BioND — Dynamics of Biological Networks

Moment Closure Approximations for Contact Processes in Adaptive Networks

Güven Demirel
PhD thesis in Physics, Dresden Technical University, 2013.

Abstract

PhD Thesis

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 de fined 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 di erent contact processes in adaptive networks. First, I explore the e ffect of cyclic dominance in opinion formation. For this, I propose an adaptive network model: the adaptive rock-paper-scissors game. The model displays four di fferent 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 speci fic to the complex phenomena under investigation.

Third, I study the collective motion in mobile animal groups, using the conceptual frame- work 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 con rmed in a fish school experiment, constituting the rst example of direct veri fication 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 nite infectiousness, the model lacks an epidemic threshold, which is a genuine adaptive network e ffect. Diseases with very low infectiousness can thus persist and prevail in growing populations.

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