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Graphical dynamical systems and their applications to bio-social systems

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Abstract

In this review paper, we discuss graphical dynamical systems (GDSs) and their applications to biological and social systems (bio-social systems). Traditionally, differential equation-based models have been central in modeling bio-social systems. GDSs provide an alternate modeling framework. This framework explicitly represents individual components of the system and captures the interactions among them via a network. The purpose of this review is to enable modelers to obtain an understanding of this basic mathematical and computational framework so that it can be used to study specific bio-social applications. The work covers the range from computational theory to simulation-based analysis. We also provide some directions for future work.

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Notes

  1. Concepts such as phase space and fixed points are defined in Sect. 3.

  2. For definitions concerning complexity classes, we refer the reader to [61].

  3. A negative threshold function is the negation of a threshold function. For example, a negative three-threshold function has the value 1 if and only if two or fewer of its inputs have the value 1.

  4. For definitions related to treewidth, we refer the reader to [63].

  5. A Boolean function is monotone if does not change from 1 to 0 when one more of the inputs is changed from 0 to 1. For example, every k-threshold function (for any integer \(k \ge 0\)) is monotone.

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Acknowledgements

We thank members of the Network Dynamics and Simulation Science Laboratory (NDSSL) for their comments and input. Specifically, we thank Chris Barrett, Christian Reidys, Daniel Rosenkrantz and Richard Stearns for their collaboration on several papers discussed in this article. We thank the computer systems administrators and managers at the Biocomplexity Institute of Virginia Tech for their help in this and many other works: Dominik Borkowski, William Miles Gentry, Jeremy Johnson, William Marmagas, Douglas McMaster, Kevin Shinpaugh and Robert Wills. This work has been partially supported by DTRA CNIMS (Contract HDTRA1-11-D-0016-0001), NSF BIG DATA Grant IIS-1633028, NSF DIBBS Grant ACI-1443054 and NSF EAGER Grant CMMI-1745207. The US Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.

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  1. Madhav V. Marathe

    Present address: Computer Science Department, University of Virginia, Charlottesville, VA, USA

Authors and Affiliations

  1. Biocomplexity Institute and Initiative, University of Virginia, 1001 Research Park Boulevard, Charlottesville, VA, 22911, USA

    Abhijin Adiga, Chris J. Kuhlman, Madhav V. Marathe, Henning S. Mortveit, S. S. Ravi & Anil Vullikanti

  2. Computer Science Department, University of Virginia, Charlottesville, VA, USA

    Anil Vullikanti

  3. University at Albany – State University of New York, Albany, NY, USA

    S. S. Ravi

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Adiga, A., Kuhlman, C.J., Marathe, M.V. et al. Graphical dynamical systems and their applications to bio-social systems. Int J Adv Eng Sci Appl Math 11, 153–171 (2019). https://doi.org/10.1007/s12572-018-0237-6

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