Large Networks, Perturbation of Block Structures - Bolla, Marianna (Spectral Clustering and Biclustering, 2013)
This chapter applies the results of the previous chapters for the spectral clustering of large networks. Networks are modeled both by edge‐weighted graphs or contingency tables, and usually subject to random errors due to their evolving and flexible nature. Asymptotic properties of SD and SVD of the involved matrices are discussed when not only the number of graph vertices or the number of rows and columns of the contingency table tends to infinity, but the cluster sizes also grow proportionally with them. Mostly, perturbation results for the SD and SVD of blown‐up matrices burdened with a Wigner‐type error matrix are investigated. Conversely, given a weight‐matrix or rectangular array of nonnegative entries, we are looking for the underlying block‐structure. The chapter shows that under general circumstances, clusters of vertices of a large graph, and simultaneously those of the rows and columns of a large contingency table, are identified with high probability.