The world is full of complex systems, with many interconnected parts interacting in some way, and the whole system cannot be understood simply by looking at each part individually. Instead, it is necessary to consider the system as a whole, which may exhibit emergent behaviour as a result of the many interconnections. Network theory is a very powerful tool for analysing complex systems, and has been applied to a wide range of phenomena with great success.This work is concerned with spatial networks, which are the ones that are naturally embedded in physical space in some way. For example, wireless sensor networks in a geographical region such as a city where the flow of information is essential, or plant populations in a crop field where it is important to understand and try to limit the spread of diseases. Another example is the rocks found deep underground in the Gulf of Mexico that are highly fractured; these fractures can clearly be thought of as spatially embedded networks through which fluids such as oil and gas can flow. There is great commercial interest in efficiently extracting the oil and gas from the rocks, and there are also serious efforts to use the depleted rocks as a means of carbon sequestration to help combat the problem of greenhouse gases.Therefore, it is clearly important to understand the nature of the structure and dynamical properties of these real-world networks.It is intuitive that the topological and dynamical properties of spatial networks depend on the shape of the space in which they are embedded. In this work we discuss the generalisation of two spatially-defined random graph models to consider nodes located in a unit rectangle. We generalise the random geometric graph (RGG) to the random rectangular graph (RRG), and the relative neighbourhood graph (RNG) to the rectangular relative neighbourhood graph (RRNG).We found an analytic expression for the expected value of the average node degree of RRGs, as well as useful bounds for the diameter, the average path length, and the algebraic connectivity, and approximations to the degree distribution, connectivity, and clustering coefficient. For the RRNGs we found an approximation to the average node degree and a bound on the diameter and algebraic connectivity.Using this generalisation, we examine the behaviour of diffusion in RRGs and find that increasing the elongation causes diffusive particles to spread more slowly. We also discuss some results relating to epidemics in crop fields, where we show that elongating the field makes it more difficult for a disease to become epidemic. Finally, we find that the relative neighbourhood graphs work well to mimic the properties of the rock fracture networks compared to other null models, and we are able to optimise the value of the elongation in the model for each fracture network.
|Date of Award||25 May 2018|
- University Of Strathclyde
|Sponsors||EPSRC (Engineering and Physical Sciences Research Council)|
|Supervisor||Ernesto Estrada (Supervisor) & Philip Knight (Supervisor)|