Large Random Networks: Route and Information-Exchange Processes

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The project concerns theoretical study of mathematical models of large randomnetworks, with a focus on the following two settings. Certain observed featuresof the U.S. road network are consistent with it being approximatelyscale-invariant over scales of 5 - 500 miles. It is proposed to studymathematical properties of models that are exactly scale-invariant. Such studyraises new foundational and substantive issues. Second, there is great currentinterest in information flow in social networks. Relevant abstract mathematicalmodels are similar to models previously studied in many different scientific andengineering contexts (epidemics; broadcasting in computer networks; interactingparticles in statistical physics). It is proposed to delineate a particularbroad class (``Finite Markov Information Exchange") of such models, record whatis already known about them, and formulate and investigate many new questions.The formulation of questions is guided by analogy with the well understood theorysurrounding mixing times for finite Markov chains.At the research level, broader impacts envisaged from this research arefacilitating the spread of techniques between different research communitiesand providing additional tools for the design, analysis and empiricalinvestigation of large networks. At the postgraduate training level thesecond topic provides a fertile field of open problems and varying apparentdifficulty, whose study will expose the student to a broad range of techniques.Both topics also fit well with the principal investigator's activities inundergraduate teaching (a "probability and the real world" course) andsupervising undergraduate research projects.