DynSyst_Special_Topics: Time-varying dynamical networks: theory and applications

本文的研究目的是研究网络结构对时变耦合网络的动力学和信息处理能力的影响。该建议侧重于一个很大程度上未开发的领域，即由动态系统组成的网络的数学分析和建模，其耦合结构随着时间的推移而变化，时间尺度从快到慢。具有开关连接的动态网络的一般严格理论的发展及其在三种特定类型的工程、生态和神经网络中的应用构成了提案的研究部分。第一类是一类随机开关工程网络，如电源变换器、通信和信息处理网络等具有快速开关连接的网络。第二种类型的网络是相互关联的生态元种群，在现实的假设下，斑块之间的迁移是零星的，并且由于罕见和短期的气象条件。第三种网络是具有时变连接的神经网络。在此资助下的研究整合了应用数学和工程不同学科领域的知识，包括稳定性和分岔理论、图论、信息论、遍历理论和平均。在许多生物、生态和工程网络中，耦合强度和连接拓扑可以随时间变化。鉴于时变网络在信息处理中的应用及其在同步和模式形成中的作用，本研究采用严格的数学技术来研究时变网络结构与整体网络动力学之间的相互作用。特别是，该项目研究了快速交换连接的增加如何提高具有静态链路的网络的性能。该项目还涉及网络脆弱性问题。具体来说，当个别系统或链路的一部分被破坏时，网络处理信息的能力如何，动态系统的时变网络比静态结构的网络有什么优势。这些结果将有助于更好地理解现实网络在自然界中的功能，并为工程中设计信息处理网络提供一般指导。本研究计划透过发展动态网路领域的跨学科研究生课程，并指导本科生研究计画，将研究活动整合到本科与研究生课程中。该补助金直接支持一名研究生和两名本科生，包括来自代表性不足群体的学生。

The research objective of this proposal is to investigate the influence of network structure on the dynamics and the information processing capabilities of networks with a time-varying coupling. This proposal focuses on a largely unexplored area, namely, mathematical analysis and modeling of networks that are composed of dynamical systems, whose coupling structure evolves over time, on a time scale that ranges from fast to slow. The development of the general rigorous theory of dynamical networks with on-off connections and its application to three specific types of engineering, ecological, and neuronal networks constitute the research component of the proposal. The first type is a class of stochastically switched engineering networks, such as power converters, communication and information processing networks with fast on-off connections. The second type of networks is interconnected ecological metapopulations, under the realistic assumption that the migration among the patches is sporadic and due to rare and short term meteorological conditions. The third type of networks is neuronal networks with time-varying connections. Research funded under this grant integrates knowledge from different disciplinary areas in applied mathematics and engineering, including stability and bifurcation theory, graph theory, information theory, ergodic theory and averaging.In many biological, ecological and engineering networks the coupling strength and the connection topology can vary in time. The proposed research applies rigorous mathematical techniques to investigate the interplay between time-varying network structure and overall network dynamics, in view of the time-varying network's use in information processing and its role in synchronization and pattern formation. In particular, the project investigates how the addition of fast switching connections can enhance the performance of networks with static links. The project also addresses the question of vulnerability of networks. Specifically, how well can networks process information when part of the individual systems or links are destroyed, and what advantages do time-varying networks of dynamical systems have over networks with static structure. The results will help to better understand the functioning of realistic networks in nature, as well as give general guidelines for designing information processing networks in engineering. The proposed research integrates research activities into the undergraduate and graduate curriculum through developing an interdisciplinary graduate course in the field of dynamical networks and mentoring undergraduate research projects. The grant directly supports one graduate and two undergraduate students, including those from under-represented groups.