| Summary: | <p>Payoff division schemes from cooperative game theory, such as the Shapley value and Banzhaf index, have recently gained popularity as measures of node centrality in networks. The general idea in this line of research is to treat nodes as players in a cooperative game, to treat a group centrality measure as a characteristic function and to apply a division scheme as a centrality measure. While this research direction is promising, the computational problems that surround it are challenging and are mostly open. On the one hand, the current research lacks sufficient breadth to understand the field and its computational intricacy as a whole. Existing literature has focused on combining a single division scheme (or class of division schemes) with a single group measure and computing it. On the other hand, the current game-theoretic measures lack sufficient depth to be applicable to certain complex but often-encountered settings, such as networks with an overlapping community structure.</p> <p>The aim of this thesis is to (1) provide a broad computational analysis of a wide class of game-theoretic measures, (2) develop a new class of measures for networks with overlapping community structures, (3) propose high-performance game-theoretic measures.</p> <p>With respect to the first issue, we present a generic framework for defining game-theoretic measures and prove that all measures that can be expressed in our framework are computable in polynomial time. The framework applies to a broad class of division schemes called semivalues in combination with a very wide range of group centrality measures, which is the first result of its kind to date.</p> <p>Our second focus is on a class of real-life networks that have an overlapping community structure. We develop a new class of solution concepts - configuration semivalues - and use them to propose a class of centrality measures based on closeness. We develop polynomial algorithms for its computation and apply them to the Warsaw public transportation network, where they produce a highly interpretable ranking that is - arguably - better than other measures.</p> <p>Finally, we propose the generalised closeness interaction index for the purposes of link prediction. We develop polynomial-time algorithms for the computation of our measure and find that it achieves the best results when compared to the state-of-the-art link prediction methods in the literature.</p>
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