CAREER: Network Centrality and Its Applications in Detection, Dynamics, and Load Balancing

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。该项目旨在发展对网络随时间演变的数学理解，以及发生在网络上的过程。关键思想是识别带有网络过去足迹的网络属性，并利用它们从当前配置重建网络的早期阶段。这可以用来检测，例如，谣言传播的起源，受欢迎的个人及其在社交网络中的影响力，或疾病爆发的来源。这一结果有望在国家安全和公共卫生方面有重大应用。另一个重要的研究方向是系统地了解大型网络中的局部相互作用如何影响其全局几何形状。这种知识可用于通过协作的本地交互来提高服务器网络的整体效率，在改进机场安全、超市配送系统以及疫苗和其他药品配送的路由方案方面具有潜在的应用前景。该项目旨在开发一个强大的工具箱，涵盖科学和工程的许多学科。该项目涉及重要的教育活动和研究与教育的整合。这些活动旨在通过对本科生和研究生的长期研究和职业指导，培养多样化的STEM劳动力。这项研究的课程材料和评论论文将免费提供给学生和研究人员。中心性驱动网络连接了动态随机网络和强化过程两个领域。该项目的第一部分研究了中心性驱动的网络，其中传入的顶点与一个或多个现有顶点的关联概率与它们的中心性得分成正比。像PageRank这样的中心性度量，它插入了全局和局部网络的几何形状，将被用于研究网络考古问题。还将分析由PageRank和意见动态驱动的新型动态网络。项目的第二部分探讨了网络的中心驱动路由策略，其中每个顶点都有一个具有单位服务速率的服务器。到达指定到达节点的顶点子集的作业会短视地探索网络，寻找不太忙的服务器，以最小化它们的等待时间。到达节点的最优布局将使用新的负载中心性概念进行分析。基于Dirichlet能量泛函的扩散限制和负载平衡将用于研究大流量系统。研究中使用的技术将依赖于离散时间和连续时间动力学之间的微妙相互作用，以超越特定模型的计算。作为副产品，该项目旨在提供分支过程极限和收敛速度的新特征。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project aims to develop a mathematical understanding of networks that evolve over time and the processes that take place on them. The key idea is to identify network attributes that carry a footprint of a network's past and to exploit them in reconstructing the early stages of a network from its current configuration. This can be used to detect, for example, the origin of a rumor spread, popular individuals and their influence in a social network, or a source of a disease outbreak. The results are expected to have significant applications in national security and public health. Another key research direction is to develop a systematic understanding of how local interactions in a large network influence its global geometry. This knowledge can be used to increase the overall efficiency of a network of servers through cooperative local interactions, with potential applications in improving routing schemes for airport security, supermarket distribution systems, and distribution of vaccines and other medicines. The project aims to develop a robust toolbox across many disciplines in science and engineering. The project involves significant educational activities and integration of research and education. The activities aim to prepare a diverse STEM workforce through long-term research and career mentoring for undergraduate and graduate students. Course material and review papers emerging from this research will be freely available online to students and researchers. Centrality driven networks connect the fields of dynamic random networks and reinforced processes. The first part of the project investigates centrality driven networks, in which incoming vertices attach to one or more existing vertices with probability proportional to their centrality scores. Centrality measures like PageRank, which interpolate the global and local network geometries, will be used to study network archaeology questions. Novel dynamic networks driven by PageRank and opinion dynamics will also be analyzed. The second part of the project explores centrality-driven routing policies for networks where each vertex has a server with a unit service rate. Jobs arriving at a subset of vertices designated arrival nodes myopically explore the network in search of less busy servers to minimize their waiting time. Optimal placement of the arrival nodes will be analyzed using a new notion of load centrality. Diffusion limits and load balancing based on Dirichlet energy functionals will be used to study systems in heavy traffic. The techniques used in the research will rely on a delicate interplay between discrete time and continuous time dynamics to move beyond model specific computations. As a byproduct, the project aims to provide new characterizations of branching process limits and rates of convergence.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.