Multistability, synchronization, and self-organization in networks of nonlinear systems with changing graph topologies

Complex network structures appear in myriad contexts. From social networks to computer networks, protein and transport networks, and neuronal networks of the mammalian brain. Furthermore, many of these networks share common structural properties. Are there general underlying mechanisms for the emergence of certain complex network structures? One such shared principle is the mutual relationship between structure and function in self-organising networks. Understanding their emergence can be decomposed into two simpler problems: (1) How does structure effect dynamics? (2) How do dynamics shape the structure? Concepts of nonlinear systems theory provide a tool-set for stability analysis of dynamics on a network. Here, stability analysis is applied to the problem of how a small change in network structure effects the dynamics. Two connectivity configurations are considered; the directed chain and the directed cycle, distinguished by a single edge. Their linear stability is first analysed, followed by the stability of interconnected nonlinear oscillators. Stability analysis reveals radical changes in the patterns of dynamics; while the directed chain possess only one stable solution (synchronization), the directed cycle possesses multistabiliy (synchronization and rotating waves). This capacity for multistability is realised by the extremal properties of the directed cycle; the slow decal of oscillations in the coupling dynamics resonates with the dynamics of the individual oscillators. This result is generalised to networks that contain modular structures and heterogeneous dynamics. For applications of evolving network structures, systems theory is limited. Computational modelling, on the other hand, provides an effcacious alternative. A well-established driving mechanism for network structure evolution is adaptive rewiring; adaptation of structure to function. Computational modelling reveals a synergy between spatial organisation and adaptive rewiring. Emergence of modular small-world network structures are more pronounced, and evolution more robust, than in models without spatial organisation. However, studies employing adaptive rewiring have been frustrated by the need to explicitly specify dynamics. To address this, explicit dynamics are replaced by an abstract representation of network diffusion (information transfer or traffic flow): shortcuts are created where traffic flow is intense, while annihilating underused connections - like pedestrians define walkways in parks. The resulting networks are a family of small-world structures; networks may be modular or centralised. Moreover, at the critical point of phase transition of network structure, hierarchical structures emerge - like those found in the brain. This thesis therefore serves to help bridge the gap between dynamical systems theory and computational modelling in the field of complex network theory; the importance of connectivity on dynamics, as detailed in systems theory, is captured using graph diffusion, and applied in the context of computational modelling. This successfully reduces the highly complex problem of complex network emergence to a much simpler one, namely, patterns of connectivity. In doing so, the generality of this machinery provides a more lucid understanding for the self-organisation of complex network structures across a broad range of contexts.