DMS-EPSRC: The Dynamics and Structure of Multiway Networks

网络科学是一个广泛应用的数学框架，用于交互系统和连接的数据。网络科学的很大一部分力量来自于将复杂的连接分解为基本结构：为实体建模的节点和同时连接两个节点的边。例如，社会关系可以被建模为网络，其中个人是节点，边对应于友谊、家庭或工作关系。交通系统、互联网和基因调控也同样可以通过网络进行建模。这个获奖项目的重点是多路网络的数学发展，它同时对两个以上实体之间的互动进行建模。这项研究将涉及多路网络上新型动力系统的发展和理论分析，包括扩散、随机行走和共识形成，以了解涉及信息或疾病传播的过程。该项目将对经验数据和新的数学模型进行动力学研究，这将反过来导致数据科学和机器学习的新算法。该奖项的大部分资金用于资助一名研究生研究该奖项的主题。图形或网络是复杂系统和关系数据的广泛使用的数学模型。该奖项的研究将促进对多路网络的动力学和结构的理解，这些网络通过直接对多路相互作用进行建模，超越了图形。该方法基于多向网络的超图和单纯复杂模型，并在这些模型的背景下研究随机行走、扩散和共识。这项研究围绕四个方向展开。首先是发展随机网络模型和收集经验数据，以激励和评估新的动力系统。第二部分是关于超图的动力学理论和分析。第三种是基于单纯复形的新的动力学类型，以及它们与超图上的类似动力学有何不同。第四项是基于这些动态为多路网络开发新的数据科学和机器学习算法，如集群和半监督学习。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Network science is a widespread mathematical framework for interacting systems and connected data. Much of the power of network science comes from breaking down complex connectivity into basic structure: nodes that model entities and edges that connect two nodes at a time. For example, social relationships can be modeled as networks where individuals are nodes and edges correspond to friendship, family, or work relationships. Transportation systems, the Internet, and gene regulation can similarly be modeled by networks. This awarded project centers on the mathematical development of multiway networks, which model interactions between more than two entities at a time. The research will involve the development and theoretical analysis of new types of dynamical systems on multiway networks, including diffusions, random walks, and consensus formation to understand processes involving the spread of information or disease. The dynamics will be studied on empirical data and new mathematical models developed in the project, which will in turn lead to new algorithms for data science and machine learning. Much of the funding of this award goes to support a graduate student working on the topics of the award.Graphs or networks are a broadly used mathematical model for complex systems and relational data. The research in this award will advance the understanding of dynamics and structure of multiway networks, which go beyond graphs by directly modeling multiway interactions. The methodology is based on hypergraph and simplicial complex models of multiway networks, and studying random walks, diffusions, and consensus in the context of these models. The research is organized around four directions. The first is the development of random network models and the collection of empirical data for motivating and evaluating new dynamical systems. The second involves the theory and analysis of dynamics on hypergraphs. The third is based on new types of dynamics for simplicial complexes and how these differ from similar dynamics on hypergraphs. The fourth is the development of new data science and machine learning algorithms for multiway networks based on these dynamics, such as clustering and semi-supervised learning.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.