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

复杂网络是由许多较小的相互作用的部分组成的大型对象，这些部分可以表示为图，并且可以使用图论的方法进行数学描述。诸如社交网络、交通网络、生态系统和神经网络之类的系统可以被归类为复杂网络，因此可以用图来表示。为了研究和量化复杂网络的特性，已经开发了各种方法，其中许多方法本质上是计算的，因此需要时间和计算资源来使用。此外，目前的复杂网络量化方法难以描述各种现象，因此，开发更好地评估复杂网络特性的新数学方法和测量方法是有余地的。本课题旨在开发描述复杂网络的“鲁棒性”（robustness）的新数学方法，即复杂网络能够承受其组成部分被移除的能力。目前用于评估鲁棒性的方法要么由于依赖于慷慨的假设而在范围上受到限制，要么依赖于在时间和计算资源上昂贵的算法。本项目将研究是否可以使用信息论领域的数学来开发量化鲁棒性的新方法。信息论关注的是有效通信背后的数学，因此复杂网络和信息论可以通过将复杂网络建模为通信网络来结合，其中通信发生在网络的组成部分之间。通过使用信息论方法来量化复杂网络中的鲁棒性，可以测量网络的鲁棒性，使得需要非常少的假设并且花费最少的资源。这种方法在真实的世界中的一个可能的应用是，在流行病期间，有效地确定交互网络的鲁棒性，从而可以制定更好的预防计划，有效地隔离网络的组成部分并阻止疾病的传播。此外，以这种方式量化鲁棒性将有助于更好地理解是什么因素使网络鲁棒，因为当前用于测量鲁棒性的方法不能识别网络中确保鲁棒性的结构因素。如果鲁棒性背后的结构因素是量化的和已知的，这将允许一个设计网络的基础上所需的鲁棒性水平，这可能有真实的世界的应用领域，如建设基础设施和恢复生态系统。为了联合收割机信息论与复杂网络，我将使用计算和数学方法，检查人工生成的网络和真实的网络。该项目的最终目标是开发几种数学工具和算法，可以量化适用于任何给定复杂网络的鲁棒性和相关网络结构。由于网络的鲁棒性也取决于其组成部分是如何被移除的，以及以什么顺序被移除，我的目标也是开发不同的方法来测量不同类型的鲁棒性。

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.