CIF: SMALL: Metric Representations of Network Data

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Network data, defined as one that encodes relationships between elements, is a pervasive component of modern data analytics. The purpose of this project is to advance our capacity to process and understand network data. The contention is that the central difficulty in analyzing large-scale complex networks comes from lack of structure. This loose nature contrasts with the rigidity of a closely related construction: the metric space. If understanding networks is challenging but understanding metric spaces is not, a route to network analysis is to project networks into metric spaces. This motivates the technical goal of designing methods and algorithms to implement these projections. The project builds on preliminary results combining a projection axiom (networks that are already metric remain unchanged after projection) and a dissimilarity-reducing axiom (smaller networks have smaller projections) to establish existence and uniqueness results. Further explorations are pursued in four thrusts: (1) Study of projection methods in symmetric networks. (2) Incorporation of asymmetric networks and asymmetric quasimetric spaces. (3) Metric representations derived from triangle inequalities written in dioid algebras; which albeit metric in an abstract sense are very different from regular metrics. (4) Generalizations to high order networks in which dissimilarities are defined for tuples other than binary. Applications to search, network comparison, and diffusion processes, complement the theoretical research. Broader impacts come through industrial partnerships and an aggressive educational agenda that leverages the University of Pennsylvania's institutional commitment to play a leading role in the education of engineers that are to exploit the opportunities afforded by the ever increasing access to data and the ever broadening scope that networks play in our society.