Advertisement

Cyclic Structure Induced by Load Fluctuations in Adaptive Transportation Networks

  • Erik Andreas MartensEmail author
  • Konstantin Klemm
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 30)

Abstract

Transport networks are crucial to the functioning of natural systems and technological infrastructures. For flow networks in many scenarios, such as rivers or blood vessels, acyclic networks (i.e., trees) are optimal structures when assuming time-independent in- and outflow. Dropping this assumption, fluctuations of net flow at source and/or sink nodes may render the pure tree solutions unstable even under a simple local adaptation rule for conductances. Here, we consider tree-like networks under the influence of spatially heterogeneous distribution of fluctuations, where the root of the tree is supplied by a constant source and the leaves at the bottom are equipped with sinks with fluctuating loads. We find that the network divides into two regions characterized by tree-like motifs and stable cycles. The cycles emerge through transcritical bifurcations at a critical amplitude of fluctuation. For a simple network structure, depending on parameters defining the local adaptation, cycles first appear close to the leaves (or root) and then appear closer towards the root (or the leaves). The interaction between topology and dynamics gives rise to complex feedback mechanisms with many open questions in the theory of network dynamics. A general understanding of the dynamics in adaptive transport networks is essential in the study of mammalian vasculature, and adaptive transport networks may find technological applications in self-organizing piping systems.

Notes

Acknowledgements

EAM and KK acknowledge travel funding from Action CA15109, European Cooperation for Statistics of Network Data Science (COSTNET). KK acknowledges funding from MINECO through the Ramón y Cajal program and through project SPASIMM, FIS2016-80067-P (AEI/FEDER, EU). We thank J. C. Brings Jacobsen for helpful discussions on circulatory physiology and E. Katifori on adaptive networks.

References

  1. 1.
    Davidsen, J., Ebel, H., Bornholdt, S: Emergence of a small world from local interactions: modeling acquaintance networks. Phys. Rev. Lett. 88, 128701 (2002)CrossRefGoogle Scholar
  2. 2.
    Solé, R.V., Pastor-Satorras, R., Smith, E., Kepler, T.B.: A model of large-scale proteome evolution. Adv. Complex Syst. 5(1), 43–54 (2002)CrossRefGoogle Scholar
  3. 3.
    Ispolatov, I., Krapivsky, P.L., Yuryev, A.: Duplication-divergence model of protein interaction network. Phys. Rev. E 71, 061911 (2005)CrossRefGoogle Scholar
  4. 4.
    Gross, T., Blasius, B.: Adaptive coevolutionary networks: a review. J. R. Soc., Interface 5(20), 259–71 (2008).CrossRefGoogle Scholar
  5. 5.
    Herrera, J.L., Cosenza, M.G., Tucci, K., González-Avella, J.C.: General coevolution of topology and dynamics in networks. Europhys. Lett. 95(5), 58006 (2011)CrossRefGoogle Scholar
  6. 6.
    Porter, M.A., Gleeson, J.P.: Dynamical Systems on Networks. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol. 4 (2016)CrossRefGoogle Scholar
  7. 7.
    Kantorovich, L.V.: On the translocation of masses. In: Doklady Akademii Nauk SSSR, vol. 37, pp. 199–201 (1942)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)Google Scholar
  9. 9.
    Corson, F.: Fluctuations and redundancy in optimal transport networks. Phys. Rev. Lett. 104(4), 048703 (2010)CrossRefGoogle Scholar
  10. 10.
    Katifori, E., Szőllősi, G.J., Magnasco, M.O.: Damage and fluctuations induce loops in optimal transport networks. Phys. Rev. Lett. 104(4), 048704 (2010)Google Scholar
  11. 11.
    Hu. D., Cai, D.: Adaptation and optimization of biological transport networks. Phys. Rev. Lett. 111, 138701 (2013)Google Scholar
  12. 12.
    Martens, E.A., Klemm, K.: Transitions from trees to cycles in adaptive flow networks. Front. Phys. 5(62), 1–10 (2017)Google Scholar
  13. 13.
    Jacobsen, J.C.B., Mulvany, M.J., Holstein-Rathlou, N.H.: A mechanism for arteriolar remodeling based on maintenance of smooth muscle cell activation. Am. J. Physiol. Regul. Integr. Comp. Physiol. 294, R1379–R1389 (2008)CrossRefGoogle Scholar
  14. 14.
    Jacobsen, J.C.B., Hornbech, M.S., Holstein-Rathlou, N.H.: A tissue in the tissue: models of microvascular plasticity. Eur. J. Pharm. Sci. 36(1), 51–61 (2009)CrossRefGoogle Scholar
  15. 15.
    Reichold, J., Stampanoni, M., Keller, A.L., Buck, A., Jenny, P., Weber, B.:Vascular graph model to simulate the cerebral blood flow in realistic vascular networks. J. Cereb. Blood Flow Metab. 29(8), 1429–1443 (2009)CrossRefGoogle Scholar
  16. 16.
    Martens, E.A., Klemm, K.: Noise-induced bifurcations in a model of vascular networks. In preparation (2019)Google Scholar
  17. 17.
    Blinder, P., Tsai, P.S., Kaufhold, P.S., Knutsen, P.M., Suhl, H., Kleinfeld, D.: The cortical angiome: an interconnected vascular network with noncolumnar patterns of blood flow. Nat. Neurosci. 16(7), 889–97 (2013)CrossRefGoogle Scholar
  18. 18.
    Poelma, C.: Exploring the potential of blood flow network data. Meccanica 52(3), 489–502 (2017)CrossRefGoogle Scholar
  19. 19.
    Alim, K.: Fluid flows shaping organism morphology. Philos. Trans. R. Soc., B 373(1747), 1–5 (2018)CrossRefGoogle Scholar
  20. 20.
    Campbell, T.A., Tibbits, S., Garrett, B.: The programmable world. Sci. Am. 311(5), 60–65 (2014)CrossRefGoogle Scholar
  21. 21.
    Papadopoulou, A., Laucks, J., Tibbits, S.: From self-assembly to evolutionary structures. Archit. Des. 87(4), 28–37 (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Kgs. LyngbyDenmark
  2. 2.IFISC (CSIC-UIB)Campus Universitat de les Illes Balears, Palma de MallorcaSpain

Personalised recommendations