Link Building

Google uses the PageRank algorithm to compute an estimate of the popularity of each page based solely on the link structure of the web graph – these estimates are the so-called PageRank values. A page will achieve one of the top spots of a search result if it has a high PageRank value and matches the search criteria for the actual Google search. For a given page t and k ∈ Z+ we study the problem of computing k new links pointing to t – so-called backlinks to t – producing the maximum increase in the PageRank value of t. The problem of obtaining optimal new backlinks in order to achieve good search engine rankings is known as Link Building and this problem attracts much attention from the Search Engine Optimization (SEO) industry. In this dissertation we concentrate on the problem of identifying optimal new backlinks and refer to this problem as the Link Building problem. We show that no FPTAS exists for Link Building under the assumption NP 6 =P and that Link Building is W[1]hard. On the more positive side we show how to solve the case with fixed k = 1 using time corresponding to a small and constant number of PageRank computations using a randomized scheme and we show that Link Building is a member of the complexity class APX. We also show how to use Mixed Integer Linear Programming to solve the problem for smaller graphs and values of k. We show how the Link Building problem is related to the problem of detecting community structures in networks. We present a community definition justified by a formal analysis of a very simple model of the evolution of a directed graph G(V,E) and show that the problem of deciding whether a community C 6= V exists such that R ⊆ C for a given set of representatives R is NP complete. In spite of the intractability result we show that a fast and simple parameter free greedy approach performs well when detecting communities in a crawl of the Danish part of the web graph. We present results from a branch of game theory dealing with so-called Hedonic Games and argue that community structures can be viewed as Nash equilibriums for Hedonic Games and in this way we provide a link to the other topics in the dissertation. To be more specific we show that computing Nash equilibriums in Additively Separable Hedonic Games is NP-hard. Finally, we present results from another branch of game theory concerning what is known as Simple Games. For several properties we study the computational complexity of deciding whether or not a given simple game has the property. Some of the proof techniques used in this final part of the dissertation are used several other places in the dissertation.