Connecting the Dots: Network Testing, Community Estimation, and Genomic Applications

Abstract

This thesis presents three self-contained chapters: a global detection test in the mixed membership Stochastic Block Model (SBM), a community estimation algorithm in the dynamic degree-corrected mixed membership SBM, and a network-based study of the transcriptional control of cellular signal sensing. Detecting the presence of structured communities is one of the most fundamental problems of statistical network analysis. In Chapter 1, we introduce a degree- and cycle count-based test statistic for global testing in the mixed-membership SBM, a common model for social networks. We derive its asymptotic null distribution and show that it is optimal for all choices of model parameters. Studying the evolving structure of complex dynamic networks is becoming increasingly popular. In Chapter 2, we propose a spectral algorithm for dynamic node embedding and mixed membership estimation. We establish explicit error rates under smoothness assumptions on the temporal evolution of mixed memberships and potentially severe degree heterogeneity, showing that our method is rate optimal across a broad parameter range. We showcase its effectiveness on a trade network and a human contact network. Network methods provide critical insights in many scientific disciplines, such as genomics. In Chapter 3, we report evidence of an important layer of transcriptional control of cellular signal sensing. Taking adhesion receptors as an example, we apply diverse network and statistical approaches to characterize the link between chromatin organization in the cell nucleus, gene co-regulation, and receptor proteins clustering.