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对象，这些部分可以表示为图，并且可以使用图论中的方法进行数学描述。诸如社会网络、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性质，已经开发了各种方法，其中许多方法本质上是计算的，因此需要花费时间和计算资源来使用。此外，目前用于量化复杂网络的方法很难描述各种现象，因此有机会开发新的数学方法和度量来更好地评估复杂网络的性质。本项目旨在开发新的数学方法来描述复杂网络的“稳健性”，即复杂网络承受其组成部分移除的能力。目前用于评估健壮性的方法要么依赖于慷慨的假设而在范围上受到限制，要么依赖于耗费时间和计算资源的算法。这个项目将调查是否可以利用信息论领域的数学来开发新的量化稳健性的方法。信息论关注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健壮性，可以测量网络的健壮性，从而只需要很少的假设和最少的资源消耗。这在现实世界中的一个可能应用是，在流行病期间有效地确定相互作用网络的健壮性，从而允许制定更好的隔离计划，以有效地隔离网络的组件并阻止疾病的传播。此外，以这种方式量化健壮性将允许更好地理解是什么因素使网络健壮，因为目前测量健壮性的方法不能识别网络中确保健壮性的结构因素。如果健壮性背后的结构因素是量化和已知的，这将使人们能够根据期望的健壮性水平来设计网络，这可能在诸如建设基础设施和恢复生态系统等领域具有现实应用。为了将信息论与复杂网络相结合，我将使用计算和数学方法，研究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，这些工具和算法可以量化适用于任何给定复杂网络的健壮性和相关网络结构。由于网络的健壮性还取决于如何移除其组成部分以及以何种顺序移除，因此我的目标也是开发不同的方法来衡量不同的健壮性。

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.