Stochastic Approach to Control of Large Scale Networks

随着生物网络、电力网络和许多社会网络等多种大型网络的出现，对这些网络的控制已成为一个越来越重要的课题。本提案的研究目标是从概率的角度来解决网络控制理论中自然产生的基本问题。特别是，调查人员将解决这样的问题：广播信号控制大型网络的可能性有多大？媒体能影响公众舆论吗？一个领导人能将公众舆论从一个州转移到任何他想要的州吗？研究人员将使用最先进的工具在概率和随机矩阵理论来研究这些基本问题。该建议的一个重点是概率论和控制论之间的协同关系。为了鼓励和帮助其他研究人员，研究人员计划调查工作，将强调特别有用的概率工具和技术，人们需要解决这些类型的问题。基于这种决斗关系，研究者讨论的许多问题都有可能导致随机矩阵理论中有趣的新问题和方向，随机矩阵理论在数学之外有许多应用，包括统计学，数学物理，组合学和理论计算机科学。该提案还有几个教育组成部分，包括支持工程和数学研究生学习，促进少数民族参与高等科学和工程教育。本研究旨在利用最新发展的概率论和随机矩阵理论的工具来解决网络控制理论中的基本问题。具体来说，研究人员计划分析大多数系统（即使是那些非常离散的系统）都是可控的普遍现象。一个具体的激励例子是大型网络中基于拉普拉斯的领导-追随者动态的可控性问题。研究人员计划通过研究底层图或网络是随机的情况来分析这类系统。为了解决这些问题，研究人员计划使用和扩展概率和随机矩阵理论中的许多不同和涉及的技术，包括解析技术（例如解决技术，测度的集中），代数工具（例如线性代数）和概率方法（例如Littlewood-Offord理论）。此外，研究人员还希望提出的工作对数学家有用，因为它将准确地强调其他科学领域存在的问题和应用类型。除了基于拉普拉斯的领导-追随者动力学外，研究者还计划解决由有向图和加权图组成的系统，以及参数随机系统的最小能量控制问题。

With the advent of multiple large-scale networks' such as biological networks, power networks, and many social networks' the control of these networks has become an increasingly important topic. The research objective of this proposal is to address fundamental questions that arise naturally in network control theory from a probabilistic perspective. In particular, the investigators will address questions such as: how likely is it that a broadcast signal can control a large network. Can media shape public opinion. Can a single leader move public opinion from one state to any arbitrary desired state? The investigators will use state-of-the-art tools in probability and random matrix theory to study such fundamental problems. One emphasis of the proposal is the synergistic relationship between probability and control theory. To encourage and assist other researchers, the investigators plan survey works that will emphasis particularly useful probabilistic tools and techniques that one needs to address these types of problems. Based on this duel relationship, many of the problems discussed by the investigators have the potential to lead to interesting new questions and directions in random matrix theory, which has many applications outside of mathematics including statistics, mathematical physics, combinatorics, and theoretical computer science. The proposal also has several educational components including support for graduate study in engineering and mathematics and promoting participation of minorities in higher education in science and engineering. The proposed research aims to address fundamental problems and questions in network control theory using recently developed tools from probability and random matrix theory. Specifically, the investigators plan to analyze the general phenomenon that most systems "even those of a very discrete nature" are controllable. One specific motivating example is the question of controllability for Laplacian-based leader-follower dynamics on a large network. The investigators plan to analyze such systems by studying the case when the underlying graph or network is random. To address such problems, the investigators plan to use and extend a number of diverse and involved techniques from probability and random matrix theory including analytic techniques (e.g. resolvent techniques, concentration of measure), algebraic tools (e.g. linear algebra), and probabilistic methods (e.g. Littlewood-Offord theory). Additionally, the investigators also expect the proposed work to be useful to mathematicians, as it will highlight precisely what types of problems and applications exist in other scientific areas. Besides for the case of Laplacian-based leader-follower dynamics, the investigators also plan to address systems formed from directed and weighted graphs as well as a minimum-energy control problem for systems whose parameters are random.