2019/03/06 by Jong Yun Hwang, Hwang, Jong Yun, Ji Oon Lee +3
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Complex Network Analysis Techniques
paper · pdf · doi:10.48550/arxiv.1903.02179
We consider the spectral properties of sparse stochastic block models, where N vertices are partitioned into K balanced communities. Under an assumption that the intra-community probability and inter-community probability are of similar order, we prove a local semicircle law up to the spectral edges, with an explicit formula on the deterministic shift of the spectral edge. We also prove that the fluctuation of the extremal eigenvalues is given by the GOE Tracy--Widom law after rescaling and centering the entries of sparse stochastic block models. Applying the result to sparse stochastic block models, we rigorously prove that there is a large gap between the outliers and the spectral edge without centering.