@misc{Donges2009,
  type      = {Master Thesis},
  author    = {Donges, Jonathan Friedemann},
  title     = {Complex networks in the climate system},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49775},
  school      = {Universit{\"a}t Potsdam},
  year      = {2009},
  abstract  = {Complex network theory provides an elegant and powerful framework to statistically investigate the topology of local and long range dynamical interrelationships, i.e., teleconnections, in the climate system. Employing a refined methodology relying on linear and nonlinear measures of time series analysis, the intricate correlation structure within a multivariate climatological data set is cast into network form. Within this graph theoretical framework, vertices are identified with grid points taken from the data set representing a region on the the Earth's surface, and edges correspond to strong statistical interrelationships between the dynamics on pairs of grid points. The resulting climate networks are neither perfectly regular nor completely random, but display the intriguing and nontrivial characteristics of complexity commonly found in real world networks such as the internet, citation and acquaintance networks, food webs and cortical networks in the mammalian brain. Among other interesting properties, climate networks exhibit the "small-world" effect and possess a broad degree distribution with dominating super-nodes as well as a pronounced community structure. We have performed an extensive and detailed graph theoretical analysis of climate networks on the global topological scale focussing on the flow and centrality measure betweenness which is locally defined at each vertex, but includes global topological information by relying on the distribution of shortest paths between all pairs of vertices in the network. The betweenness centrality field reveals a rich internal structure in complex climate networks constructed from reanalysis and atmosphere-ocean coupled general circulation model (AOGCM) surface air temperature data. Our novel approach uncovers an elaborately woven meta-network of highly localized channels of strong dynamical information flow, that we relate to global surface ocean currents and dub the backbone of the climate network in analogy to the homonymous data highways of the internet. This finding points to a major role of the oceanic surface circulation in coupling and stabilizing the global temperature field in the long term mean (140 years for the model run and 60 years for reanalysis data). Carefully comparing the backbone structures detected in climate networks constructed using linear Pearson correlation and nonlinear mutual information, we argue that the high sensitivity of betweenness with respect to small changes in network structure may allow to detect the footprints of strongly nonlinear physical interactions in the climate system. The results presented in this thesis are thoroughly founded and substantiated using a hierarchy of statistical significance tests on the level of time series and networks, i.e., by tests based on time series surrogates as well as network surrogates. This is particularly relevant when working with real world data. Specifically, we developed new types of network surrogates to include the additional constraints imposed by the spatial embedding of vertices in a climate network. Our methodology is of potential interest for a broad audience within the physics community and various applied fields, because it is universal in the sense of being valid for any spatially extended dynamical system. It can help to understand the localized flow of dynamical information in any such system by combining multivariate time series analysis, a complex network approach and the information flow measure betweenness centrality. Possible fields of application include fluid dynamics (turbulence), plasma physics and biological physics (population models, neural networks, cell models). Furthermore, the climate network approach is equally relevant for experimental data as well as model simulations and hence introduces a novel perspective on model evaluation and data driven model building. Our work is timely in the context of the current debate on climate change within the scientific community, since it allows to assess from a new perspective the regional vulnerability and stability of the climate system while relying on global and not only on regional knowledge. The methodology developed in this thesis hence has the potential to substantially contribute to the understanding of the local effect of extreme events and tipping points in the earth system within a holistic global framework.},
  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}
}