Riemannian Geometric and Stochastic Methods for Robust

and High Performance Network Communications

The network becomes more complex with the increase of networked entities and their types, and the complicatedness of the deployment environment. The terrains, where the networks and sensors of military are deployed, are often non-flat and with obstacles and holes. The conventional network algorithm and protocol often fail or work at low performance in these types of terrain. In addition, the locations of communication devices or targets may often be needed while GPS or conventional localization devices are not readily available or costly to deploy. There also need methodologies to predict and improve the network performance, and detect the network problems at low cost.

The objective of this work is to investigate and understand the mathematics principles underlying the predictable operation of networks and systems, and provide some fundamental methodologies and mathematical tools that can facilitate the analysis, design and management of networks. Our proposed program focuses on applying mathematical theories to novelly address some fundamental challenges in network design to ensure robust and efficient network communications. Our theoretical framework and practical tools will guide the network deployment, provisioning and management, localization of network nodes and targets, routing, and abnormality detection. Our design will specifically address the challenge of enabling network functions on complex terrain with non-flat interface, holes, and obstacles. If successfully implemented, we expect to see a set of network tools that build upon rigorous mathematics principles to facilitate the design of future network algorithms and protocols, and a set of network algorithms and protocols that leverage the tools to enable robust, flexible and high performance network communications. We will base our design on Ricci flow theory and stochastic process on Riemannian manifolds, and generalize these theories from smooth manifolds to networks.

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