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

推进数据科学的关键是我们能够探测、概念化、解释和可视化复杂数据集中的信息，以便将数据转化为知识。 该项目开发计算方法和工具，用于网络和3D形状中结构变化的调查和可视化，以及对这种变化的功能结果的推断。这些问题出现在战略利益的领域，如卫生和医学，在这些领域，重要的是要了解生物形状和生物网络结构变化的形态变化模式，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，这些案例研究调查：（i）面部形状发展的机制;（ii）大脑形状，头骨形态和行为之间的相互作用;（iii）消化道中微生物群落的组织;（iv）社交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影响。潜在的应用包括动态社交网络的分析，探索性发现生物网络和表型性状或疾病之间的关联，形态性状的进化，发育和遗传的定量研究，以及诸如索引和组织网络或3D形状的数据库以进行有效的数据管理，搜索和检索等挑战。该项目整合了拓扑数据分析、积分几何、谱几何和统计学等技术，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关联。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。形状或网络由希尔伯特空间上的Borel概率测度表示，希尔伯特空间的元素表示编码丰富几何和拓扑性质的欧拉特征曲线。降维和概率度量的离散化导致表示，允许将复杂数据集组织成紧凑的字典，以促进数据分析，处理，可视化和搜索。这使得新方法与现有的多元统计分析技术的阵列，以解决这样的问题，如基于回归的模型在网络上的发展。

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.