Das erste Ziel dieser Arbeit ist eine umfassende Analyse der Wasserflüsse durchzuführen und Quellen und Senken des kontinentalen Niederschlags zu identifizieren. Als Analysemethode werden komplexe Netzwerke verwendet, ein Ansatz, mit dessen Hilfe das neuartige Konzept des “cascading moisture recycling'''' (CMR) eingeführt wird. CMR wird als vielfache Verdunstung von Niederschlag während des Feuchtigkeitstransports über bewaldeten Gebieten definiert. Dieses Verfahren ermöglicht es, den Anteil von CMR an der Menge des regionalen Niederschlags zu quantifizieren und Schlüsselregionen des CMR zu identifizieren. Die Analyse zeigt, dass der südliche Bereich des Amazonasbeckens nicht nur eine direkte Quelle für Niederschlag im La-Plata Becken ist, sondern auch als ``Brückenregion'''' dient, über die die verdunstete Feuchtigkeit des ganzen Amazonasbeckens auf dem Weg in die Subtropen transportiert wird. Die Ergebnisse zeigen, dass eine Neubewertung der Vulnerabilität des Amazonasregenwalds unter Umweltveränderungen unabdingbar ist. Dies ist das zweite Ziel der vorliegenden Arbeit. Durch diese Veränderungen könnten große Teile des Regenwaldes in eine Savanne umgewandelt werden. Dies würde wiederum den Niederschlag reduzieren, was sich negativ auf die Stabilität der verbleibenden Waldgebiete auswirken und ein Waldsterben verursachen kann. Für die Analyse dieser Zusammenhänge werden ebenfalls komplexe Netzwerke verwendet, um das Konzept der Ökosystem-Resilienz und CMR basierend auf Beobachtungsdaten zu kombinieren. Es werden die Schlüsselregionen, in denen Entwaldung zu einer Destabilisierung der verbleibenden Wald führt, identifiziert und die Möglichkeit eines großflächigen Absterben des Regenwaldes aufgrund von verlängerter Trockenzeit untersucht. Die Ergebnisse zeigen, dass die Diversität des Regenwaldes und die durch den Feuchtigkeitstransport gegebene Konnektivität der Waldgebiete eine wichtige Rolle für die Stabilität und ökologische Integrität dieses Ökosystems spielen. / The first aim of this thesis is to improve the understanding of vegetation-atmosphere interactions by means of complex network analysis of water fluxes from the sources to the sinks of rainfall in South America. This novel approach allows to introduce the concept of “cascading moisture recycling” defined as moisture recycling on the continent involving re-evaporation cycles along the way. A methodological framework is developed to quantify the importance of cascading moisture recycling and to identify key regions that sustain this process. It reveals, for instance, that the southern part of the Amazon basin is not only a direct source of rainfall for the La Plata basin as previously thought but also an intermediary region that re-distribute moisture evaporating from the entire Amazon basin towards the subtropics. This new concept lays the foundation for evaluating the vulnerability of the Amazon forest to environmental perturbations, which is the second aim of this thesis. Land-use and rainfall variability are expected to be intensified at the end of the twenty-first century and may push the south-eastern part of the Amazon forest towards a grass-dominated ecosystem. Such a forest loss would reduce local dry-season evapotranspiration and the resulting moisture supply for down-wind rainfall. In turn, this might erode the resilience of the remaining forest and lead to further forest losses. Using a complex network approach, the concepts of forest resilience and cascading moisture recycling are combined in a data-driven modeling framework. Key regions are identified where deforestation would greatly destabilize the remaining forest, as well as tipping points in dry-season intensification for large-scale self-amplified Amazon forest loss. The findings highlight the need to maintain the diversity and connectivity of forest patches in order to sustain the ecological integrity of the largest remaining tropical forest on Earth.

Complex networks have been used to represent the fundamental structure of a multitude of complex systems from various fields. In the network representation, the system is reduced to a set of nodes and links that denote the elements of the system and the connections between them respectively. Complex networks are commonly adaptive such that the structure of the network and the states of nodes evolve dynamically in a coupled fashion. Adaptive networks lead to peculiar complex dynamics and network topologies, which can be investigated by moment-closure approximations, a coarse-graining approach that enables the use of the dynamical systems theory. In this thesis, I study several contact processes in adaptive networks that are defined by the transmission of node states. Employing moment-closure approximations, I establish analytical insights into complex phenomena emerging in these systems. I provide a detailed analysis of existing alternative moment-closure approximation schemes and extend them in several directions. Most importantly, I consider developing analytical approaches for models with complex update rules and networks with complex topologies. I discuss four different contact processes in adaptive networks. First, I explore the effect of cyclic dominance in opinion formation. For this, I propose an adaptive network model: the adaptive rock-paper-scissors game. The model displays four different dynamical phases (stationary, oscillatory, consensus, and fragmented) with distinct topological and dynamical properties. I use a simple moment-closure approximation to explain the transitions between these phases. Second, I use the adaptive voter model of opinion formation as a benchmark model to test and compare the performances of major moment-closure approximation schemes in the literature. I provide an in-depth analysis that leads to a heightened understanding of the capabilities of alternative approaches. I demonstrate that, even for the simple adaptive voter model, highly sophisticated approximations can fail due to special dynamic correlations. As a general strategy for targeting such problematic cases, I identify and illustrate the design of new approximation schemes specific to the complex phenomena under investigation. Third, I study the collective motion in mobile animal groups, using the conceptual framework of adaptive networks of opinion formation. I focus on the role of information in consensus decision-making in populations consisting of individuals that have conflicting interests. Employing a moment-closure approximation, I predict that uninformed individuals promote democratic consensus in the population, i.e. the collective decision is made according to plurality. This prediction is confirmed in a fish school experiment, constituting the first example of direct verification for the predictions of adaptive network models. Fourth, I consider a challenging problem for moment-closure approximations: growing adaptive networks with strongly heterogeneous degree distributions. In order to capture the dynamics of such networks, I develop a new approximation scheme, from which analytical results can be obtained by a special coarse-graining procedure. I apply this analytical approach to an epidemics problem, the spreading of a fatal disease on a growing population. I show that, although the degree distribution has a finite variance at any finite infectiousness, the model lacks an epidemic threshold, which is a genuine adaptive network effect. Diseases with very low infectiousness can thus persist and prevail in growing populations.

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.

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

Complex networks have been used to represent the fundamental structure of a multitude of complex systems from various fields. In the network representation, the system is reduced to a set of nodes and links that denote the elements of the system and the connections between them respectively. Complex networks are commonly adaptive such that the structure of the network and the states of nodes evolve dynamically in a coupled fashion. Adaptive networks lead to peculiar complex dynamics and network topologies, which can be investigated by moment-closure approximations, a coarse-graining approach that enables the use of the dynamical systems theory. In this thesis, I study several contact processes in adaptive networks that are defined by the transmission of node states. Employing moment-closure approximations, I establish analytical insights into complex phenomena emerging in these systems. I provide a detailed analysis of existing alternative moment-closure approximation schemes and extend them in several directions. Most importantly, I consider developing analytical approaches for models with complex update rules and networks with complex topologies. I discuss four different contact processes in adaptive networks. First, I explore the effect of cyclic dominance in opinion formation. For this, I propose an adaptive network model: the adaptive rock-paper-scissors game. The model displays four different dynamical phases (stationary, oscillatory, consensus, and fragmented) with distinct topological and dynamical properties. I use a simple moment-closure approximation to explain the transitions between these phases. Second, I use the adaptive voter model of opinion formation as a benchmark model to test and compare the performances of major moment-closure approximation schemes in the literature. I provide an in-depth analysis that leads to a heightened understanding of the capabilities of alternative approaches. I demonstrate that, even for the simple adaptive voter model, highly sophisticated approximations can fail due to special dynamic correlations. As a general strategy for targeting such problematic cases, I identify and illustrate the design of new approximation schemes specific to the complex phenomena under investigation. Third, I study the collective motion in mobile animal groups, using the conceptual framework of adaptive networks of opinion formation. I focus on the role of information in consensus decision-making in populations consisting of individuals that have conflicting interests. Employing a moment-closure approximation, I predict that uninformed individuals promote democratic consensus in the population, i.e. the collective decision is made according to plurality. This prediction is confirmed in a fish school experiment, constituting the first example of direct verification for the predictions of adaptive network models. Fourth, I consider a challenging problem for moment-closure approximations: growing adaptive networks with strongly heterogeneous degree distributions. In order to capture the dynamics of such networks, I develop a new approximation scheme, from which analytical results can be obtained by a special coarse-graining procedure. I apply this analytical approach to an epidemics problem, the spreading of a fatal disease on a growing population. I show that, although the degree distribution has a finite variance at any finite infectiousness, the model lacks an epidemic threshold, which is a genuine adaptive network effect. Diseases with very low infectiousness can thus persist and prevail in growing populations.:1. Introduction .................................................................................. 1 2. Moment-closure approximations of complex networks ................. 5 3. Cyclic dominance in adaptive network models of opinion formation .......... 25 4. Performance of moment-closure approximations of adaptive networks .... 35 5. Information and consensus in a fish school ................................. 65 6. Epidemic spreading on growing heterogeneous adaptive networks ......... 83 7. Conclusions ................................................................................. 101 Appendix A: Moment expansion for node update rules ................... 107

Diese Dissertation beschäftigt sich mit regionaler Klimamodellierung über Südamerika, der Analyse von Modellsensitivitäten bezüglich Wolkenparametrisierungen und der Entwicklung neuer Methoden zur Modellevaluierung mithilfe von Klimanetzwerken. Im ersten Teil untersuchen wir Simulationen mit dem COnsortium for Small scale MOdeling model in CLimate Mode (COSMO-CLM) und stellen die erste umfassende Evaluierung dieses dynamischen regionalen Klimamodells über Südamerika vor. Dabei untersuchen wir insbesondere die Abhängigkeit simulierter tropischer Niederschläge von Parametrisierungen subgitterskaliger cumuliformer und stratiformer Wolken und finden starke Sensitivitäten bezüglich beider Wolkenparametrisierungen über Land. Durch einen simultanen Austausch der entsprechenden Schemata gelingt uns eine beträchtliche Reduzierung von Fehlern in klimatologischen Niederschlags- und Strahlungsmitteln, die das COSMO-CLM über tropischen Regionen für lange Zeit charakterisierten. Im zweiten Teil führen wir neue Metriken für die Evaluierung von Klimamodellen bezüglich räumlicher Kovariabilitäten ein. Im Kern bestehen diese Metriken aus Unähnlichkeitsmaßen für den Vergleich von simulierten mit beobachteten Klimanetzwerken. Wir entwickeln lokale und globale Unähnlichkeitsmaße zum Zwecke der Darstellung lokaler Unähnlichkeiten in Form von Fehlerkarten sowie der Rangordnung von Modellen durch Zusammenfassung lokaler zu globalen Unähnlichkeiten. Die neuen Maße werden dann für eine vergleichende Evaluierung regionaler Klimasimulationen mit COSMO-CLM und dem Statistical Analogue Resampling Scheme über Südamerika verwendet. Dabei vergleichen wir die sich ergebenden Modellrangfolgen mit solchen basierend auf mittleren quadratischen Abweichungen klimatologischer Mittelwerte und Varianzen und untersuchen die Abhängigkeit dieser Rangfolgen von der betrachteten Jahreszeit, Variable, dem verwendeten Referenzdatensatz und Klimanetzwerktyp. / This dissertation is about regional climate modeling over South America, the analysis of model sensitivities to cloud parameterizations, and the development of novel model evaluation techniques based on climate networks. In the first part we examine simulations with the COnsortium for Small scale MOdeling weather prediction model in CLimate Mode (COSMO-CLM) and provide the first thorough evaluation of this dynamical regional climate model over South America. We focus our analysis on the sensitivity of simulated tropical precipitation to the parameterizations of subgrid-scale cumuliform and stratiform clouds. It is shown that COSMO-CLM is strongly sensitive to both cloud parameterizations over tropical land. Using nondefault cumulus and stratus parameterization schemes we are able to considerably reduce long-standing precipitation and radiation biases that have plagued COSMO-CLM across tropical domains. In the second part we introduce new performance metrics for climate model evaluation with respect to spatial covariabilities. In essence, these metrics consist of dissimilarity measures for climate networks constructed from simulations and observations. We develop both local and global dissimilarity measures to facilitate the depiction of local dissimilarities in the form of bias maps as well as the aggregation of those local to global dissimilarities for the purposes of climate model intercomparison and ranking. The new measures are then applied for a comparative evaluation of regional climate simulations with COSMO-CLM and the STatistical Analogue Resampling Scheme (STARS) over South America. We compare model rankings obtained with our new performance metrics to those obtained with conventional root-mean-square errors of climatological mean values and variances, and analyze how these rankings depend on season, variable, reference data set, and climate network type.

Menschliche Systeme werden seit einiger Zeit modelliert und analysiert auf der Basis der Theorie komplexer Netzwerke. Dies erlaubt es quantitativ zu untersuchen, welche strukturellen und zeitlichen Merkmale eines Systems Ausbreitungsprozesse beeinflussen, z.B. von Informationen oder von Infektionskrankheiten. Im ersten Teil der Arbeit wird untersucht, wie eine modular-hierarchische Struktur von statischen Netzwerken eine schnelle Verbreitung von Signalen ermöglicht. Es werden neue Heuristiken entwickelt um die Random-Walk-Observablen “First Passage Time” und “Cover Time” auf lokal geclusterten Netzwerken zu ermitteln. Vergleiche mit der Approximation eines gemittelten Mediums zeigen, dass das Auftreten der beobachteten Minima der Observablen ein reiner Netzwerkeffekt ist. Es wird weiterhin dargelegt, dass nicht alle modular-hierarchischen Netzwerkmodelle dieses Phänomen aufweisen. Im zweiten Teil werden zeitlich veränderliche face-to-face Kontaktnetzwerke auf ihre Anfälligkeit für Infektionskrankheiten untersucht. Mehrere Studien belegen, dass Menschen vornehmlich Zeit in Isolation oder kleinen, stark verbundenen Gruppen verbringen, und dass ihre Kontaktaktivität einem zirkadianen Rhythmus folgt. Inwieweit diese beiden Merkmale die Ausbreitung von Krankheiten beeinflussen, ist noch unklar. Basierend auf einem neuen Modell wird erstmals gezeigt, dass zirkadian variierende Netzwerke Trajektorien folgen in einem Zustandsraum mit einer strukturellen und einer zeitlichen Dimension. Weiterhin wird dargelegt, dass mit zunehmender Annäherung der zeitlichen Dimension von System und Krankheit die systemische Infektionsanfälligkeit sinkt. Dies steht in direktem Widerspruch zu Ergebnissen anderer Studien, die eine zunehmende Anfälligkeit vorhersagen, eine Diskrepanz, die auf die Ungültigkeit einer weit verbreiteten Approximation zurückzuführen ist. Die hier vorgestellten Ergebnisse implizieren, dass auf dem Gebiet die Entwicklung neuer theoretischer Methoden notwendig ist. / Human systems have been modeled and analyzed on the basis of complex networks theory in recent time. This abstraction allows for thorough quantitative analyses to investigate which structural and temporal features of a system influence the evolution of spreading processes, such as the passage of information or of infectious diseases. The first part of this work investigates how the ubiquitous modular hierarchical structure of static real-world networks allows for fast delivery of messages. New heuristics are developed to evaluate random walk mean first passage times and cover times on locally clustered networks. A comparison to average medium approximations shows that the emergence of these minima are pure network phenomena. It is further found that not all modular hierarchical network models provide optimal message delivery structure. In the second part, temporally varying face-to-face contact networks are investigated for their susceptibility to infection. Several studies have shown that people tend to spend time in small, densely-connected groups or in isolation, and that their connection behavior follows a circadian rhythm. To what extent both of these features influence the spread of diseases is as yet unclear. Therefore, a new temporal network model is devised here. Based on this model, circadially varying networks can for the first time be interpreted as following trajectories through a newly defined systemic state space. It is further revealed that in many temporally varying networks the system becomes less susceptible to infection when the time-scale of the disease approaches the time-scale of the network variation. This is in direct conflict with findings of other studies that predict increasing susceptibility of temporal networks, a discrepancy which is attributed to the invalidity of a widely applied approximation. The results presented here imply that new theoretical advances are necessary to study the spread of diseases in temporally varying networks.