Complex systems can often be modelled as networks, in which their basic units are represented by abstract nodes and the interactions among them by abstract links. This network of interactions is the key to understanding emergent collective phenomena in such systems. In most cases, it is an adaptive network, which is defined by a feedback loop between the local dynamics of the individual units and the dynamical changes of the network structure itself. This feedback loop gives rise to many novel phenomena. Adaptive networks are a promising concept for the investigation of collective phenomena in different systems. However, they also present a challenge to existing modelling approaches and analytical descriptions due to the tight coupling between local and topological degrees of freedom. In this thesis, I present a simple rule-based framework for the investigation of adaptive networks, using which a wide range of collective phenomena can be modelled and analysed from a common perspective. In this framework, a microscopic model is defined by the local interaction rules of small network motifs, which can be implemented in stochastic simulations straightforwardly. Moreover, an approximate emergent-level description in terms of macroscopic variables can be derived from the microscopic rules, which we use to analyse the system\'s collective and long-term behaviour by applying tools from dynamical systems theory. We discuss three adaptive-network models for different collective phenomena within our common framework. First, we propose a novel approach to collective motion in insect swarms, in which we consider the insects\' adaptive interaction network instead of explicitly tracking their positions and velocities. We capture the experimentally observed onset of collective motion qualitatively in terms of a bifurcation in this non-spatial model. We find that three-body interactions are an essential ingredient for collective motion to emerge. Moreover, we show what minimal microscopic interaction rules determine whether the transition to collective motion is continuous or discontinuous. Second, we consider a model of opinion formation in groups of individuals, where we focus on the effect of directed links in adaptive networks. Extending the adaptive voter model to directed networks, we find a novel fragmentation mechanism, by which the network breaks into distinct components of opposing agents. This fragmentation is mediated by the formation of self-stabilizing structures in the network, which do not occur in the undirected case. We find that they are related to degree correlations stemming from the interplay of link directionality and adaptive topological change. Third, we discuss a model for the evolution of cooperation among self-interested agents, in which the adaptive nature of their interaction network gives rise to a novel dynamical mechanism promoting cooperation. We show that even full cooperation can be achieved asymptotically if the networks\' adaptive response to the agents\' dynamics is sufficiently fast.

Complex systems can often be modelled as networks, in which their basic units are represented by abstract nodes and the interactions among them by abstract links. This network of interactions is the key to understanding emergent collective phenomena in such systems. In most cases, it is an adaptive network, which is defined by a feedback loop between the local dynamics of the individual units and the dynamical changes of the network structure itself. This feedback loop gives rise to many novel phenomena. Adaptive networks are a promising concept for the investigation of collective phenomena in different systems. However, they also present a challenge to existing modelling approaches and analytical descriptions due to the tight coupling between local and topological degrees of freedom. In this thesis, I present a simple rule-based framework for the investigation of adaptive networks, using which a wide range of collective phenomena can be modelled and analysed from a common perspective. In this framework, a microscopic model is defined by the local interaction rules of small network motifs, which can be implemented in stochastic simulations straightforwardly. Moreover, an approximate emergent-level description in terms of macroscopic variables can be derived from the microscopic rules, which we use to analyse the system\'s collective and long-term behaviour by applying tools from dynamical systems theory. We discuss three adaptive-network models for different collective phenomena within our common framework. First, we propose a novel approach to collective motion in insect swarms, in which we consider the insects\' adaptive interaction network instead of explicitly tracking their positions and velocities. We capture the experimentally observed onset of collective motion qualitatively in terms of a bifurcation in this non-spatial model. We find that three-body interactions are an essential ingredient for collective motion to emerge. Moreover, we show what minimal microscopic interaction rules determine whether the transition to collective motion is continuous or discontinuous. Second, we consider a model of opinion formation in groups of individuals, where we focus on the effect of directed links in adaptive networks. Extending the adaptive voter model to directed networks, we find a novel fragmentation mechanism, by which the network breaks into distinct components of opposing agents. This fragmentation is mediated by the formation of self-stabilizing structures in the network, which do not occur in the undirected case. We find that they are related to degree correlations stemming from the interplay of link directionality and adaptive topological change. Third, we discuss a model for the evolution of cooperation among self-interested agents, in which the adaptive nature of their interaction network gives rise to a novel dynamical mechanism promoting cooperation. We show that even full cooperation can be achieved asymptotically if the networks\' adaptive response to the agents\' dynamics is sufficiently fast.

Active matter describes systems that convert energy from their environment into directed motion. Therefore, these systems are in intrinsic nonequilibrium, unlike their passive counterparts. From a theoretical point of view, such active systems have been modeled by agent-based models, as well as hydrodynamic approaches, which allowed for the investigation of a wide range of observed collective phenomena characterizing active matter. In this thesis we develop a microscopic field-theoretical approach to describe generic properties of active systems. This description combines the phase field crystal model with a polar order parameter and a self-propulsion term. First, we validate this approach by reproducing results obtained with corresponding agent-based models, such as binary collisions, collective migration and vortex formation. We also perform a direct comparison between our model and a microscopic phase field description of active matter. Next, we use this continuum approach to simulate some larger active systems and to analyze the coarsening process in active crystals, as well as the mechanisms leading to mobile clusters. We show the generality of our approach by extending it to binary mixtures of interacting active and passive particles. Also in this case, we first validate the model by reproducing known results, such as enhanced crystallization via active doping and the suppression of collective migration in an active bath in the presence of fixed obstacles. Interestingly, for the regime of mobile passive particles in an active bath a laning state is found, which is characterized by an alignment of the active particles that is globally nematic, but polar within each lane. This state represents a theoretical prediction feasible to be validated experimentally. Finally, we explore the field of topological active matter. We develop an agent-based model to describe self-propelled particles on curved surfaces and study the complex spatiotemporal patterns that emerge. / Aktive Materie beschreibt Systeme, die Energie aus ihrer Umgebung in gerichtete bewegung umwandeln. Im Gegensatz zur passiven Materie befinden sich diese Systeme nie im physikalischen Gleichgewicht und offenbaren dadurch interessante physikalische Phänomene. Vom theoretischen Standpunkt her wurde aktive Materie bereits simuliert, typischerweise durch agenten-basierte Modelle oder hydrodynamische Ansätze, die es ermöglichen eine Vielzahl der auftretenden kollektiven Bewegungsprinzipien zu untersuchen. In dieser Doktorarbeit entwickeln wir einen mikroskopischen Kontinuumsansatz um die generischen Eigenschaften von aktiven Systemen zu untersuchen. Unsere Beschreibung kombiniert das Phasenfeld-Kristall Modell mit einem polaren Ordnungsparameter und einem Antriebsterm. Zuerst validieren wir den Ansatz durch Reproduktion bekannter Ergebnisse agenten-basierter Modelle, wie binäre Kollisionen, kollektive Bewegung und Wirbelformationen. Des Weiteren führen wir einen direkten Vergleich zwischen unserem Modell und einer mikroskopischen Phasenfeldbeschreibung aktiver Materie durch. Danach nutzen wir den kontinuierlichen Ansatz um große aktive Systeme zu simulieren und analysieren den Vergröberungsprozess in aktiven Kristallen und Mechanismen der mobilen Aggregatbildung. Wir illustrieren die Allgemeingültigkeit unseres Simulationsansatzes durch die Erweiterung auf binäre Systeme, in denen sowohl aktive als auch passive Partikel enthalten sind. Auch in diesem Fall validieren wir das Modell durch Vergleiche mit bekannten Resultaten, wie zum Beispiel die verstärkte Kristallisation durch aktives Doping oder die Unterdrückung kollektiver Bewegung durch die Einführung von Hindernissen in einem aktiven Bad. Interessanterweise finden wir bei der Präsenz mobiler passiver Partikel in einem aktiven Bad einen Fahrspur-Zustand, in welchem die aktiven Partikel nematische Fahrspuren bilden und sich nur jeweils innerhalb einer Fahrspur nematisch polar anordnen. Dieser bisher unbekannte Zustand stellt eine theoretische Vorhersage dar, die experimentell geprüft werden kann. Schließlich begeben wir uns auf das Gebiet der topologischen aktiven Materie. Wir entwickeln ein agenten-basiertes Modell um selbst-angetriebene Partikel auf gekrümmten Oberflächen zu beschreiben und untersuchen die dabei auftretenden zeitlich und räumlich komplexen Muster.%, die dabei auftreten.

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