@phdthesis{Katzmann2023,
  author    = {Katzmann, Maximilian},
  title     = {About the analysis of algorithms on networks with underlying hyperbolic geometry},
  doi       = {10.25932/publishup-58296},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-582965},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xi, 191},
  year      = {2023},
  abstract  = {Many complex systems that we encounter in the world can be formalized using networks. Consequently, they have been in the focus of computer science for decades, where algorithms are developed to understand and utilize these systems. Surprisingly, our theoretical understanding of these algorithms and their behavior in practice often diverge significantly. In fact, they tend to perform much better on real-world networks than one would expect when considering the theoretical worst-case bounds. One way of capturing this discrepancy is the average-case analysis, where the idea is to acknowledge the differences between practical and worst-case instances by focusing on networks whose properties match those of real graphs. Recent observations indicate that good representations of real-world networks are obtained by assuming that a network has an underlying hyperbolic geometry. In this thesis, we demonstrate that the connection between networks and hyperbolic space can be utilized as a powerful tool for average-case analysis. To this end, we first introduce strongly hyperbolic unit disk graphs and identify the famous hyperbolic random graph model as a special case of them. We then consider four problems where recent empirical results highlight a gap between theory and practice and use hyperbolic graph models to explain these phenomena theoretically. First, we develop a routing scheme, used to forward information in a network, and analyze its efficiency on strongly hyperbolic unit disk graphs. For the special case of hyperbolic random graphs, our algorithm beats existing performance lower bounds. Afterwards, we use the hyperbolic random graph model to theoretically explain empirical observations about the performance of the bidirectional breadth-first search. Finally, we develop algorithms for computing optimal and nearly optimal vertex covers (problems known to be NP-hard) and show that, on hyperbolic random graphs, they run in polynomial and quasi-linear time, respectively. Our theoretical analyses reveal interesting properties of hyperbolic random graphs and our empirical studies present evidence that these properties, as well as our algorithmic improvements translate back into practice.},
  language  = {en}
}
@phdthesis{Fischer2022,
  author    = {Fischer, Jens Walter},
  title     = {Random dynamics in collective behavior - consensus, clustering \& extinction of populations},
  doi       = {10.25932/publishup-55372},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-553725},
  school      = {Universit{\"a}t Potsdam},
  pages     = {242},
  year      = {2022},
  abstract  = {The echo chamber model describes the development of groups in heterogeneous social networks. By heterogeneous social network we mean a set of individuals, each of whom represents exactly one opinion. The existing relationships between individuals can then be represented by a graph. The echo chamber model is a time-discrete model which, like a board game, is played in rounds. In each round, an existing relationship is randomly and uniformly selected from the network and the two connected individuals interact. If the opinions of the individuals involved are sufficiently similar, they continue to move closer together in their opinions, whereas in the case of opinions that are too far apart, they break off their relationship and one of the individuals seeks a new relationship. In this paper we examine the building blocks of this model. We start from the observation that changes in the structure of relationships in the network can be described by a system of interacting particles in a more abstract space. These reflections lead to the definition of a new abstract graph that encompasses all possible relational configurations of the social network. This provides us with the geometric understanding necessary to analyse the dynamic components of the echo chamber model in Part III. As a first step, in Part 7, we leave aside the opinions of the inidividuals and assume that the position of the edges changes with each move as described above, in order to obtain a basic understanding of the underlying dynamics. Using Markov chain theory, we find upper bounds on the speed of convergence of an associated Markov chain to its unique stationary distribution and show that there are mutually identifiable networks that are not apparent in the dynamics under analysis, in the sense that the stationary distribution of the associated Markov chain gives equal weight to these networks. In the reversible cases, we focus in particular on the explicit form of the stationary distribution as well as on the lower bounds of the Cheeger constant to describe the convergence speed. The final result of Section 8, based on absorbing Markov chains, shows that in a reduced version of the echo chamber model, a hierarchical structure of the number of conflicting relations can be identified. We can use this structure to determine an upper bound on the expected absorption time, using a quasi-stationary distribution. This hierarchy of structure also provides a bridge to classical theories of pure death processes. We conclude by showing how future research can exploit this link and by discussing the importance of the results as building blocks for a full theoretical understanding of the echo chamber model. Finally, Part IV presents a published paper on the birth-death process with partial catastrophe. The paper is based on the explicit calculation of the first moment of a catastrophe. This first part is entirely based on an analytical approach to second degree recurrences with linear coefficients. The convergence to 0 of the resulting sequence as well as the speed of convergence are proved. On the other hand, the determination of the upper bounds of the expected value of the population size as well as its variance and the difference between the determined upper bound and the actual value of the expected value. For these results we use almost exclusively the theory of ordinary nonlinear differential equations.},
  language  = {en}
}