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Applications of the Helmholtz-Hodge Decomposition to Networks and Random Processes

Strang, Alexander
http://orcid.org/0000-0001-6618-631X
http://rave.ohiolink.edu/etdc/view?acc_num=case1595596768356487

Abstract Details

2020, Doctor of Philosophy, Case Western Reserve University, Applied Mathematics.
Cycle and potential decompositions are widely used mathematical tools for analyzing systems across fields. The discrete Hodge-Helmholtz Decomposition (HHD) decomposes an edge flow on a graph into two components. The first component is conservative, and associated with the gradient of a potential function defined on the vertices. The second component is cyclic, and is associated with the adjoint curl of a set of vorticities defined on the loops of the network. We explore applications of the HHD to problems arising in a variety of fields. We provide examples where the HHD is used as a descriptive tool for characterizing structure, and as a predictive tool for understanding dynamics. To show that the HHD can be used to describe the structure we apply it to tournaments arising in politics, animal behavior, and sports. To show that the HHD can be used to predict dynamics we apply the decomposition to discrete-space continuous-time Markov models motivated by biophysical and ecological examples. It is shown that the HHD has a natural thermodynamic interpretation, and can be used to construct analogous thermodynamics for a generic class of Markov chains. We show that the HHD can be applied to understand steady-state dynamics in either the strong noise, or weak rotation, limit and controls the long-term production rate of observables. A formal expansion of steady-state distributions and steady-state fluxes in the cyclic component of the HHD is introduced. Comparisons to existing potential theories and cycle decompositions are made, and it is shown that the HHD is a complementary decomposition to the quasipotential in the continuum limit.
Peter Thomas, PhD (Committee Chair)
Daniela Calvetti, PhD (Committee Member)
Wojbor Woyczynski, PhD (Committee Member)
Karen Abbott, PhD (Committee Member)
Michael Hinczewksi, PhD (Committee Member)
735 p.
Ecology; Mathematics; Physics; Political Science
Helmholtz-Hodge Decomposition; HHD; Markov process; Tournament; Competition; Transitivity; Nonequilibrium Thermodynamics; Networks; Graph Theory

Recommended Citations

Citations

  • Strang, A. (2020). Applications of the Helmholtz-Hodge Decomposition to Networks and Random Processes [Doctoral dissertation, Case Western Reserve University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=case1595596768356487

    APA Style (7th edition)

  • Strang, Alexander. Applications of the Helmholtz-Hodge Decomposition to Networks and Random Processes. 2020. Case Western Reserve University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=case1595596768356487.

    MLA Style (8th edition)

  • Strang, Alexander. "Applications of the Helmholtz-Hodge Decomposition to Networks and Random Processes." Doctoral dissertation, Case Western Reserve University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=case1595596768356487

    Chicago Manual of Style (17th edition)

case1595596768356487
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© 2020, all rights reserved.
This open access ETD is published by Case Western Reserve University School of Graduate Studies and OhioLINK.