The project concerns theoretical study of mathematical models of large random networks, with a focus on the following two settings. Certain observed features of the U.S. road network are consistent with it being approximately scale-invariant over scales of 5 - 500 miles. It is proposed to study mathematical properties of models that are exactly scale-invariant. Such study raises new foundational and substantive issues. Second, there is great current interest in information flow in social networks. Relevant abstract mathematical models are similar to models previously studied in many different scientific and engineering contexts (epidemics; broadcasting in computer networks; interacting particles in statistical physics). It is proposed to delineate a particular broad class (``Finite Markov Information Exchange") of such models, record what is already known about them, and formulate and investigate many new questions. The formulation of questions is guided by analogy with the well understood theory surrounding mixing times for finite Markov chains.
At the research level, broader impacts envisaged from this research are facilitating the spread of techniques between different research communities and providing additional tools for the design, analysis and empirical investigation of large networks. At the postgraduate training level the second topic provides a fertile field of open problems and varying apparent difficulty, whose study will expose the student to a broad range of techniques. Both topics also fit well with the principal investigator's activities in undergraduate teaching (a "probability and the real world" course) and supervising undergraduate research projects.