Severability of mesoscale components and local time scales in dynamical networks
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A major goal of dynamical systems theory is the search for simplified descriptions of the dynamics of a large number of interacting states. For overwhelmingly complex dynamical systems, the derivation of a reduced description on the entire dynamics at once is computationally infeasible. Other complex systems are so expansive that despite the continual onslaught of new data only partial information is available. To address this challenge, we define and optimise for a local quality function severability for measuring the dynamical coherency of a set of states over time. The theoretical underpinnings of severability lie in our local adaptation of the Simon-Ando-Fisher time-scale separation theorem, which formalises the intuition of local wells in the Markov landscape of a dynamical process, or the separation between a microscopic and a macroscopic dynamics. Finally, we demonstrate the practical relevance of severability by applying it to examples drawn from power networks, image segmentation, social networks, metabolic networks, and word association.
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