Understanding, Modelling and Improving the Robustness of Complex Networks for Varying Degrees of Structural Information

A complex network is a large object made up of many smaller, interacting parts which may be represented as a graph, and may be described mathematically using methods from graph theory. Systems such as social networks, traffic networks, ecosystems and neural networks may be classed as complex networks and can therefore be represented as graphs. In order to investigate and quantify the properties of complex networks, various methods have been developed, many of which are computational in nature and therefore take time and computational resources to use. Furthermore, current methods for quantifying complex networks struggle to describe various phenomena, and so there is scope to develop new mathematical methods and measurements that are better at evaluating the properties of complex network.

This project aims to develop new mathematical methods for describing the "robustness" of complex networks, i.e. the ability of complex networks to withstand the removal of their constituent parts. Current methods for evaluating robustness are either limited in scope by relying upon generous assumptions, or they are reliant upon algorithms that are expensive in time and computational resources. This project will investigate whether new methods for quantifying robustness may be developed using mathematics from the field of information theory. Information theory is concerned with the mathematics behind efficient communication, so complex networks and information theory may be combined by modelling a complex network as though it is a communication network, with communication occurring between the component parts of the network.

By using information theoretic methods to quantify robustness in complex networks, it may be possible to measure the robustness of networks such that very few assumptions are required and minimal resources are expended. A possible real world application of this would be, during an epidemic, the efficient determination of how robust an interaction network is, allowing for better quarantining plans to be developed that can effectively isolate the components of the network and stop the spread of disease.

Furthermore, quantifying robustness in this way would allow for a better understanding of what factors make a network robust, as current methods for measuring robustness are unable to identify the structural factors in a network that ensure robustness. If the structural factors behind robustness are quantified and known, this would allow one to design networks based upon a desired level of robustness, which may have real world applications in areas such as constructing infrastructure and restoring ecosystems.

In order to combine information theory with complex networks, I will be using both computational and mathematical methods, examining both artificially generated networks and real networks. The end goal of the project is to develop several mathematical tools and algorithms that can quantify robustness and the relevant network structure that are applicable to any given complex network. Since the robustness of a network is also dependent upon how its component parts are removed and in what order, I also aim to develop different methods for measuring different "types" of robustness.