Mathematics Colloquia and Seminars

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Network structures are pervasive in the natural and engineered world, from biological networks to the Internet. What advantages do network structures offer over regular topology? How does network topology affect function, and in turn, feedback from the function affect topological change? Can we use properties observed in natural networks to steer engineering implementations? Such questions guide our thinking as we strive to build a mathematical framework for understanding network phenomena.

One common observation in many classes of networks is the existence of "scale-free" probability distributions. We introduce an underlying mathematical framework for network growth, based on optimization of tradeoffs, and show that the much studied mechanisms of Preferential Attachment (PA) and of saturation emerge from these more basic considerations. Previously these mechanisms had been assumed as fundamental axioms. Time permitting, applications to self-organizing sensor networks, and the larger issue of the interplay between statistical physics, computer simulation and probability theory will be presented.