
dependency structures; 3. application: we apply our method to large COVID-19
network data sets and discover meaningful results.
