Morphogenesis of Complex Networks: A Reaction Diffusion Framework for Spatial Graphs

  • Michele TiricoEmail author
  • Stefan Balev
  • Antoine Dutot
  • Damien Olivier
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 812)


A large variety of real systems are composed by entities in relationships which can be represented by networks. In many of these systems, elements are embedded in the space and location information impacts properties and evolution. Local interactions between elements generate different kinds of equilibrium and often indicate a self-organized behaviour. In this paper we are interested in essential mechanisms behind morphogenesis of spatial networks such as street networks. We propose a multi-layer model, where a reaction-diffusion mechanism governs the growth of spatial networks. We study its evolution with some metrics.


Spatial networks Reaction diffusion model Morphogenesis Gray-Scott model Network models 



This work is supported by the project “AMED” co-funded by ERDF and the region Normandy.


  1. 1.
    Achibet, M., Balev, S., Dutot, A., Olivier, D.: A model of road network and buildings extension co-evolution. Procedia Comput. Sci. 32, 828–833 (2014)CrossRefGoogle Scholar
  2. 2.
    Adamatzky, A.: Generative complexity of Gray-Scott model. Commun. Nonlinear Sci. Numer. Simul. 56, 457–466 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barthelemy, M.: Spatial networks. Phys. Rep. 499(1), 1–101 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barthelemy, M.: Morphogenesis of spatial networks. In: Lecture Notes in Morphogenesis. Springer International Publishing (2018)Google Scholar
  5. 5.
    Barthelemy, M., Flammini, A.: Modeling urban street patterns. Phys. Rev. Lett. 100(13), 138,702 (2008)Google Scholar
  6. 6.
    Barthélemy, M., Flammini, A.: Optimal traffic networks. J. Stat. Mech. Theory Exp. 2006(07), L07,002 (2006)CrossRefGoogle Scholar
  7. 7.
    Batty, M.: Cities and Complexity: Understanding Cities With Cellular Automata, Agent-Based Models, and Fractals. MIT Press, Cambridge, MA (2007)Google Scholar
  8. 8.
    Bird, R.B.: Theory of diffusion. In: T.B. Drew, J.W. Hoopes (eds.) Advances in Chemical Engineering, vol. 1, pp. 155–239. Academic Press (1956)Google Scholar
  9. 9.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: structure and dynamics. Phys. Rep. 424(4), 175–308 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Brede, M.: Coordinated and uncoordinated optimization of networks. Phys. Rev. E 81(6), 066,104 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Buhl, J., et al.: Topological patterns in street networks of self-organized urban settlements. Eur. Phys. J. B Condens. Matter Complex Syst. 49(4), 513–522 (2006)Google Scholar
  12. 12.
    Courtat, T., Gloaguen, C., Douady, S.: Mathematics and morphogenesis of cities: a geometrical approach. Phys. Rev. E 83(3), 036,106 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Diestel, R.: Graph Theory, 4 edizione edn. Springer, Heidelberg; New York (2010)CrossRefGoogle Scholar
  14. 14.
    Gastner, M.T., Newman, M.E.J.: The spatial structure of networks. Eur. Phys. J. B Condens. Matter Complex Syst. 49(2), 247–252 (2006)CrossRefGoogle Scholar
  15. 15.
    Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38(1), 29–43 (1983)CrossRefGoogle Scholar
  16. 16.
    Guillier, S., Muñoz, V., Rogan, J., Zarama, R., Valdivia, J.A.: Optimization of spatial complex networks. Phys. A Stat. Mech. Its Appl. 467, 465–473 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Holme, P.: Modern temporal network theory: a colloquium. Eur. Phys. J. B 88(9), 1–30 (2015)CrossRefGoogle Scholar
  18. 18.
    Jiang, B., Claramunt, C.: Topological analysis of urban street networks. Environ. Plan. B Plan. Des. 31(1), 151–162 (2004)CrossRefGoogle Scholar
  19. 19.
    Kari, J.: Theory of cellular automata: a survey. Theor. Comput. Sci. 334(1), 3–33 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Katifori, E., Magnasco, M.O.: Quantifying loopy network architectures. PLOS ONE 7(6), e37,994 (2012)CrossRefGoogle Scholar
  21. 21.
    Kondo, S., Miura, T.: Reaction-diffusion model as a framework for understanding biological pattern formation. Science (New York, N.Y.) 329(5999), 1616–1620 (2010)Google Scholar
  22. 22.
    Latora, V., Nicosia, V., Russo, G.: Complex Networks: Principles, Methods and Applications. Cambridge University Press, Cambridge, United Kingdom; New York, NY (2017)CrossRefGoogle Scholar
  23. 23.
    Lion, B., Barthelemy, M.: Central loops in random planar graphs. Phys. Rev. E 95(4), 042,310 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Masucci, A.P., Smith, D., Crooks, A., Batty, M.: Random planar graphs and the London street network. Eur. Phys. J. B 71(2), 259–271 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nicolaides, C., Juanes, R., Cueto-Felgueroso, L.: Self-organization of network dynamics into local quantized states. Sci. Rep. 6, 21,360 (2016)Google Scholar
  26. 26.
    Perna, A., Kuntz, P., Douady, S.: Characterization of spatial network like patterns from junction geometry. Phys. Rev. E 83(6), 066,106 (2011)CrossRefGoogle Scholar
  27. 27.
    Pigne, Y., Dutot, A., Guinand, F., Olivier, D.: GraphStream: a tool forbridging the gap between complex systems and dynamic graphs. Emergent properties in natural and artificial complex systems. In: Satellite Conference within the 4th European Conference on Complex Systems (2008)Google Scholar
  28. 28.
    Porta, S., Crucitti, P., Latora, V.: The network analysis of urban streets: a primal approach. Environ. Plan. B Plan. Des. 33(5), 705–725 (2006)CrossRefGoogle Scholar
  29. 29.
    Rui, Y., Ban, Y., Wang, J., Haas, J.: Exploring the patterns and evolution of self-organized urban street networks through modeling. Eur. Phys. J. B 86(3), 74 (2013)CrossRefGoogle Scholar
  30. 30.
    Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. London. Ser. B Biol. Sci. 237(641), 37–72 (1952)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Viana, M.P., Strano, E., Bordin, P., Barthelemy, M.: The simplicity of planar networks. Sci. Rep. 3, 3495 (2013)CrossRefGoogle Scholar
  32. 32.
    Xie, Y.B., Zhou, T., Bai, W.J., Chen, G., Xiao, W.K., Wang, B.H.: Geographical networks evolving with an optimal policy. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 75(3 Pt 2), 036,106 (2007)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Michele Tirico
    • 1
    Email author
  • Stefan Balev
    • 1
  • Antoine Dutot
    • 1
  • Damien Olivier
    • 1
  1. 1.Normandy University, UNIHAVRE, LITISLe HavreFrance

Personalised recommendations