Random Structures and Dynamics Arising from Questions in Social, Biological, and Physical Sciences

随机结构和随机动力学的分析一直是许多学科感兴趣的话题。研究各种空间和非空间随机图模型、随机空间模型和随机图上发生的随机过程，以及基于网络的算法是这些工作的关键。尽管这些问题在文献中得到了极大的关注，但在许多领域还需要进行严格的分析和基础工作，以获得可靠的结果，并应对即将到来的挑战和真实世界数据的复杂性。这项研究解决了几个理论和一些经验上的挑战，涉及物理学、社会科学和生物科学中出现的广泛类别的问题。将从这项研究中受益的具体研究领域包括确定大脑中神经网络的功能连通性、大流行管理、虫害控制战略和社会互动的演变。这项研究将为不同社会和生物现象的潜在复杂结构和相关协变量的机械基础提供新的见解，这是许多学科研究的中心目标。这一分析将有助于科学家理解生物科学和社会科学中的实验和观测数据，并制定适当的控制策略。对于时间网络的社区检测将增强对多个空间和时间尺度上的大脑功能的理解。拟议的工作还具有巨大的潜力，可以提高我们开发有效和成本效益高的干预战略的能力，以减轻一些最具潜在破坏性和传播速度最快的人、牲畜和植物(包括小麦、大豆、柑橘和啤酒花)的入侵流行病。为了确保结果与科学家相关，该项目的部分内容将与不同领域的科学家合作进行。将在利益攸关方会议和社区外联活动中介绍相关研究。本科生和研究生将参与数学和其他学科交叉的研究活动。有机会学习各种概率模型的理论和经验分析，以及与科学家的互动，将增强他们在数学和各种相关领域之间工作的能力。这个项目的主要目标之一是了解和分析各种空间和非空间随机图模型的不同结构性质和极限行为，以及图上发生的几种随机动力学。随机图模型包括开放渗流簇及相关模型、多层模型和时间网络模型。渗流模型起源于物理学文献，是一种多孔介质的模型。有许多有用的工具和成熟的理论来研究二维晶格上的渗流模型。然而，对于高维晶格，几个关键方面，包括近临界状态和模型在子图中的行为，如扇区，却鲜为人知。该项目将解决其中一些问题。多层网络是各种科学领域中产生的大量数据集的自然模型，包括基因组学、生物医学科学、神经科学、经济学、社会学、生态学、流行病学和技术网络。根据上下文的不同，各种参数概率模型已被用于形成多层、多路和时间网络。为了从数据中估计这些参数，并出于模型选择的目的，了解一些关键泛函的行为(例如，子图计数或聚合邻接矩阵的集中度)是非常重要的。这些泛函将针对各种时态网络模型进行分析，特别是在快照具有某些关联结构的情况下。随机动力学包括在存在附加限制(例如，临时隔离)的情况下感染传播和意见演变的标准模型的一些重要变体。该项目还包括研究各种检测问题的几个理论和经验方面(例如，可检测性阈值的界、有效的检测算法)，其中包括异常结构的检测、从相关网络序列中的社区检测、流行病源头的检测。对上述模型的严格分析将需要开发新的数学技术和研究工具。这些技术和工具将有助于更深入地了解在多个科学学科中出现的更广泛类别的随机结构和动力学。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。

Analysis of random structures and stochastic dynamics has been a topic of interest across many disciplines. Study of various spatial and non-spatial random graph models, stochastic spatial models, and stochastic processes taking place on random graphs, and network-based algorithms are key to these efforts. Although these problems have received significant attention in the literature, rigorous analysis and foundational work is needed in many areas to have reliable results and to cope with upcoming challenges and the complexity of real-world data. This research addresses several theoretical and some empirical challenges involving a wide class of questions arising in physics, social sciences, and biosciences. Specific research areas that would benefit from this research include determining functional connectivity within neural networks in brains, pandemic management, pest control strategies, and the evolution of social interactions. This research will provide new insights into the mechanistic underpinning of the underlying complex structures and associated covariates on different social and biological phenomena which are the central objective of research in many disciplines. This analysis will help scientists to understand experimental and observational data in biosciences and social sciences and develop suitable control strategies. Community detection for temporal networks would enhance the understanding of brain function across multiple spatial and temporal scales. The proposed work also has significant potential to increase our capacity to develop efficient and cost-effective intervention strategies to mitigate some of the most potentially damaging and fastest-spreading invasive epidemics of humans, livestock, and plants (including wheat, soybean, citrus, and hop). To guarantee that the results are relevant to scientists, parts of the project will be carried out in collaboration with scientists of different fields. Presentations on the relevant research will be given in stakeholder meetings and community outreach events. Undergraduate and graduate students will be involved in the research activities at the intersection of mathematics and other disciplines. Opportunities to learn theoretical and empirical analysis of various probabilistic models, and interactions with scientists will enhance their ability to work at the interface between mathematics and various related areas. One of the primary goals of this project is to understand and analyze different structural properties and limiting behavior of various spatial and non-spatial random graph models, and several stochastic dynamics taking place on graphs. The random graph models include open clusters of percolation and related models, multi-layer, and temporal network models. Percolation models originated in the physics literature as a model for a porous medium. There are many useful tools and a well-developed theory for studying the percolation models on two-dimensional lattices. However, for higher dimensional lattices, several key aspects, including the near-critical regime and the behavior of the model in subgraphs such as sectors, are poorly understood. This project will address some of these issues. Multi-layer networks are natural models for numerous datasets arising in various scientific fields, including genomics, biomedical sciences, neuroscience, economics, sociology, ecology, epidemiology, and technological networks. Depending on the context, various parametric probabilistic models have been used for the formation of multi-layer, multiplex, and temporal networks. For the estimation of those parameters from data, and for model selection purposes, it is very important to understand the behavior of some key functionals (e.g., subgraph counts or the concentration of aggregated adjacency matrices). These functionals will be analyzed for a wide variety of temporal network models, particularly where the snapshots have some correlation structure. The stochastic dynamics include some important variants of the standard models for infection spreading and opinion evolution in presence of additional restrictions (e.g., temporary isolation). The project also includes studying several theoretical and empirical aspects (e.g., bounds for detectability threshold, efficient detection algorithms) of various detection problems, which includes detection of anomalous structures, community detection from a correlated sequence of networks, detection of the source of the epidemic. The rigorous analyses of the models discussed above will require the development of novel mathematical techniques and research tools. These techniques and tools would be instrumental to obtain a deeper understanding of a broader class of random structures and dynamics arising in multiple scientific disciplines.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.