The global phenomena observed in complex systems are not inherent in the individual constituents but arise from their local interactions. It is often useful to model such systems as networks, which retain these important local interactions while abstracting the detail away. Adaptive networks, in particular, are well suited to the modelling of such systems. The feedback loop between state and topology can give rise to the self-organising patterns observed in complex systems. In the mathematical exploration of such models the central challenge is often to map the problem onto a tractable set of equations. This thesis aims to address this challenge by coupling heterogeneous moment expansions and generating functions in order to model adaptive networks. In the first part of the thesis, we use heterogeneous moment expansions to describe stochastic agent-based models by infinite-dimensional systems of ordinary differential equations (ODEs). We then convert the infinite-dimensional systems of ODEs into low-dimensional systems of partial differential equations (PDEs) using generating functions. We finally solve the PDEs using methods from the literature. When analytic solutions are not possible we provide a method to obtain an accurate approximation to the solution using a Taylor series expansion of the generating function. In the second part of the thesis, we use the methodology of heterogeneous expansions and generating functions to address the challenge of designing networks that self-organise towards target degree distributions. Beginning with a set of processes acting on a network and a target steady-state degree distribution, we investigate the generating function PDEs produced under their action. From this we determine the combination of process rates required to produce such a target distribution. Where the first half of the thesis focuses on modelling real-world systems using adaptive networks, the second half instead looks at reverse engineering such systems in order to produce some desired global phenomenon.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:702113 |
| Date | January 2016 |
| Creators | Silk, Holly |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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