Essays on the economics of networks

Networks (collections of nodes or vertices and graphs capturing their linkages) are a common object of study across a range of fields includ- ing economics, statistics and computer science. Network analysis is often based around capturing the overall structure of the network by some reduced set of parameters. Canonically, this has focused on the notion of centrality. There are many measures of centrality, mostly based around statistical analysis of the linkages between nodes on the network. However, another common approach has been through the use of eigenfunction analysis of the centrality matrix. My the- sis focuses on eigencentrality as a property, paying particular focus to equilibrium behaviour when the network structure is fixed. This occurs when nodes are either passive, such as for web-searches or queueing models or when they represent active optimizing agents in network games. The major contribution of my thesis is in the applica- tion of relatively recent innovations in matrix derivatives to centrality measurements and equilibria within games that are function of those measurements. I present a series of new results on the stability of eigencentrality measures and provide some examples of applications to a number of real world examples

Exact Algorithms for NP-hard Problems on Networks: Design, Analysis, and Implementation

Die Arbeit befasst sich mit dem Entwurf von exakten Algorithmen für verschiedene NP-vollständige Optimierungsprobleme auf Graphen, wie etwa Vertex Cover, Independent Set, oder Dominating Set. Viele praxisbezogene Aufgaben, beispielsweise sogenannte 'facility location'-Probleme aus dem Bereich der Entscheidungsanalyse (decision analysis), sind durch entsprechende Netzwerk-Modellierung auf derartige Fragestellungen zurückzuführen. Im Vordergrund der Arbeit stehen Lösungsverfahren mit beweisbaren Laufzeitschranken. Wegen der solchen Problemen inhärenten, hohen kombinatorischen Komplexität müssen wir exponentielles Laufzeitverhalten unserer Algorithmen in Kauf nehmen, wollen dieses jedoch kleinstmöglich halten. Wir verfolgen dabei den jüngst vorgeschlagenen Ansatz sogenannter 'parametrisierter Algorithmen'. Vereinfacht gesagt handelt es sich hierbei um eine zweidimensionale Herangehensweise, bei welcher die Laufzeit nicht ausschlie3lich in der Grö3e der Eingabeinstanz, sondern überdies auch in der Grö3e eines sogenannten 'Problemparameters' gemessen wird. Dabei untersuchen wir sowohl von theoretischer, als auch von praktischer Seite unterschiedliche Methoden des Algorithmen-Designs: Datenreduktion, beschränkte Suchbäume, Separation von Graphen und das Konzept von Baumzerlegungen. Schlie3lich stellen wir ein Software-Paket vor, welches im Rahmen dieses Projektes entwickelt wurde und eine Vielzahl der entwickelten Algorithmen implementiert. Wir berichten über eine Reihe von empirischen Studien zur Auswertung der Praxistauglichkeit dieser Algorithmen.This thesis deals with the design of exact algorithms for various NP-complete optimization problems on graphs like Vertex Cover, Independent Set, or Dominating Set. We encounter such problems in a broad variety of application ranges, e.g., when modelling so-called facility location tasks in the area of decision analysis and network design. The main focus of this work is on solving algorithms with provable bounds on the running time. Due to the seemingly unavoidable inherent high combinatorial complexity of the problems under consideration, we are forced to deal with exponential running times; our goal, however, is to keep this exponential part as low as possible. To this end, we follow a recent approach of so-called 'fixed-parameter algorithms,' where the running time of an algorithm solving this problem shall be measured not only in the size of the input instance, but also in the size of a so-called 'problem parameter.' In this sense, whereas classical complexity theory offers a one-dimensional approach, parameterized complexity theory is a two-dimensional study of combinatorial problems. We investigate from both, a theoretical, as well as a practical point of view, various methods in the design of fixed-parameter algorithms: data reduction, bounded search trees, graph separation, and the concept of tree-decompositions. In addition, a software-package is presented, which was developed in our project and which implements most of our algorithms. Finally, we report on a first series of empirical studies underpinning the practical strength and usefulness of our algorithms

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum