CAREER: Network Geometry for Analyzing Complex Dynamical Systems

在现代科技世界，我们越来越依赖于可靠性、稳健性、服务质量和及时性的超大互联动态系统，包括电力分配、运输和通信系统。在过去的二十年里，我们见证了信息的急剧增长，其中对此类系统的分析总是呈现出具有挑战性的大数据复杂性问题。例如，在传递资源和信息时，一个关键要求是能够适应和重新配置以响应结构和动态变化，同时避免中断服务。为此，我们建议发展网络功能与相应图的某些拓扑和几何性质之间的基本关系。这个项目的主题愿景是基于最近发现的事实，即曲率的几何概念（或物体如何偏离平面）与系统功能稳健性或其适应动态变化的能力呈正相关。虽然开发的理论和工具将适用于广泛的动态系统，但我们将重点关注癌症生物学和人为分布式恶意软件。这项研究的成果将通过K-12外展、出版物、开源软件形式的教程和开发的研讨会来提高对拟议研究的工程意识，从而传播开来。具体来说，由于数据（网络）科学的兴起，对工程师的需求不断增加，特别是在美国退伍军人可能发挥不可或缺作用的国家安全领域。因此，我们将努力直接让退伍军人参与拟议研究的各个方面，同时也为更好的大学准备提供兽医相关的指导。也就是说，为了实现这些目标，提出的研究计划旨在研究离散几何、最佳质量传递、熵和控制之间的密切联系，以制定量化和预测不同尺度复杂网络动态特性的新方法。系统属性包括但不限于健壮性、异构性和拥塞。因此，本研究填补了通过几何和控制的融合研究来研究网络功能的需求，这依赖于具有深远物理（统计力学）和信息论意义的几个观察。具体来说，通过在图上放置概率结构，相对于现有方法的优势在于图上的概率空间比单独的底层离散空间具有更好的性质。然后，相关的概率度量可以被赋予黎曼结构，从而使测地线（最短距离）路径随之而来，并且沿着路径的熵的凹凸性反映了图的几何性质。有了这个基础，我们就可以开始开发基于最优和随机控制的强大几何网络工具。总的来说，由于网络的几何和控制仍处于起步阶段，该计划解决了日益增长的需求，即开发这些领域作为进一步理解复杂系统的工具。

In modern technological world, we increasingly depend upon the reliability, robustness, quality of service and timeliness of exceedingly large interconnected dynamical systems including those of power distribution, transportation, and communication. Over the past twenty years, we have witness a dramatic rise of information in which the analysis of such systems invariably present challenging big data complexity issues. For example, in transferring resources and information, a key requirement is the ability to adapt and reconfigure in response to structural and dynamic changes while avoiding disruption of service. To this end, we propose to develop fundamental relationships between network functionality and certain topological and geometric properties of the corresponding graph. The thematic vision for this program is based on the recently discovered fact that the geometric notion of curvature (or how objects deviate from being flat) is positively correlated with a systems functional robustness or its ability to adapt to dynamic changes. While the developed theory and tools will be applicable to a broad set of dynamical systems, we will focus on cancer biology and man-made distributed malware. The accomplishments of this research will be disseminated through raising engineering awareness of the proposed research through K-12 outreach, publications, tutorials in the form of open-source software, and developed workshops. Specifically, given the rise of data (network) science, there is an increasing demand for engineers especially in national security areas where U.S. Veterans may serve an integral role. As such, we will work to directly engage veterans to engage in aspects of the proposed research while also providing vet dependent mentoring for better university preparation.This said, to accomplish these goals, the proposed research program aims to study the intimate connections between discrete geometry, optimal mass transport, entropy, and control to formulate new approaches to quantify and predict dynamical properties of complex networks at varying scales. System properties include, but not limited to, robustness, heterogeneity, and congestion. As such, this research fills a need by investigating network functionality through the confluent study of geometry and control that relies upon several observations with far reaching physical (statistical mechanics) and information theoretic significance. Specifically, by placing a probability structure on a graph, the advantage over existing methods is that the space of probabilities on graphs has much nicer properties than the underlying discrete space alone. The associated probability measures can then be endowed with a Riemannian structure whereby geodesic (shortest distance) paths ensue and convexity properties of the entropy along paths reflect on geometric qualities of the graph. Given this foundation, we can then begin to develop powerful geometric network tools rooted in optimal and stochastic control. Altogether, as geometry and control of networks is still in its infancy, this program addresses the increasing need to develop such areas as a further tool to understanding complex systems.