Multilevel Methods for Sparsification and Linear Arrangement Problems on Networks

Abstract

The computation of network properties such as diameter, centrality indices, and paths on networks may become a major bottleneck in the analysis of network if the network is large. Scalable approximation algorithms, heuristics and structure preserving network sparsification methods play an important role in modern network analysis. In the first part of this thesis, we develop a robust network sparsification method that enables filtering of either, so called, long- and short-range edges or both. Edges are first ranked by their algebraic distances and then sampled. Furthermore, we also combine this method with a multilevel framework to provide a multilevel sparsification framework that can control the sparsification process at different coarse-grained resolutions. Experimental results demonstrate an effectiveness of the proposed methods without significant loss in a quality of computed network properties. In the second part of the thesis, we introduce asymmetric coarsening schemes for multilevel algorithms developed for linear arrangement problems. Effectiveness of the set of coarse variables, and the corresponding interpolation matrix is the central problem in any multigrid algorithm. We are pushing the boundaries of fast maximum weighted matching algorithms for coarsening schemes on graphs by introducing novel ideas for asymmetric coupling between coarse and fine variables of the problem