The main aim of this proposal is a systematic mathematical study of a number of random network models arising from applications in computer science, biology and statistical physics, understanding dynamics on these network models, and developing mathematical methodology to glean information from real world networks. Using branching process embeddings and local weak convergence techniques, we propose to develop a set of robust tools that can be used to analyze one of the most important family of network models (the attachment family) which arises in a diverse range of applications. These mathematical techniques will give information on the asymptotics of not only local functionals such as degree distributions but global functionals such as the maximal degree and the spectral distribution of (random) adjacency matrices. Using continuous time branching process techniques we also propose to analyze models of network flow and first passage percolation, in order to understand the effect of disorder on the geometry of random network models and the propagation of congestion across edges in flow carrying networks. Preliminary computations suggest that for a wide array of models, macroscopic order emerges from microscopic rules of transport as the size of the network increases, and the project will attempt to understand this phenomenon and explore connections between these models and stable age distribution theory and the Malthusian rate of growth of branching process models in biology. New models of random trees motivated by statistical physics and biology will also be studied wherein using random walk constructions and conditioned branching processes, we aim to understand the scaling limits of such models. Finally we propose to develop mathematical methodology to analyze, understand and quantify sources of variation in the structure of real world complex networks such as trees arising as blood circulatory networks in the brain.
Over the last few years the availability of empirical data on many real world networks including social networks, data transmission networks such as the Internet and various biological networks, has stimulated an explosion in the array of mathematical models proposed to understand these networks. Researchers in a wide array of fields are interested in understanding properties of such networks, the evolution and change of such networks over time, as well as the dynamics of various processes on these networks such as transporting flow or traffic through these networks and epidemic models on these networks. An understanding of the behavior of these mathematical models would allow practitioners to glean important information and insight about such processes in the real world, ranging from the design of more efficient networks, understanding the factors that influence the rate of spread of congestion of flow processes or other dynamics through the network, to the significant factors that contribute to the actual emergence of the structure of the network itself. A mathematical analysis of such problems leads to interesting connections between these models and wide areas of mathematical probability including branching process models in biology and random fractals. The aim of this project is to develop mathematical methodology to understand properties of such network models and in particular understand what happens when the system size grows large. At the same time the project will also develop techniques to accurately understand data arising from various biological networks such as vascular networks in the brain and the factors that significantly affect functional properties of such networks. The techniques developed will be of use to a wide community of researchers, and we anticipate that the project will foster interdisciplinary collaborative projects with both national and international research groups and facilitate the training of students and expose them to this rapidly emerging field of research.
