Collaborative Research: Topological Methods for Parsing Shapes and Networks and Modeling Variation in Structure and Function

推动数据科学的关键是我们探索、概念化、解释和可视化驻留在复杂数据集中的信息的能力，以便将数据转化为知识。该项目开发了用于研究和可视化网络和3D形状中的结构变化以及推断这种变化的功能结果的计算方法和工具。这些问题出现在卫生和医学等具有战略意义的领域，在这些领域，重要的是了解生物形状的形态变化模式和生物网络的结构变化，以及它们在行为、健康和疾病中的作用。为了强调这一重要的跨学科方面，该项目得到了几个案例研究的支持，这些研究调查了：(I)面部形状发育的潜在机制；(Ii)脑形状、头骨形态和行为之间的相互作用；(Iii)消化道微生物群落的组织；以及(Iv)社会网络和微生物组网络之间的联系。我们设想了该项目的许多长期后果。潜在的应用包括动态社会网络的分析，生物网络与表型特征或疾病之间的关联的探索性发现，形态特征的进化、发展和遗传的定量研究，以及为有效的数据管理、搜索和检索而索引和组织网络或3D形状的数据库等挑战。该项目综合了拓扑数据分析、积分几何、光谱几何和统计学的技术，以开发计算方法和工具，用于对不同形状和网络集合中的结构变化进行建模和可视化，并探索结构和功能变化之间的联系。该项目涉及理论基础、计算方法、统计分析和可视化工具的实施以及方法的验证。形状或网络由Hilbert空间上的Borel概率度量来表示，其元素表示具有丰富几何和拓扑性质的欧拉特征曲线。概率度量的降维和离散化导致了允许将复杂的数据集组织成便于数据分析、处理、可视化和搜索的紧凑词典的表示。这使得新的方法能够与一系列现有的多变量统计分析技术相结合，以解决在网络上开发基于回归的模型等问题。

Critical to advancing data-enabled science is our ability to probe, conceptualize, interpret, and visualize information residing in complex datasets in order to transform data into knowledge. This project develops computational methods and tools for investigation and visualization of structural variation in networks and 3D shapes, as well as inference of functional outcomes of such variation. These problems arise in areas of strategic interest such as health and medicine, where it is important to understand patterns of morphological variation in biological shapes and structural changes in biological networks, and their roles in behavior, health and disease. To emphasize this important interdisciplinary facet, the project is supported by several case studies that investigate: (i) mechanisms underlying the development of facial shape; (ii) interactions between brain shape, skull morphology, and behavior; (iii) organization of microbial communities in the digestive tract; and (iv) connections between social networks and microbiome networks. We envision many long-term ramifications of the project. Potential applications include analyses of dynamical social networks, exploratory discovery of associations between biological networks and phenotypic traits or diseases, quantitative studies of evolution, development, and inheritance of morphological traits, and challenges such as indexing and organizing databases of networks or 3D shapes for efficient data management, search, and retrieval. The project integrates techniques from topological data analysis, integral geometry, spectral geometry, and statistics to develop computational methods and tools for modeling and visualizing structural variation in diverse collections of shapes and networks, and exploring associations between variation in structure and function. The project addresses theoretical foundations, computational methods, implementation of tools for statistical analysis and visualization, and validation of methodology. A shape or network is represented by a Borel probability measure on a Hilbert space whose elements represent Euler characteristic curves that encode rich geometric and topological properties. Dimension reduction and discretization of the probability measure lead to representations that allow organization of complex datasets into compact dictionaries that facilitate data analytics, processing, visualization, and search. This enables integration of the new methods with an array of existing techniques of multivariate statistical analysis to address such problems as development of regression-based models over networks.