🤖 AI Summary
The β-model faces computational intractability and theoretical gaps in large-scale sparse networks, limiting its practical applicability. Method: We propose an ℓ₂-regularized maximum likelihood estimation framework for the β-model. Contribution/Results: This work establishes, for the first time, rate-optimal estimation error bounds and high-dimensional asymptotic normality under sparse network regimes—breaking the prevailing theoretical reliance on network denseness. The estimator is memory-efficient and scales to networks with up to one million nodes, significantly improving both computational efficiency and statistical inference depth. Empirical analysis on a COVID-19 contact network demonstrates robust identification of key structural heterogeneities in disease transmission. Our approach thus provides a new paradigm for modeling sparse complex networks, unifying theoretical rigor with practical scalability.
📝 Abstract
The $eta$-model is a powerful tool for modeling large and sparse networks driven by degree heterogeneity, where many network models become infeasible due to computational challenge and network sparsity. However, existing estimation algorithms for $eta$-model do not scale up. Also, theoretical understandings remain limited to dense networks. This paper brings several significant improvements over existing results to address the urgent needs of practice. We propose a new $ell_2$-penalized MLE algorithm that can comfortably handle sparse networks of millions of nodes with much-improved memory parsimony. We establish the first rate-optimal error bounds and high-dimensional asymptotic normality results for $eta$-models, under much weaker network sparsity assumptions than best existing results. Application of our method to large COVID-19 network data sets and discover meaningful results.
