Link Patterns in Complex Networks

Abstract

Network theorists define patterns in complex networks in various ways to make them accessible to human beholders. Prominent definitions are thereby based on the partition of the network's nodes into groups such that underlying patterns in the link structure become apparent. Clustering and blockmodeling are two well-known approaches of this kind. In this thesis, we treat pattern search problems as discrete mathematical optimization problems. From this viewpoint, we develop a new mathematical classification of clustering and blockmodeling approaches, which unifies these two fields and replaces several NP-hardness proofs by a single one. We furthermore use this classification to develop integer mathematical programming formulations for pattern search problems and discuss new linearization techniques for polynomial functions therein. We apply these results to a model for a new pattern search problem. Even though it is the most basic problem in combinatorial terms, we can prove its NP-hardness. In fact, we show that it is a generalization of well-known problems including the Traveling Salesman and the Quadratic Assignment Problem. Our derived exact pattern search procedure is up to 10,000 times faster than comparable methods from the literature. To demonstrate its practicability, we finally apply the procedure to the world trade network from the United Nations' database and show that the network deviates by less than 0.14% from the patterns we found.