A Tipping Point in the Structural Formation of Interconnected Networks

  • Alex ArenasEmail author
  • Filippo Radicchi
Part of the Understanding Complex Systems book series (UCS)


The interaction substrate of many natural and synthetic systems is well represented by a complex mesh of networks where information, people and energy flows. These networks are interconnected with each other, and present structural and dynamical features different from those observed in isolated networks. While examples of such dissimilar properties are becoming more abundant, for example diffusion, robustness and competition, it is not yet clear where these differences are rooted. Here we show that the composition of independent networks into an interconnected network of networks undergoes a structurally sharp transition, a tipping point, as the interconnections are formed. Depending on the relative importance of inter- and intra- layer connections, we find that the entire interconnected system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act essentially as independent entities; in the other regime, strong structural correlation arise, and network layers are indistinguishable i.e. the whole system behaves as a single-level network. We analytically show that the transition between the two regimes is discontinuous even for finite size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure, which may manifest new dynamical properties.


Travel Salesman Problem Fitness Landscape Interconnected System Percolation Transition Algebraic Connectivity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Modern Phys. 74(1), 47 (2002)CrossRefADSMathSciNetzbMATHGoogle Scholar
  2. 2.
    Albert, R., Jeong, H., Barabási, A.-L.: Error and attack tolerance of complex networks. Nature 406(6794), 378–382 (2000)CrossRefADSGoogle Scholar
  3. 3.
    Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Barthélemy, M.: Spatial networks. Phys. Rep. 499(1), 1–101 (2011)CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Baxter, G.J., Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Avalanche collapse of interdependent networks. Phys. Rev. Lett. 109(24), 248701 (2012)CrossRefADSGoogle Scholar
  6. 6.
    Bıyıkoglu, T., Leydold, J., Stadler, P.F.: Laplacian Eigenvectors of Graphs. Lecture Notes in Mathematics, vol. 1915. Springer, Berlin/Heidelberg (2007)Google Scholar
  7. 7.
    Blundell, S., Blundell, K.M.: Concepts in Thermal Physics. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  8. 8.
    Bollobás, B., Borgs, C., Chayes, J., Riordan, O.: Percolation on dense graph sequences. Ann. Probab. 38(1), 150–183 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Brummitt, C.D., D’Souza, R.M., Leicht, E.A.: Suppressing cascades of load in interdependent networks. Proc. Natl. Acad. Sci. 109(12), E680–E689 (2012)CrossRefADSGoogle Scholar
  10. 10.
    Buldyrev, S.V., Parshani, R., Paul, G., Stanley, H.E., Havlin, S.: Catastrophic cascade of failures in interdependent networks. Nature 464(7291), 1025–1028 (2010)CrossRefADSGoogle Scholar
  11. 11.
    Castellano, C., Pastor-Satorras, R.: Thresholds for epidemic spreading in networks. Phys. Rev. Lett. 105(21), 218701 (2010)CrossRefADSGoogle Scholar
  12. 12.
    Cavers, M., Fallat, S., Kirkland, S.: On the normalized laplacian energy and general randić index r −1 of graphs. Linear Algebra Appl. 433(1), 172–190 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Chung, F.R.K.: Spectral Graph Theory, vol. 92. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  14. 14.
    Chung, F.R.K., Lu, L., Vu, V.: Spectra of random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA 100(11), 6313–6318 (2003)CrossRefADSMathSciNetzbMATHGoogle Scholar
  15. 15.
    Cohen, R., Ben-Avraham, D., Havlin, S.: Percolation critical exponents in scale-free networks. Phys. Rev. E 66(3), 036113 (2002)CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Courant, R.: Über die eigenwerte bei den differentialgleichungen der mathematischen physik. Mathematische Zeitschrift 7(1), 1–57 (1920)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    De Domenico, M., Solè-Ribalta, A., Cozzo, E., Kivelä, M., Moreno, Y., Porter, M.A., Gòmez, S., Arenas, A.: Mathematical formulation of multi-layer networks. Phys. Rev. X 3, 041022 (2013)Google Scholar
  18. 18.
    De Domenico, M., Solè-Ribalta, A., Gòmez, S., Arenas, A.: Navigability of interconnected networks under random failures. PNAS 111(23), 8351–8356 (2014)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Dickison, M., Havlin, S., Stanley, H.E.: Epidemics on interconnected networks. Phys. Rev. E 85(6), 066109 (2012)CrossRefADSGoogle Scholar
  20. 20.
    Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.: Critical phenomena in complex networks. Rev. Mod. Phys. 80(4), 1275–1335 (2008)CrossRefADSGoogle Scholar
  21. 21.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks. Adv. Phys. 51(4), 1079–1187 (2002)CrossRefADSGoogle Scholar
  22. 22.
    P.G. Doyle, Snell, J.L.: Random Walks and Electric Networks. Carus Mathematical Monographs, vol. 22. Mathematical Association of America, Washington, DC (1984)Google Scholar
  23. 23.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23(2), 298–305 (1973)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Fiedler, M.: A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslov. Math. J. 25(4), 619–633 (1975)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Fiedler, M.: Laplacian of graphs and algebraic connectivity. Banach Cent. Publ. 25(1), 57–70 (1989)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Fischer, E.: Über quadratische formen mit reellen koeffizienten. Monatshefte für Mathematik 16(1), 234–249 (1905)CrossRefzbMATHGoogle Scholar
  27. 27.
    Gao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Robustness of a network of networks. Phys. Rev. Lett. 107(19), 195701 (2011)CrossRefADSGoogle Scholar
  28. 28.
    Gao, J., Buldyrev, S.V., Stanley, H.E., Havlin, S.: Networks formed from interdependent networks. Nat. Phys. 8(1), 40–48 (2012)CrossRefGoogle Scholar
  29. 29.
    Goltsev, A.V., Dorogovtsev, S.N., Oliveira, J.G., Mendes, J.F.F.: Localization and spreading of diseases in complex networks. Phys. Rev. Lett. 109(12), 128702 (2012)CrossRefADSGoogle Scholar
  30. 30.
    Gomez, S., Diaz-Guilera, A., Gomez-Gardeñes, J., Perez-Vicente, C.J., Moreno, Y., Arenas, A.: Diffusion dynamics on multiplex networks. Phys. Rev. Lett. 110(2), 028701 (2013)CrossRefADSGoogle Scholar
  31. 31.
    Granell, C., Gómez, S., Arenas, A.: Dynamical interplay between awareness and epidemic spreading in multiplex networks. Phys. Rev. Lett. 111, 128701 (2013)CrossRefADSGoogle Scholar
  32. 32.
    Granell, C., Gómez, S., Arenas, A.: Competing spreading processes on multiplex networks: awareness and epidemics. Phys. Rev. E 90, 012808 (2014)CrossRefADSGoogle Scholar
  33. 33.
    Grover, L.K.: Local search and the local structure of NP-complete problems. Oper. Res. Lett. 12(4), 235–243 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Harris, T.E.: The Theory of Branching Processes. Courier Dover Publications, New York (2002)zbMATHGoogle Scholar
  35. 35.
    Hashimoto, K.: Zeta functions of finite graphs and representations of p-adic groups. In: Hashimoto, K., Namikawa, Y. (eds.) Automorphic Forms and Geometry of Arithmetic Varieties, pp. 211–280. Academic, Boston (1989)CrossRefGoogle Scholar
  36. 36.
    Hu, Y., Ksherim, B., Cohen, R., Havlin, S.: Percolation in interdependent and interconnected networks: abrupt change from second-to first-order transitions. Phys. Rev. E 84(6), 066116 (2011)CrossRefADSGoogle Scholar
  37. 37.
    Jamakovic, A., Van Mieghem, P.: On the robustness of complex networks by using the algebraic connectivity. In: Das, A., Pung, H.K., Lee, F.B.S., Wong, L.W.C. (eds.) NETWORKING 2008 Ad Hoc and Sensor Networks, Wireless Networks, Next Generation Internet, 7th International IFIP-TC6 Networking Conference. Proceedings, Singapore, 5–9 May 2008. Lecture Notes in Computer Science, pp. 183–194. Springer (2008)Google Scholar
  38. 38.
    Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. J. Complex Netw. 2(3), 203–271 (2014)CrossRefGoogle Scholar
  39. 39.
    Klein, D.J., Randić, M.: Resistance distance. J. Math. Chem. 12(1), 81–95 (1993)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Kolmogorov, V., Zabin, R.: What energy functions can be minimized via graph cuts? IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 147–159 (2004)CrossRefGoogle Scholar
  41. 41.
    Krzakala, F., Moore, C., Mossel, E., Neeman, J., Sly, A., Zdeborová, L., Zhang, P.: Spectral redemption in clustering sparse networks. Proc. Natl. Acad. Sci. 110(52), 20935–20940 (2013)CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Liu, Y.-Y., Slotine, J.-J., Barabási, A.-L.: Controllability of complex networks. Nature 473(7346), 167–173 (2011)CrossRefADSGoogle Scholar
  43. 43.
    Merris, R.: Laplacian graph eigenvectors. Linear Algebra Appl. 278(1), 221–236 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond, vol. 9. World Scientific, Singapore (1987)zbMATHGoogle Scholar
  45. 45.
    Mohar, B., Alavi, Y.: The Laplacian spectrum of graphs. Graph Theory Comb. Appl. 2, 871–898 (1991)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)CrossRefADSMathSciNetzbMATHGoogle Scholar
  47. 47.
    Newman, M.E.J.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. 103(23), 8577–8582 (2006)CrossRefADSGoogle Scholar
  48. 48.
    Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64(2), 026118 (2001)CrossRefADSGoogle Scholar
  49. 49.
    Ng, A.Y., Jordan, M.I., Weiss, Y.: On spectral clustering analysis and an algorithm. In: Proceedings of Advances in Neural Information Processing Systems, vol. 14, pp. 849–856. MIT Press, Cambridge, MA (2001)Google Scholar
  50. 50.
    Parshani, R., Buldyrev, S.V., Havlin, S.: Interdependent networks: reducing the coupling strength leads to a change from a first to second order percolation transition. Phys. Rev. Lett. 105(4), 048701 (2010)CrossRefADSGoogle Scholar
  51. 51.
    Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200 (2001)CrossRefADSGoogle Scholar
  52. 52.
    Radicchi, F.: Driving interconnected networks to supercriticality. Phys. Rev. X 4, 021014 (2014)Google Scholar
  53. 53.
    Radicchi, F., Arenas, A.: Abrupt transition in the structural formation of interconnected networks. Nat. Phys. 9, 717–720 (2013)CrossRefGoogle Scholar
  54. 54.
    Reidys, C.M., Stadler, P.F.: Combinatorial landscapes. SIAM Rev. 44(1), 3–54 (2002)CrossRefADSMathSciNetzbMATHGoogle Scholar
  55. 55.
    Saumell-Mendiola, A., Serrano, M.Á., Boguñá, M.: Epidemic spreading on interconnected networks. Phys. Rev. E 86(2), 026106 (2012)CrossRefADSGoogle Scholar
  56. 56.
    Sole-Ribalta, A., De Domenico, M., Kouvaris, N.E., Diaz-Guilera, A., Gomez, S., Arenas, A.: Spectral properties of the Laplacian of multiplex networks. Phys. Rev. E 88(3), 032807 (2013)CrossRefADSGoogle Scholar
  57. 57.
    Son, S.-W., Bizhani, G., Christensen, C., Grassberger, P., Paczuski, M.: Percolation theory on interdependent networks based on epidemic spreading. EPL (Europhys. Lett.) 97(1), 16006 (2012)Google Scholar
  58. 58.
    Son, S.-W., Grassberger, P., Paczuski, M.: Percolation transitions are not always sharpened by making networks interdependent. Phys. Rev. Lett. 107(19), 195702 (2011)CrossRefADSGoogle Scholar
  59. 59.
    Sood, V., Redner, S.: Voter model on heterogeneous graphs. Phys. Rev. Lett. 94(17), 178701 (2005)CrossRefADSGoogle Scholar
  60. 60.
    Szell, M., Lambiotte, R., Thurner, S.: Multirelational organization of large-scale social networks in an online world. Proc. Natl. Acad. Sci. 107(31), 13636–13641 (2010)CrossRefADSGoogle Scholar
  61. 61.
    Vazquez, F., Eguíluz, V.M.: Analytical solution of the voter model on uncorrelated networks. New J. Phys. 10(6), 063011 (2008)CrossRefGoogle Scholar
  62. 62.
    von Neumann, J., Wigner, E.: Non crossing rule. Zeitschrift für Physik 30, 467–470 (1929)Google Scholar
  63. 63.
    Wilf, H.S.: generatingfunctionology. Academic (1990). Available at Google Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain
  2. 2.Center for Complex Networks and Systems Research, School of Informatics and ComputingIndiana UniversityBloomingtonUSA

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