@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}
}