A General Model of Dynamics on Networks with Graph Automorphism Lumping

  • Jonathan A. WardEmail author
  • John Evans
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 812)


In this paper we introduce a general Markov chain model of dynamical processes on networks. In this model, nodes in the network can adopt a finite number of states and transitions can occur that involve multiple nodes changing state at once. The rules that govern transitions only depend on measures related to the state and structure of the network and not on the particular nodes involved. We prove that symmetries of the network can be used to lump equivalent states in state-space. We illustrate how several examples of well-known dynamical processes on networks correspond to particular cases of our general model. This work connects a wide range of models specified in terms of node-based dynamical rules to their exact continuous-time Markov chain formulation.


Dynamics on networks Markov chains Graph automorphisms Lumping Epidemic models Opinion dynamics Social physics 


  1. 1.
    Axelrod, R.: The dissemination of culture: a model with local convergence and global polarization. J. Confl. Resolut. 41(2), 203–226 (1997)Google Scholar
  2. 2.
    Banisch, S., Lima, R.: Markov chain aggregation for simple agent-based models on symmetric networks: the voter model. Adv. Complex Syst. 18(03n04), 1550011 (2015)Google Scholar
  3. 3.
    Banisch, S., Lima, R., Araújo, T.: Agent based models and opinion dynamics as markov chains. Soc. Netw. 34(4), 549–561 (2012)Google Scholar
  4. 4.
    Baronchelli, A., Felici, M., Loreto, V., Caglioti, E., Steels, L.: Sharp transition towards shared vocabularies in multi-agent systems. J. Stat. Mech. 2006(06), P06014 (2006)Google Scholar
  5. 5.
    Barrat, A., Barthelemy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008)Google Scholar
  6. 6.
    Bass, F.M.: A new product growth for model consumer durables. Manag. Sci. 15(5), 215–227 (1969)Google Scholar
  7. 7.
    Bernardes, A.T., Stauffer, D., Kertész, J.: Election results and the sznajd model on Barabasi network. Eur. Phys. J. B 25(1), 123–127 (2002)Google Scholar
  8. 8.
    Boguná, M., Pastor-Satorras, R.: Epidemic spreading in correlated complex networks. Phys. Rev. E 66(4), 047104 (2002)Google Scholar
  9. 9.
    Bonabeau, E., Theraulaz, G., Deneubourg, J.L.: Phase diagram of a model of self-organizing hierarchies. Phys. A 217(3–4), 373–392 (1995)Google Scholar
  10. 10.
    Castellano, C., Marsili, M., Vespignani, A.: Nonequilibrium phase transition in a model for social influence. Phys. Rev. Lett. 85(16), 3536 (2000)Google Scholar
  11. 11.
    Castellano, C., Muñoz, M.A., Pastor-Satorras, R.: Nonlinear q-voter model. Phys. Rev. E 80(4), 041129 (2009)Google Scholar
  12. 12.
    Castellano, C., Pastor-Satorras, R.: Thresholds for epidemic spreading in networks. Phys. Rev. Lett. 105(21), 218701 (2010)Google Scholar
  13. 13.
    Castellano, C., Vilone, D., Vespignani, A.: Incomplete ordering of the voter model on small-world networks. Eur. Lett. 63(1), 153 (2003)Google Scholar
  14. 14.
    Castelló, X., Eguíluz, V.M., San Miguel, M.: Ordering dynamics with two non-excluding options: bilingualism in language competition. New J. Phys. 8(12), 308 (2006)Google Scholar
  15. 15.
    Chen, P., Redner, S.: Majority rule dynamics in finite dimensions. Phys. Rev. E 71(3), 036101 (2005)Google Scholar
  16. 16.
    Clifford, P., Sudbury, A.: A model for spatial conflict. Biometrika 60(3), 581–588 (1973)Google Scholar
  17. 17.
    Daley, D.J., Kendall, D.G.: Epidemics and rumours. Nature 204(4963), 1118 (1964)Google Scholar
  18. 18.
    Deffuant, G., Neau, D., Amblard, F., Weisbuch, G.: Mixing beliefs among interacting agents. Adv. Complex Syst. 3(01n04), 87–98 (2000)Google Scholar
  19. 19.
    Eames, K.T., Keeling, M.J.: Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. Proc. Natl. Acad. Sci. 99(20), 13330–13335 (2002)Google Scholar
  20. 20.
    Epstein, J.M., Axtell, R.: Growing Artificial Societies: Social Science from the Bottom Up. Institution Press, Brookings (1996)Google Scholar
  21. 21.
    Fennell, P.G., Gleeson, J.P.: Multistate dynamical processes on networks: analysis through degree-based approximation frameworks. arXiv preprint arXiv:1709.09969 (2017)
  22. 22.
    Fraleigh, J.B.: A First Course in Abstract Algebra. Pearson Education, India (2003)Google Scholar
  23. 23.
    Galam, S.: Minority opinion spreading in random geometry. Eur. Phys. J. B 25(4), 403–406 (2002)Google Scholar
  24. 24.
    Glauber, R.J.: Time-dependent statistics of the Ising model. J. Math. Phys. 4(2), 294–307 (1963)Google Scholar
  25. 25.
    Gleeson, J.P.: High-accuracy approximation of binary-state dynamics on networks. Phys. Rev. Lett. 107(6), 68701 (2011)Google Scholar
  26. 26.
    Gleeson, J.P.: Binary-state dynamics on complex networks: pair approximation and beyond. Phys. Rev. X 3(2), 021004 (2013)Google Scholar
  27. 27.
    Gleeson, J.P., Hurd, T., Melnik, S., Hackett, A.: Systemic risk in banking networks without Monte Carlo simulation. Advances in Network Analysis and its Applications, pp. 27–56. Springer, Berlin (2012)Google Scholar
  28. 28.
    Gleeson, J.P., Melnik, S., Ward, J.A., Porter, M.A., Mucha, P.J.: Accuracy of mean-field theory for dynamics on real-world networks. Phys. Rev. E 85(2), 026106 (2012)Google Scholar
  29. 29.
    Gleeson, J.P., Ward, J.A., Osullivan, K.P., Lee, W.T.: Competition-induced criticality in a model of meme popularity. Phys. Rev. Lett. 112(4), 048701 (2014)Google Scholar
  30. 30.
    Godsil, C., Royle, G.F.: Algebraic Graph Theory, vol. 207. Springer Science & Business Media, Berlin (2013)Google Scholar
  31. 31.
    Goldenberg, J., Libai, B., Muller, E.: Talk of the network: a complex systems look at the underlying process of word-of-mouth. Mark. Lett. 12(3), 211–223 (2001)Google Scholar
  32. 32.
    Haldane, A.G., May, R.M.: Systemic risk in banking ecosystems. Nature 469(7330), 351 (2011)Google Scholar
  33. 33.
    Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul. 5(3) (2002)Google Scholar
  34. 34.
    Holley, R.A., Liggett, T.M.: Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 643–663 (1975)Google Scholar
  35. 35.
    Holme, P.: Shadows of the susceptible-infectious-susceptible immortality transition in small networks. Phys. Rev. E 92(1), 012804 (2015)Google Scholar
  36. 36.
    Holme, P., Tupikina, L.: Epidemic extinction in networks: insights from the 12,110 smallest graphs. arXiv preprint arXiv:1802.08849 (2018)
  37. 37.
    Kemeny, J.G., Snell, J.L., et al.: Finite Markov Chains, vol. 356. van Nostrand, Princeton (1960)Google Scholar
  38. 38.
    Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146. ACM (2003)Google Scholar
  39. 39.
    Kijima, M.: Markov processes for stochastic modeling, vol. 6. CRC Press, Boston (1997)Google Scholar
  40. 40.
    Kirman, A.: Ants, rationality, and recruitment. Q. J. Econ. 108(1), 137–156 (1993)Google Scholar
  41. 41.
    Kiss, I.Z., Miller, J.C., Simon, P.L.: Mathematics of Epidemics on Networks. Springer, Berlin (2017)Google Scholar
  42. 42.
    Lambiotte, R.: How does degree heterogeneity affect an order-disorder transition? Eur. Lett. 78(6), 68002 (2007)Google Scholar
  43. 43.
    López-García, M.: Stochastic descriptors in an SIR epidemic model for heterogeneous individuals in small networks. Math. Biosci. 271, 42–61 (2016)Google Scholar
  44. 44.
    MacArthur, B.D., Sánchez-García, R.J.: Spectral characteristics of network redundancy. Phys. Rev. E 80(2), 026117 (2009)Google Scholar
  45. 45.
    MacArthur, B.D., Sánchez-García, R.J., Anderson, J.W.: Symmetry in complex networks. Discret. Appl. Math. 156(18), 3525–3531 (2008)Google Scholar
  46. 46.
    Mellor, A., Mobilia, M., Redner, S., Rucklidge, A.M., Ward, J.A.: Influence of Luddism on innovation diffusion. Phys. Rev. E 92(1), 012806 (2015)Google Scholar
  47. 47.
    Melnik, S., Ward, J.A., Gleeson, J.P., Porter, M.A.: Multi-stage complex contagions. Chaos 23(1), 013124 (2013)Google Scholar
  48. 48.
    Mobilia, M., Petersen, A., Redner, S.: On the role of zealotry in the voter model. J. Stat. Mech. 2007(08), P08029 (2007)Google Scholar
  49. 49.
    Mobilia, M., Redner, S.: Majority versus minority dynamics: phase transition in an interacting two-state spin system. Phys. Rev. E 68(4), 046,106 (2003)Google Scholar
  50. 50.
    Motter, A.E., Lai, Y.C.: Cascade-based attacks on complex networks. Phys. Rev. E 66(6), 065102 (2002)Google Scholar
  51. 51.
    Newman, M.: Networks. Oxford University Press, Oxford (2010)Google Scholar
  52. 52.
    Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys. 87(3), 925 (2015)Google Scholar
  53. 53.
    Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200 (2001)Google Scholar
  54. 54.
    Pastor-Satorras, R., Vespignani, A.: Immunization of complex networks. Phys. Rev. E 65(3), 036104 (2002)Google Scholar
  55. 55.
    Porter, M.A., Gleeson, J.P.: Dynamical Systems on Networks. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol. 4. Springer, Berlin (2016)Google Scholar
  56. 56.
    Sanchez-Garcia, R.J.: Exploiting symmetry in network analysis. arXiv preprint arXiv:1803.06915 (2018)
  57. 57.
    Schelling, T.C.: Models of segregation. Am. Econ. Rev. 59(2), 488–493 (1969)Google Scholar
  58. 58.
    Schelling, T.C.: Dynamic models of segregation. J. Math. Sociol. 1(2), 143–186 (1971)Google Scholar
  59. 59.
    Simon, P.L., Taylor, M., Kiss, I.Z.: Exact epidemic models on graphs using graph-automorphism driven lumping. J. Math. Biol. 62(4), 479–508 (2011)Google Scholar
  60. 60.
    Sood, V., Redner, S.: Voter model on heterogeneous graphs. Phys. Rev. Lett. 94(17), 178701 (2005)Google Scholar
  61. 61.
    Sznajd-Weron, K., Sznajd, J.: Opinion evolution in closed community. Int. J. Mod. Phys. C 11(06), 1157–1165 (2000)Google Scholar
  62. 62.
    Vazquez, F., Krapivsky, P.L., Redner, S.: Constrained opinion dynamics: freezing and slow evolution. J. Phys. A 36(3), L61 (2003)Google Scholar
  63. 63.
    Ward, J.A.: Instability in heterogeneous traffic. In: Proceedings of the Traffic Flow Theory and Characteristics Committee (2010)Google Scholar
  64. 64.
    Watts, D.J.: A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. 99(9), 5766–5771 (2002)Google Scholar

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Authors and Affiliations

  1. 1.University of LeedsLeedsUK

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