Abstract
Power grids, transportation systems, neural circuits and gene regulatory networks are just some of the many examples of networks in action. To understand mechanisms underlying collective network dynamics, typically a forward perspective is taken and mathematical models of given systems are explored as a function of their parameters. One question, for instance, might be how the collective dynamics undergoes a bifurcation when the network connectivity is changed. Here, we propose an inverse perspective on. We determine, based on the units’ time series, the set of all networks that generate a given collective dynamics. In particular, we show how the dynamics of a network may be parametrized in the phase portrait. Interestingly, even networks with very different connection topologies may generate identical dynamics. As an example, we rewire networks of Kuramoto-like oscillators with random network topologies into different networks that display the same collective time evolution. The results offer an alternative view on studying the interplay between the structure and dynamics of complex networks.
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References
Newman, M.: Networks: An Introduction. Oxford University Press, New York (2010)
Strogatz, S.H., Nature 410(6825), 268–276 (2001)
Timme, M., Casadiego, J.: J. Phys. A Math. Theor. 47(34), 343001 (2014)
Rohden, M., Sorge, A., Timme, M., Witthaut, D.: Phys. Rev. Lett. 109(6), 064101 (2012)
Filatrella, G., Nielsen, A.H., Pedersen, N.F.: Eur. Phys. J. B 61(4), 485–491 (2008)
Guimerà, R., Mossa, S., Turtschi, A., Amaral, L.A.N.: Proc. Natl. Acad. Sci. U.S.A. 102(22), 7794-7799 (2005)
Mirollo, R.E., Strogatz, S.H.: SIAM J. Appl. Math. 50(6), 1645–1662 (1990)
Gardner, T.S., di Bernardo, D., Lorenz, D., Collins, J.J.: Science 301(5629), 102-105 (2003)
Braess, D.: Unternehmensforschung 12, 258–268 (1968)
Youn, H., Gastner, M., Jeong, H.; Phys. Rev. Lett. 101(12), 128701 (2008)
Witthaut, D., Timme, M.: New J. Phys. 14(8), 083036 (2012)
Deco, G., Jirsa, V.K., McIntosh, A.R.: Nat. Rev. Neurosci. 12(1), 43–56 (2011)
Bullmore, E., Sporns, O.: Nat. Rev. Neurosci. 13(5), 336–349 (2012)
Prinz, A.A., Bucher, D., Marder, E.: Nat. Neurosci. 7(12), 1345–1352 (2004)
Makarov, V.A., Panetsos, F., de Feo, O.; J. Neurosci. Methods 144(2), 265–279 (2005)
Memmesheimer, R.M., Timme, M.: Phys. Rev. Lett. 97(18), 188101 (2006)
Memmesheimer, R.M., Timme, M.: Phys. D 224, 182–201 (2006)
Van Bussel, F., Kriener, B., Timme, M.: Front. Comput. Neurosci. 5, 3 (2011)
Strogatz, S.H.: Phys. D Nonlinear Phenom. 143(1–4), 1–20 (2000)
Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA (2000)
Yeung, M.K.S., Tegnér, J., Collins, J.J.: Proc. Natl. Acad. Sci. U.S.A. 99(9), 6163–6168 (2002)
Timme, M.; Phys. Rev. Lett. 98(22), 224101 (2007)
Shandilya, S.G., Timme, M.: New J. Phys. 13(1), 013004 (2011)
Hansel, D., Mato, G., Meunier, C.: Phys. Rev. E 48(5), 3470–3477 (1993)
Bandyopadhyay, S., Mehta, M., Kuo, D., Sung, M.K., Chuang, R., Jaehnig, E.J., Bodenmiller, B., Licon, K., Copeland, W., Shales, M., Fiedler, D., Dutkowski, J., Guénolé, A., van Attikum, H., Shokat, K.M., Kolodner, R.D., Huh, W.K., Aebersold, R., Keogh, M.C., Krogan, N.J., Ideker, T.: Science 330(6009), 1385–1389 (2010)
Acknowledgements
We thank Benedict Lünsmann, Fabio Schittler-Neves and Sarah Hallerberg for valuable discussions and constructive comments. Partially supported by the International Max Planck Research School (IMPRS) Physics of Biological and Complex Systems (JC), the Ministry for Education and Science (BMBF), Germany, through the Bernstein Center for Computational Neuroscience, grant no. 01GQ1005B (MT) as well as by a grant by the Max Planck Society to MT.
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Casadiego, J., Timme, M. (2015). Network Dynamics as an Inverse Problem. In: Mugnolo, D. (eds) Mathematical Technology of Networks. Springer Proceedings in Mathematics & Statistics, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-319-16619-3_4
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DOI: https://doi.org/10.1007/978-3-319-16619-3_4
Publisher Name: Springer, Cham
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