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Complex Networks

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Statistical Physics of Complex Systems

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

Complex networks have raised a lot of attention in the last decades, due in part to the availability of large data sets describing real systems.

This chapter introduces the basis types of random networks and some of their elementary statistical properties. Dynamics on complex networks is discussed on the example of epidemic as well as rumor propagations on complex networks. The chapter ends with a brief discussion of formal neural networks.

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Notes

  1. 1.

    Unfortunately, these rather natural denominations lead to invert the role of the notations S and I with respect to the epidemic case, possibly leading to some confusion.

References

  1. Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47 (2002)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  2. Amaral, L.A.N., Scala, A., Barthélémy, M., Stanley, H.E.: Classes of small-world networks. Proc. Natl. Acad. Sci. USA 97, 11149 (2000)

    Article  ADS  Google Scholar 

  3. Amit, D.J.: Modeling Brain Function: The World of Attractor Neural Networks. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  4. Amit, D.J., Gutfreund, H., Sompolinsky, H.: Spin-glass models of neural networks. Phys. Rev. A 32, 1007 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  5. Amit, D.J., Gutfreund, H., Sompolinsky, H.: Storing infinite numbers of patterns in a spin-glass model of neural networks. Phys. Rev. Lett. 55, 1530 (1985)

    Article  ADS  Google Scholar 

  6. Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford (1992)

    Google Scholar 

  7. Angel, A.G., Hanney, T., Evans, M.R.: Condensation transitions in a model for a directed network with weighted links. Phys. Rev. E 73, 016105 (2006)

    Google Scholar 

  8. Aron, J.L., Gove, R.A., Azadegan, S., Schneider, M.C.: The benefits of a notification process in addressing the worsening computer virus problem: results of a survey and a simulation model. Comput. Secur. 20, 693 (2001)

    Article  Google Scholar 

  9. Bailey, N.T.: The mathematical theory of infectious diseases. Griffin (1975)

    Google Scholar 

  10. Barrat, A., Barthélemy, M., Pastor-Satorras, R., Vespignani, A.: The architecture of complex weighted networks. Proc. Natl. Acad. Sci. USA 101, 3747 (2004)

    Article  MATH  ADS  Google Scholar 

  11. Barrat, A., Barthélémy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  12. Barthélemy, M., Barrat, A., Pastor-Satorras, R., Vespignani, A.: Velocity and hierarchical spread of epidemic outbreaks in complex scale-free networks. Phys. Rev. Lett. 92, 178701 (2004)

    Google Scholar 

  13. Barthélemy, M., Barrat, A., Pastor-Satorras, R., Vespignani, A.: Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. J. Theor. Biol. 235, 275 (2005)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. Boguñá, M., Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in complex networks with degree correlations. Lect. Notes Phys. 652, 127 (2003)

    Article  MATH  ADS  Google Scholar 

  15. Bornholdt, S., Ebel, H.: World wide web scaling exponent from Simon’s 1955 model. Phys. Rev. E 64, 035104(R) (2001)

    Article  ADS  Google Scholar 

  16. Braunstein, A., Mézard, M., Zecchina, R.: Survey propagation: an algorithm for satisfiability. Rand. Struct. Algor. 27, 201 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Braunstein, A., Zecchina, R.: Learning by message passing in networks of discrete synapses. Phys. Rev. Lett. 96, 030201 (2006)

    Google Scholar 

  18. Colizza, V., Barrat, A., Barthélemy, M., Vespignani, A.: The role of the airline transportation network in the prediction and predictability of global epidemics. Proc. Natl. Acad. Sci. USA 103, 2015 (2006)

    Article  MATH  ADS  Google Scholar 

  19. Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on the Theory of Computing, p. 151 (1971)

    Google Scholar 

  20. Daley, D.J., Gani, J.: Epidemic Modelling: An Introduction. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  21. Daley, D.J., Kendall, D.G.: Epidemics and rumours. Nature 204, 1118 (1964)

    Article  ADS  Google Scholar 

  22. Daley, D.J., Kendall, D.G.: Stochastic rumours. IMA J. Appl. Math. 1, 42 (1965)

    Article  MathSciNet  Google Scholar 

  23. Derrida, B., Gardner, E., Zippelius, A.: An exactly solvable asymmetric neural network model. EPL 4, 167 (1987)

    Article  ADS  Google Scholar 

  24. Evans, M.R., Hanney, T.: Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A Math. Gen. 38, R195 (2005)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  25. Friedgut, E.: Sharp thresholds of graph properties, and the K-Sat problem. J. Amer. Math. Soc. 12, 1017 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Harko, T., Lobo, F.S.N., Mak, M.K.: Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates. Appl. Math. Comput. 236, 184 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Hethcote, H.W., Yorke, J.A.: Gonorrhea: transmission and control. Lect. Notes Biomath. 56, 1 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79, 2554 (1982)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  29. Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81, 3088 (1984)

    Article  MATH  ADS  Google Scholar 

  30. Kephart, J.O., White, S.R., Chess, D.M.: Computers and epidemiology. IEEE Spect. 30, 20 (1993)

    Article  Google Scholar 

  31. Kirkpatrick, S., Selman, B.: Critical behavior in the satisfiability of random Boolean expressions. Science 264, 1297 (1994)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  32. Krauth, W., Mézard, M.: Storage capacity of memory networks with binary couplings. J. Phys. France 50, 3057 (1989)

    Article  Google Scholar 

  33. Kröger, M., Schlickeiser, R.: Analytical solution of the SIR-model for the temporal evolution of epidemics. Part A: time-independent reproduction factor. J. Phys. A: Math. Theor. 53, 505601 (2020)

    Google Scholar 

  34. Kschischang, F.R., Frey, B.J., Loeliger, H.A.: Factor graphs and the sumproduct algorithm. IEEE Trans. Inf. Theory 47, 498 (2001)

    Article  MATH  Google Scholar 

  35. Little, W.A.: The existence of persistent states in the brain. Math. Biosci. 19, 101 (1974)

    Article  MATH  Google Scholar 

  36. Maki, D.P., Thompson, M.: Mathematical Models and Applications, with Emphasis on the Social, Life and Management Sciences. Prentice-Hall, Hoboken (1973)

    Google Scholar 

  37. May, R.M., LLoyd, A.L.: Infection dynamics on scale-free networks. Phys. Rev. E 64, 066112 (2001)

    Google Scholar 

  38. Mézard, M., Montanari, A.: Information, Physics and Computation. Oxford University Press, Oxford (2009)

    Book  MATH  Google Scholar 

  39. Mézard, M., Mora, T.: Constraint satisfaction problems and neural networks: a statistical physics perspective. J. Physiol. Paris 103, 107 (2008)

    Article  Google Scholar 

  40. Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812 (2003)

    Article  ADS  Google Scholar 

  41. Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity from characteristic phase transitions. Nature 400, 133 (1999)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  42. Moreno, Y., Pastor-Satorras, R., Vespignani, A.: Epidemic outbreaks in complex heterogeneous networks. Eur. Phys. J. B 26, 521 (2002)

    Article  ADS  Google Scholar 

  43. Newman, M.E.J.: Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002)

    Google Scholar 

  44. Newmann, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118 (2001)

    Google Scholar 

  45. O’Connor, D.H., Wittenberg, G.M., Wang, S.S.H.: Graded bidirectional synaptic plasticity is composed of switch-like unitary events. Proc. Natl. Acad. Sci. USA 102, 9679 (2005)

    Article  ADS  Google Scholar 

  46. Parisi, G.: Asymmetric neural networks and the process of learning. J. Phys. A 19, L675 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  47. Parisi, G.: Attractor neural networks (1994). arxiv:cond-mat/9412030

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

    Article  MathSciNet  ADS  Google Scholar 

  49. Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200 (2001)

    Article  ADS  Google Scholar 

  50. Petersen, C.C.H., Malenka, R.C., Nicoll, R.A., Hopfield, J.J.: All-or-none potentiation at Ca3-Ca1 synapses. Proc. Natl. Acad. Sci. USA 95, 4732 (1998)

    Article  ADS  Google Scholar 

  51. Rosenblatt, F.: Principles of Neurodynamics: Perceptions and the Theory of Brain Mechanisms. Spartan Books, New York (1962)

    Google Scholar 

  52. Saichev, A., Malevergne, Y., Sornette, D.: Theory of Zipf’s law and beyond. Lecture Notes in Economics and Mathematical Systems, vol. 632. Springer, Berlin (2009)

    Google Scholar 

  53. Selman, B., Kirkpatrick, S.: Critical behavior in the computational cost of satisfiability testing. Artif. Intell. 81, 273 (1996)

    Article  MathSciNet  Google Scholar 

  54. Simon, H.A.: On a class of skew distribution functions. Biometrika 42, 425 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  55. Sudbury, A.: The proportion of the population never hearing a rumor. J. Appl. Prob. 22, 443 (1985)

    Article  MATH  Google Scholar 

  56. Watts, D.J., Strogatz, S.H.: Collective dynamics of “small-world” networks. Nature 393, 440 (1998)

    Google Scholar 

  57. Yedidia, J.S., Freeman, W.T., Weiss, Y.: Advances in Neural Information Processing Systems (NIPS), chap. Generalized belief propagation, p. 689. MIT Press, Cambridge (2001)

    Google Scholar 

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Authors and Affiliations

  1. LIPhy, CNRS and Université Grenoble Alpes, Grenoble, France

    Eric Bertin

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  1. Eric Bertin
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Bertin, E. (2021). Complex Networks. In: Statistical Physics of Complex Systems. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-030-79949-6_6

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