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Alexander Strang, University of Chicago: From local to global structure in random edge flows

Apr 4, 2022 - 11:00 AM
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Presenter: Dr. Alexander Strang, University of Chicago

Time: 11:00 AM Central Time, Monday, April 4, 2022

Title: From Local to Global Structure in Random Edge Flows

Abstract: Edge flows, the analog to vector fields on graphs, are used to model a variety of network structures that occur naturally across fields as diverse as economics, biophysics, and data science. Typically, an edge flow is characterized by decomposition into meaningful components, much as a vector field may be decomposed into circulating and conservative components via the Helmholtz Decomposition. The sizes of these components measure global flow structure. It is an open question how global structure arises from local processes, particularly when the edge flow is generated randomly. Such random edge flows are useful for comparison, hypothesis testing, act as null models, and can be used to study mechanisms that promote or suppress specific structures. We show that power series expansions of locally acting graph operators provide a rich framework for building random flow ensembles and for introducing sequences of local correlations that explain global behavior. We study the expressivity of these power series, and approximation theory under partial expansion. This approximation theory reveals deep connections with graph spectra and symmetries, and illustrates why some topologies are amenable to approximation while others aren’t . We show that for a wide family of common graphs only a few terms in the expansions are required, thus establishing a minimal set of local correlations that explain global structure. 

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