Spectral methods in networks: hierarchical structures and risk estimation

Although spectral clustering has been extensively studied in network analysis, some important issues, such as hierarchical clustering via eigenvectors and determining the number of communities via eigenvalues, have been far less investigated thus far. The first part of the talk is  about the theoretical analysis of hierarchical community detection. We show that graph-Laplacian based spectral hierarchical clustering is consistent under general tree structures and broad ranges of connectivity probabilities. Our analysis relies on a careful exploitation of the algebraic properties of graph Laplacian and statistical properties of hierarchical SBM. The second part is concerning risk estimation for spectral hard thresholding based graphon estimation. We combine Efron's Steinian risk estimation framework and a divergence formula of spectral functions to derive a close-form approximate formula, which is further used to determine the number of communities by the spectrum of the adjacency matrix. Our approach enjoys desirable empirical properties which are illustrated in some real world datasets.

Dr. Xiaodong Li is an assistant professor in the statistics department at UC Davis. Prior to that, he worked in the statistics department of Wharton School at University of Pennsylvania for two years. He got PhD of mathematics at Stanford University in 2013, and BS at Peking University in 2008. He has general research interests in network analysis, theory of machine learning, and mathematical/statistical signal processing. His papers have been published in various journals of statistics, mathematics and engineering such as AoS, ACHA, FOCM, JACM, IEEE TIT, etc. He is awarded the NSF Career Award in 2019.