This project takes a novel look at Complex Networks and related problems in data mining and signal processing. Issues arising in Complex Networks are formulated as intrinsically topological problems which, when studied using algebraic topological tools, affords highly efficient and effective solutions. The algebraic approach yields distributed computations, and naturally unveils topological information about the underlying network for strategic planning of mitigation. In particular, this unified view allows tackling problems of dynamics and evolution of networks, where time-varying properties may in turn be used for network healing.
The efficient developed solutions will have an impact on data mining analysis, and provide more insight into the associated strengths and limitations, as well as other network related applications in a broad variety of networks.
In this project we utilized a previously, relatively unused branch of mathematics called Algebraic Topology in a setting which is both mathematically sound and application friendly. The results of our work can be summarized into the following three categories: a) A biomedical application (wearable sensor to monitor for abnormal breathing) b) A framework to analyze large mobile sensor networks. c) A means to explore and downsize social networks with an emphasis on collaborations. Using topology we were able to reduce the complexity of the algorithms involved, increase the robustness to errors and noise and even work with limited information and energy. Using a concrete representation, called a persistent barcode, we were able to find commonalities in all the projects discussed above and give a unified mathematical formulation of the problems and their solutions. While such a representation was well fit to answer the posed questions, the computational complexity involved was high originally; For instance, in the biomedical application, the complexity was equal to the standard methods used and the wearable device we are implementing this algorithm on, does not even have a battery. In order for our methods to succeed they must be economic while also maintaining an "on-line" status (a detection with a 30 minute gap is not very informative and useful). Thus our work on creating an even more compact representation was not done only as a mathematical curiosity but also as a means to enable applications, where complexity and energy consumption need to be minimal. So we worked both on refining our methods using a more economical mathematics model using Information Theoretic tools and to reduce the complexity of the problem using distributive algorithms and dimension reduction techniques.
