L-2 Regularized maximum likelihood for $\beta$-model in large and sparse networks

arXiv2110.11856MaRDI QIDQ6381047

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Publication date: 22 October 2021

Abstract: The $$-model is a powerful tool for modeling network generation driven by degree heterogeneity. Its simple yet expressive nature particularly well-suits large and sparse networks, where many network models become infeasible due to computational challenge and observation scarcity. However, existing estimation algorithms for $$-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 $$-model to address urgent needs of practical applications. Our contributions include: 1. method: we propose a new $el{l}_2$ penalized MLE scheme; we design a novel fast algorithm that can comfortably handle sparse networks of millions of nodes, much faster and more memory-parsimonious than all existing algorithms; 2. theory: we present new error bounds on $$-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 $el{l}_2$ 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.

Has companion code repository: https://github.com/MjiaShao/L2-beta-model

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