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Dynamical Systems on Dynamical Networks

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Dynamical Systems on Networks

Part of the book series: Frontiers in Applied Dynamical Systems: Reviews and Tutorials ((FIADS,volume 4))

Abstract

The study of dynamical systems on time-dependent (i.e., “temporal” or “dynamical”) networks has become extremely popular recently, but there are also much older quantitative studies of such situations. For example, Farmer et al. [92] and Bagley et al. [13] used such a framework more than two decades ago in studies of chemical reactions. Moreover, even in the early part of the 20th century, biostatistician Ronald Fisher posited that one could describe the seemingly random fluttering of a colony of butterflies as a dynamical network of information [97].

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Notes

  1. 1.

    In the physics literature, such situations with extremely slow structural dynamics are sometimes called “quenched,” because the networks are almost frozen.

  2. 2.

    In the physics literature, such an idea is invoked to justify certain approximations in models, and the word “annealed” is sometimes used to describe such a situation.

References

  1. R.J. Bagley, J.D. Farmer, S.A. Kauffman, N.H. Packard, A.S. Perelson, I.M. Stadnyk, Modeling adaptive biological systems. Biosystems 23(2–3), 113–137 (1989)

    Article  Google Scholar 

  2. I.V. Belykh, V.N. Belykh, M. Hasler, Blinking model and synchronization in small-world networks with a time-varying coupling. Physica D 195(1–2), 188–206 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Bick, M. Field, Asynchronous networks and event driven dynamics (2015). arXiv:1509.04045

    Google Scholar 

  4. M. Boguñá, L.F. Lafuerza, R. Toral, M.A. Serrano, Simulating non-Markovian stochastic processes. Phys. Rev. E 90(4), 042108 (2014)

    Google Scholar 

  5. G. Demirel, F. Vázquez, G.A. Bhöme, T. Gross, Moment-closure approximations for discrete adaptive networks. Physica D 267(1), 68–80 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. B.A. Desmarais, S.J. Cranmer, Statistical mechanics of networks: Estimation and uncertainty. Physica A 391(4), 1865–1876 (2012)

    Article  Google Scholar 

  7. R. Durrett, J.P. Gleeson, A.L. Lloyd, P.J. Mucha, F. Shi, D. Sivakoff, J.E. Socolar, C. Varghese, Graph fission in an evolving voter model. Proc. Natl. Acad. Sci. U. S. A. 109(10), 3682–3687 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. J.D. Farmer, S.A. Kauffman, N.H. Packard, Autocatalytic replication of polymers. Physica D 22(1), 50–67 (1986)

    Article  MathSciNet  Google Scholar 

  9. R.A. Fisher, The Genetical Theory of Natural Selection, Complete Varorium Edition (Oxford University Press, Oxford, 1999)

    Google Scholar 

  10. T. Gross, B. Blasius, Adaptive coevolutionary networks: A review. J. R. Soc. Interface 5(20), 259–271 (2008)

    Article  Google Scholar 

  11. T. Gross, C.J. Dommar D’Lima, B. Blasius, Epidemic dynamics on an adaptive network. Phys. Rev. Lett. 96(20), 208701 (2006)

    Google Scholar 

  12. T. Hoffmann, M.A. Porter, R. Lambiotte, Generalized master equations for non-Poisson dynamics on networks. Phys. Rev. E 86(4), 046102 (2012)

    Google Scholar 

  13. T. Hoffmann, M.A. Porter, R. Lambiotte, Random walks on stochastic temporal networks, in Temporal Networks (Springer, New York, 2013), pp. 295–314

    Google Scholar 

  14. P. Holme, Modern temporal network theory: A colloquium. Eur. Phys. J. B 88(9), 234 (2015)

    Google Scholar 

  15. P. Holme, M.E.J. Newman, Nonequilibrium phase transition in the coevolution of networks and opinions. Phys. Rev. E 74(5), 056108 (2006)

    Google Scholar 

  16. P. Holme, J. Saramäki, Temporal networks. Phys. Rep. 519(3), 97–125 (2012)

    Article  Google Scholar 

  17. P. Holme, J. Saramäki (eds.), Temporal Networks (Springer, New York, 2013)

    Google Scholar 

  18. D.X. Horváth, J. Kertész, Spreading dynamics on networks: The role of burstiness, topology and non-stationarity. New J. Phys. 16(7), 073037 (2014)

    Google Scholar 

  19. J. Ito, K. Kaneko, Spontaneous structure formation in a network of chaotic units with variable connection strengths. Phys. Rev. Lett. 88(2), 028701 (2002)

    Google Scholar 

  20. H.-H. Jo, J.I. Perotti, K. Kaski, J. Kertész, Analytically solvable model of spreading dynamics with non-Poissonian processes. Phys. Rev. X 4(1), 011041 (2014)

    Google Scholar 

  21. F. Karimi, P. Holme, Threshold model of cascades in empirical temporal networks. Physica A 392(16), 3476–3483 (2013)

    Article  Google Scholar 

  22. M. Karsai, M. Kivelä, R.K. Pan, K. Kaski, J. Kerész, A.-L. Barabási, J. Saramäki, Small but slow world: How network topology and burstiness slow down spreading. Phys. Rev. E 83(2), 025102(R) (2011)

    Google Scholar 

  23. M. Karsai, N. Perra, A. Vespignani, Time-varying networks and the weakness of strong ties. Sci. Rep. 4, 4001 (2014)

    Article  Google Scholar 

  24. S. Liu, N. Perra, M. Karsai, A. Vespignani, Controlling contagion processes in activity driven networks. Phys. Rev. Lett. 112(11), 118702 (2014)

    Google Scholar 

  25. D. Lusher, J. Koskinen, G. Robins, Exponential Random Graph Models for Social Networks (Cambridge University Press, Cambridge, 2013)

    Google Scholar 

  26. N. Malik, P.J. Mucha, Role of social environment and social clustering in spread of opinions in coevolving networks. Chaos 23(4), 043123 (2013)

    Google Scholar 

  27. V. Marceau, P.-A. Noël, L. Hébert-Dufresne, A. Allard, L.J. Dubé, Adaptive networks: Coevolution of disease and topology. Phys. Rev. E 82(3), 036116 (2010)

    Google Scholar 

  28. N. Masuda, K. Klemm, V.M. Eguíluz, Temporal networks: Slowing down diffusion by long lasting interactions. Phys. Rev. Lett. 111(18), 188701 (2013)

    Google Scholar 

  29. N. Masuda, L.E.C. Rocha, A Gillespie algorithm for non-Markovian stochastic processes: Laplace transform approach (2016). arXiv:1601.01490

    Google Scholar 

  30. J.C. Miller, E.M. Volz, Model hierarchies in edge-based compartmental modeling for infectious disease spread. J. Math. Biol. 67(4), 869–899 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. L. Moreau, Stability of multiagent systems with time-dependent communication links. IEEE Trans. Autom. Control 50(2), 169–182 (2005)

    Article  MathSciNet  Google Scholar 

  32. M. Ogura, V.M. Preciado, Stability of spreading processes over time-varying large-scale networks. IEEE Trans. Netw. Sci. Eng. 3(1), 44–57 (2016)

    Article  MathSciNet  Google Scholar 

  33. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, Epidemic processes in complex networks. Rev. Mod. Phys. 87(4), 925–979 (2015)

    Article  MathSciNet  Google Scholar 

  34. N. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani, Activity driven modeling of time varying networks. Sci. Rep. 2, 469 (2012)

    Article  Google Scholar 

  35. N. Perra, A. Baronchelli, D. Mocanu, B. Gonçalves, R. Pastor-Satorras, A. Vespignani, Random walks and search in time-varying networks. Phys. Rev. Lett. 109(23), 238701 (2012)

    Google Scholar 

  36. R. Pfitzner, I. Scholtes, A. Garas, C.J. Tessone, F. Schweitzer, Betweenness preference: Quantifying correlations in the topological dynamics of temporal networks. Phys. Rev. Lett. 110(19), 198701 (2013)

    Google Scholar 

  37. L.E.C. Rocha, N. Masuda, Individual-based approach to epidemic processes on arbitrary dynamic contact networks (2015). arXiv:1510.09179

    Google Scholar 

  38. H. Sayama, I. Pestov, J. Schmidt, B. J. Bush, C. Wong, J. Yamanoi, T. Gross, Modeling complex systems with adaptive networks. Comput. Math. Appl. 65(10), 1645–1664 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. I. Scholtes, N. Wider, R. Pfitzner, A. Garas, C.J. Tessone, F. Schweitzer, Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks. Nat. Commun. 5, 5024 (2014)

    Article  Google Scholar 

  40. F. Shi, P.J. Mucha, R. Durrett, Multiopinion coevolving voter model with infinitely many phase transitions. Phys. Rev. E 88(6), 062818 (2013)

    Google Scholar 

  41. J.D. Skufca, E.M. Bollt, Communication and synchronization in disconnected networks with dynamic topology: Moving neighborhood networks. Math. Biosci. Eng. 1(2), 347–359 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. T.A.B. Snijders, The statistical evaluation of social network dynamics. Sociol. Methodol. 40(1), 361–395 (2001)

    Article  Google Scholar 

  43. T.A.B. Snijders, G.G. Van de Bunt, C.E.G. Steglich, Introduction to stochastic actor-based models for network dynamics. Soc. Networks 32(1), 44–60 (2010)

    Article  Google Scholar 

  44. M. Starnini, A. Baronchelli, A. Barrat, R. Pastor-Satorras, Random walks on temporal networks. Phys. Rev. E 85(5), 056115 (2012)

    Google Scholar 

  45. T. Takaguchi, N. Masuda, P. Holme, Bursty communication patterns facilitate spreading in a threshold-based epidemic dynamics. PLoS ONE 8(7), e68629 (2013)

    Google Scholar 

  46. H.G. Tanner, A. Jadbabaie, G.J. Pappas, Stable flocking of mobile agents, part ii: Dynamic topology, in Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, pp. 2016–2021 (2003)

    Google Scholar 

  47. E. Valdano, L. Ferreri, C. Poletto, V. Colizza, Analytical computation of the epidemic threshold on temporal networks. Phys. Rev.X 5(2), 021005 (2015)

    Google Scholar 

  48. C.L. Vestergaard, M. Génois, Temporal Gillespie algorithm: Fast simulation of contagion processes on time-varying networks. PLoS Comput. Biol. 11(10), e1004579 (2015)

    Google Scholar 

  49. E. Volz, L.A. Meyers, Susceptible–infected–recovered epidemics in dynamic contact networks. Proc. R. Soc. Lond. B Biol. Sci. 274(1628), 2925–2934 (2007)

    Article  Google Scholar 

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Authors and Affiliations

  1. Math Institute, Radcliffe Observatory Quarter, Andrew Wiles Building, Woodstock Road, Oxford, United Kingdom

    Mason A. Porter

  2. MACSI Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland

    James P. Gleeson

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  1. Mason A. Porter
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  2. James P. Gleeson
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Porter, M.A., Gleeson, J.P. (2016). Dynamical Systems on Dynamical Networks. In: Dynamical Systems on Networks. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-26641-1_6

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