In modern technological world, we increasingly depend upon the reliability, robustness, quality of service and timeliness of exceedingly large interconnected dynamical systems including those of power distribution, transportation, and communication. Over the past twenty years, we have witness a dramatic rise of information in which the analysis of such systems invariably present challenging big data complexity issues. For example, in transferring resources and information, a key requirement is the ability to adapt and reconfigure in response to structural and dynamic changes while avoiding disruption of service. To this end, we propose to develop fundamental relationships between network functionality and certain topological and geometric properties of the corresponding graph. The thematic vision for this program is based on the recently discovered fact that the geometric notion of curvature (or how objects deviate from being flat) is positively correlated with a systems functional robustness or its ability to adapt to dynamic changes. While the developed theory and tools will be applicable to a broad set of dynamical systems, we will focus on cancer biology and man-made distributed malware. The accomplishments of this research will be disseminated through raising engineering awareness of the proposed research through K-12 outreach, publications, tutorials in the form of open-source software, and developed workshops. Specifically, given the rise of data (network) science, there is an increasing demand for engineers especially in national security areas where U.S. Veterans may serve an integral role. As such, we will work to directly engage veterans to engage in aspects of the proposed research while also providing vet dependent mentoring for better university preparation.
This said, to accomplish these goals, the proposed research program aims to study the intimate connections between discrete geometry, optimal mass transport, entropy, and control to formulate new approaches to quantify and predict dynamical properties of complex networks at varying scales. System properties include, but not limited to, robustness, heterogeneity, and congestion. As such, this research fills a need by investigating network functionality through the confluent study of geometry and control that relies upon several observations with far reaching physical (statistical mechanics) and information theoretic significance. Specifically, by placing a probability structure on a graph, the advantage over existing methods is that the space of probabilities on graphs has much nicer properties than the underlying discrete space alone. The associated probability measures can then be endowed with a Riemannian structure whereby geodesic (shortest distance) paths ensue and convexity properties of the entropy along paths reflect on geometric qualities of the graph. Given this foundation, we can then begin to develop powerful geometric network tools rooted in optimal and stochastic control. Altogether, as geometry and control of networks is still in its infancy, this program addresses the increasing need to develop such areas as a further tool to understanding complex systems.
