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

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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.