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Singular value decomposition methods for understanding long term dynamics on networks

Warren Lord

Applied Mathematics,Ìý

Date and time:Ìý

Thursday, April 17, 2014 - 2:00pm

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ECCR 257

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A diverse group of scientists, including sociologists, biologists, economists, physicists, engineers, and applied mathematicians are interested in understanding how network topology influences dynamics that occur on networks. Recently, researchers have developed approaches that are based on the largest eigenvalue of the adjacency matrix that describes a network's topology. Largest eigenvalue methods work well for a wide variety of degree distributions on random networks, but do not perform as well when structures that are common to real world networks, such as communities and degree correlations, are introduced. We extend the largest eigenvalue methods to operate on the more detailed description of network topology given by the first few singular values of the adjacency matrix. The new methods reduce complex dynamics in high dimensional state spaces to low dimensional systems. We test these methods by predicting steady states of both networks of excitable elements in discrete time systems and networks of oscillators in continuous time systems, and by predicting rates of convergence to steady state in the continuous time networks. The predictions are tested on both random networks and networks derived from real world datasets. We interpret the dimension of the system obtained from the singular value decomposition approach as a metric of the complexity of long term dynamics on networks.