In large and complex systems, failures can have dramatic consequences, such as black-outs, pandemics or the loss of entire classes of an ecosystem. Nevertheless, it is a centuries-old intuition that by using networks to capture the core of the complexity of such systems, one might understand in which part of a system a phenomenon originates. I investigate this intuition using spectral methods to decouple the dynamics of complex systems near stationary states into independent dynamical modes. In this description, phenomena are tied to a specific part of a system through localized eigenvectors which have large amplitudes only on a few nodes of the system's network.
Studying the occurrence of localized eigenvectors, I find that such localization occurs exactly for a few small network structures, and approximately for the dynamical modes associated with the most prominent failures in complex systems. My findings confirm that understanding the functioning of complex systems generally requires to treat them as complex entities, rather than collections of interwoven small parts. Exceptions to this are only few structures carrying exact localization, whose functioning is tied to the meso-scale, between the size of individual elements and the size of the global network.
However, while understanding the functioning of a complex system is hampered by the necessary global analysis, the prominent failures, due to their localization, allow an understanding on a manageable local scale. Intriguingly, food webs might exploit this localization of failures to stabilize by causing the break-off of small problematic parts, whereas typical attempts to optimize technological systems for stability lead to delocalization and large-scale failures. Thus, this thesis provides insights into the interplay of complexity and localization, which is paramount to ascertain the functioning of the ever-growing networks on which we humans depend.:1 Introduction
2 Concepts and Tools
2.1 Networks
2.2 Food webs
2.3 Dynamics on networks
2.4 Steady state operating modes
2.5 Bifurcations affecting operating modes
2.6 Dynamical modes
2.7 Generalized models for food webs
3 Perturbation Impact
3.1 Impact of perturbations on food webs
3.2 Examples
3.3 Impact formulation with dynamical modes
3.4 Influence and sensitivity of species
3.5 Localized dynamical modes
3.6 Iterative parameter estimation
3.7 Most important parameters and species
3.8 Discussion
4 Exact Localization
4.1 Graph symmetries
4.2 Localized dynamics on symmetries
4.3 Exactly localized dynamics
4.4 Symmetry reduction in networks
4.5 Application to food webs
4.6 Localization on asymmetric structures
4.7 Nearly-exact localization
4.8 Other systems
4.9 Discussion
5 Approximate Localization
5.1 Spread of a dynamical mode
5.2 Examples for localized instabilities
5.3 Localization of extreme eigenvalues
5.4 Dependence on the system size
5.5 Localization in the model of R. May
5.6 Finding motifs that carry localization
5.7 (Self-)stabilization of food webs
5.8 Repairing localized instabilities
5.9 Discussion
6 Conclusions
Acknowledgments
Appendix
A Parametrization of the Gatun Lake food web
B The Master Stability Function approach
C Approximate localization on larger structures
Bibliography
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:28263 |
| Date | 08 September 2014 |
| Creators | Aufderheide, Helge E. |
| Contributors | Gross, Thilo, Jülicher, Frank, Radons, Günther, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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