Markov chain theory and its applications

我对我的研究计划的长期愿景是通过在实际应用的驱动下进行基础理论研究，扩大马尔可夫链理论在工业中的应用。特别是，我将进行原创性研究，以开发和完善具有广泛应用的指标，算法和创新计算工具的工具包。马尔可夫链是一种数学模型，已被用于描述无数类型的动力系统，从城市道路网络中的交通行为，到分子动力学及其在药物设计中的作用。马尔可夫链用于表示可以占据许多状态中的任何一个的系统，系统的行为是通过一个概率转移矩阵来确定的，它描述了系统在状态之间的随机运动。给定一个马尔可夫链，有许多指标决定了它的“良好连接”程度。了解每个邦的“中心”程度也是可取的。例如：在传染病的接触网络中，连通性指标决定了图的密集程度（对应于疾病的快速传播），为了控制传播，应该从网络中删除（隔离）最中心的节点以减少连通性。我的项目的一个目标是研究连接性和中心性的度量，这些度量来源于图上随机游走的动态过程。一个关键的连通性指标被称为Kemeny常数，它代表了系统的全球“连通性”；即系统在状态之间移动的平均速度。对凯梅尼常数的考虑是相对较近的，还有很多有待理解的地方。本提案的一个目标是提供对Kemeny常数计算对过渡矩阵误差的敏感程度的理解，以便在这些应用之一的上下文中给出Kemeny常数计算值的置信度度量。我还计划开发更新，更有效的计算凯梅尼常数的方法，而不是目前最先进的方法。最后的目标是考虑有向复杂网络；也就是说，这种关系可能只是单向的网络或数据集（例如，社交网络Twitter包含不一定相互回报的友谊联系）。定向网络历来难以分析；例如，识别和分析集群是一个非常困难的问题。基于我过去在该领域的工作，我计划开发和实施马尔可夫链理论的新技术来解决这个问题。作为该计划的一部分，将培训4名硕士生、1名博士生和几名理学士。他们将从事理论和应用研究，这将有助于推动马尔可夫链理论在广泛的工业环境中的理论理解和适用性的长期目标，如车辆交通模型，数据科学和社会网络。

My long-term vision for my research program is to expand the use of Markov chain theory in industry by performing fundamental theoretical research that is driven by practical applications. In particular, I will undertake original research to develop and refine a toolkit of metrics, algorithms, and innovative computational tools with a wide range of applications. A Markov chain is a type of mathematical model which has been used to describe countless types of dynamical systems, from traffic behaviour in an urban road network, to molecular dynamics and their role in drug design. A Markov chain is used to represent systems which can occupy any of a number of states, and the behaviour of the system is determined via a probability transition matrix, which describes the random movement of the system between the states. Given a Markov chain, there are many metrics which determine how 'well-connected' it is. It is also desirable to know how 'central' each state is. For example: in a contact network for a communicable disease, connectivity metrics determine how dense the graph is (corresponding to fast spread of the disease) and to control the spread, the most central nodes should be removed (quarantined) from the network to reduce connectivity. One objective of my program is to research measures of connectivity and centrality that derive from the dynamical process of a random walk on a graph. One key connectivity metric is called Kemeny's constant, and it represents the global 'connectedness' of the system; i.e. how quickly the system moves between states on average. The consideration of Kemeny's constant is relatively recent, and much remains to be understood. One objective of this proposal is to provide an understanding of how sensitive the calculation of Kemeny's constant is to errors in the transition matrix, in order to give a measure of confidence in the computed value of Kemeny's constant in the context of one of these applications. I also plan to develop newer, more efficient methods of computing Kemeny's constant than the current state-of-the-art. A final objective is to consider directed complex networks; that is, networks or datasets in which the relationship may be one-way only (for example, the social network Twitter involves friendship connections which are not necessarily reciprocated). Directed networks have historically been difficult to analyse; for example, identifying and analysing clusters is a very hard problem. Based on my past work in the area, I plan to develop and implement novel techniques from Markov chain theory to solve this problem. As part of this proposal, 4 MSc students, 1 PhD student, and several BSc students will be trained. They will engage in theoretical and applied research that will contribute towards the long-term goal of advancing the theoretical understanding and applicability of Markov chain theory in a wide range of industrial settings, such as vehicle traffic models, data science, and social networks.