Mathematics Colloquia and Seminars

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Collections of networks are at the core of modern society, spanning technological, biological and social systems. Over the past decade a science of networks has been emerging and providing insights into the structure and function of many diverse types of systems, such as protein-interactions in a cell, collaboration networks of scientists, and the World Wide Web. Random graphs provide a framework for modeling network phenomena, especially phase transitions, such as the sudden emergence of large-scale connectivity. This talk will give an overview and present a variant of the classic Erdos-Renyi model of network formation (using the power of two choices), showing that we can alter the location and also the nature of the phase transition, making for an explosive onset of connectivity. We also develop random graph models of interacting networks, motivated by the fact that individual networks are increasingly interdependent (e.g., the Internet and the power grid, globalization of financial markets). We show that interactions between different types of networks can actually lower critical thresholds and provide stabilizing effects with respect to cascades.