Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Branching Processes

  • Mikko J. Alava
  • Kent Bækgaard Lauritsen
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_43-3

Definition of the Subject

Consider the fate of a human population on a small, isolated island. It consists of a certain number of individuals, and the most obvious question, of importance in particular for the inhabitants of the island, is whether this number will go to zero. Humans die and reproduce in steps of one, and therefore one can try to analyze this fate mathematically by writing down what is called master equations, to describe the dynamics as a “branching process” (BP). The branching here means that if at time t = 0 there are N humans, at the next step t = 1 there can be N − 1 (or N + 1 or N + 2 if the only change from t = 0 was that a pair of twins was born). The outcome will depend in the simplest case on a “branching number” or the number of offsprings λ that a human being will have (Harris 1989; Kimmel and Axelrod 2002; Athreya and Ney 2004; Haccou et al. 2005).

If the offsprings created are too few, then the population will decay or reach an “absorbing state” out of...

Keywords

Acoustic Emission Event Tauberian Theorem Sandpile Model Branch Process Avalanche Size Distribution 
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.
This is a preview of subscription content, log in to check access.

Bibliography

Primary Literature

  1. Alava M (2003) Self-organized criticality as a phase transition. In: Korutcheva E, Cuerno R (eds) Advances in condensed matter and statistical physics. arXiv:cond-mat/0307688; (2004) Nova Publishers, p 45Google Scholar
  2. Alava MJ, Dorogovtsev SN (2005) Complex networks created by aggregation. Phys Rev E 71:036107ADSCrossRefGoogle Scholar
  3. Alava MJ, Nukala PKNN, Zapperi S (2006) Statistical models of fracture. Adv Phys 55:349–476ADSCrossRefGoogle Scholar
  4. Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47Google Scholar
  5. Alstrøm P (1988) Mean field exponents for self-organized critical phenomena. Phys Rev A 38:4905ADSCrossRefGoogle Scholar
  6. Asmussen S, Hering H (1983) Branching processes. Birkhäuser, BostonCrossRefzbMATHGoogle Scholar
  7. Athreya KB, Ney PE (2004) Branching processes. Dover, MineolaGoogle Scholar
  8. Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality: An explanation of the 1/f noise. Phys Rev Lett 59:381; (1988) Self-organized criticality Phys Rev A 38:364Google Scholar
  9. Barrat A, Barthelemy M, Pastor-Satorras R, Vespignani A (2004a) The architecture of complex weighted network, 2004b: Weighted evolving networks: coupling topology and weights dynamics. Proc Natl Acad Sci U S A 101:3747ADSCrossRefGoogle Scholar
  10. Barrat A, Barthelemy M, Vespignani A (2004b) Phys Rev Lett 92:228701ADSCrossRefGoogle Scholar
  11. Bonachela JA, Chate H, Dornic I, Munoz MA (2007) Absorbing States and Elastic Interfaces in Random Media: Two Equivalent Descriptions of Self-Organized Criticality. Phys Rev Lett 98:115702CrossRefGoogle Scholar
  12. Chialvo DR (2010) Emergent complex neural dynamics. Nat Phys 6:744CrossRefGoogle Scholar
  13. Christensen K, Olami Z (1993) Sandpile models with and without an underlying spatial structure. Phys Rev E 48:3361ADSCrossRefGoogle Scholar
  14. Colizza V, Barrat A, Barthelemy M, Vespignani A (2006) The role of the airline transportation network in the prediction and predictability of global epidemics. Proc Natl Acad Sci U S A 103:2015ADSCrossRefzbMATHGoogle Scholar
  15. Corral A, Font-Clos F (2012) Criticality and Self-Organization in Branching Processes: Application to Natural Hazards, in Self-Organized Criticality Systems, Markus J. Aschwanden (Ed.), Open Academic Press, Berlin Warsaw (2013) arXiv:1207.2589Google Scholar
  16. Dhar D, Majumdar SN (1990) Abelian sandpile model on the bethe lattice. J Phys A 23:4333ADSMathSciNetCrossRefGoogle Scholar
  17. Dickman R, Munoz MA, Vespignani A, Zapperi S (2000) Paths to self-organized criticality. Braz J Phys 30:27ADSCrossRefGoogle Scholar
  18. Dickman R, Alava MJ, Munoz MA, Peltola J, Vespignani A, Zapperi S (2001) Critical behavior of a one-dimensional fixed-energy stochastic sandpile. Phys Rev E64:056104ADSGoogle Scholar
  19. Dorogovtsev SN, Mendes JFF (2003) Evolution of networks: from biological nets to the Internet and WWW. Oxford University Press, Oxford; (2002) Adv Phys 51:1079; (2004) arXiv:cond-mat/0404593CrossRefzbMATHGoogle Scholar
  20. Dorogovtsev SN, Goltsev AV, Mendes JFF (2007) Critical phenomena in complex networks. Rev Mod Phys 80, 1275 (2008). arXiv:cond-mat/0750.0110Google Scholar
  21. Feller W (1971) An introduction to probability theory and its applications, vol 2, 2nd edn. Wiley, New YorkGoogle Scholar
  22. Flyvbjerg H, Sneppen K, Bak P (1993) Mean field theory for a simple model of evolution. Phys Rev Lett 71:4087; de Boer J, Derrida B, Flyvbjerg H, Jackson AD, Wettig T (1994) Simple model of self-organized biological evolution. Phys Rev Lett 73:906Google Scholar
  23. Garcia-Pelayo R (1994) Dimension of branching processes and self-organized criticality. Phys Rev E 49:4903ADSCrossRefGoogle Scholar
  24. Hui Z, Zi-You G, Gang Y, Wen-Xu W (2006) Self-organization of topology and weight dynamics on networks from merging and regeneration. Chin Phys Lett 23:275ADSCrossRefGoogle Scholar
  25. Janowsky SA, Laberge CA (1993) Exact solutions for a mean-field Abelian sand- pile. J Phys A 26:L973ADSMathSciNetCrossRefGoogle Scholar
  26. Kello CT (2013) Critical branching neural networks. Psychol Rev 120:230CrossRefGoogle Scholar
  27. Kim BJ, Trusina A, Minnhagen P, Sneppen K (2005) Self organized scale-free networks from merging and regeneration. Eur Phys J B43:369ADSCrossRefGoogle Scholar
  28. Lauritsen KB, Zapperi S, Stanley HE (1996) Self-organized branching processes: Avalanche models with dissipation. Phys Rev E 54:2483ADSCrossRefGoogle Scholar
  29. Laurson L, Illa X, Santucci S, Tallakstad KT, Måløy KJ, Alava MJ (2013) Evolution of the average avalanche shape with the universality class. Nature Communications 4:2927Google Scholar
  30. Lippidello E, Godano C, de Arcangelis L (2007) Influence of time and space correlations on earthquake magnitude. Phys Rev Lett 98:098501ADSCrossRefGoogle Scholar
  31. Lubeck S (2004) Universal scaling behavior of non-equilibrium phase transitions. Int J Mod Phys B18:3977ADSCrossRefzbMATHGoogle Scholar
  32. Manna SS (1991) Two-state model of self-organized criticality. J Phys A 24:L363. In this two-state model, the energy takes the two stable values, z i = 0(empty) and z i = 1(particle). When z iz c , with z c = 2, the site relaxes by distributing two particles to two randomly chosen neighborsGoogle Scholar
  33. Manna SS, Kiss LB, Kertész J (1990) Cascades and self-organized criticalityu. J Stat Phys 61:923ADSCrossRefGoogle Scholar
  34. Ogata Y (1988) Statistical models for earthquake occurrences and residual analysis for point processes. J Am Stat Assoc 83:9CrossRefGoogle Scholar
  35. Pastor-Satorras R, Vespignani A (2001) Epidemic Spreading in Scale-Free Networks. Phys Rev Lett 86:3200ADSCrossRefGoogle Scholar
  36. Pastor-Satorras R, Vespignani A (2004) Evolution and structure of the internet: a statistical physics approach. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  37. Saichev A, Helmstetter A, Sornette D (2005) Anomalous Scaling of offspring and generation numbers in branching processes. Pure Appl Geophys 162:1113ADSCrossRefGoogle Scholar
  38. Stella AL, De Menech M (2001) Mechanisms of avalanche dynamics and forms of scaling in sandpiles. Physica A 295:1001CrossRefzbMATHGoogle Scholar
  39. Tadic B, Ramaswamy R (1996) Defects in self-organized criticality: A directed coupled map lattice model. Phys Rev E 54:3157ADSCrossRefGoogle Scholar
  40. Tadić B, Nowak U, Usadel KD, Ramaswamy R, Padlewski S (1992) Scaling behavior in disordered sandpile automata. Phys Rev A 45:8536ADSCrossRefGoogle Scholar
  41. Tang C, Bak P (1988) J Stat Phys 51:797ADSMathSciNetCrossRefGoogle Scholar
  42. Tebaldi C, De Menech M, Stella AL (1999) Multifractal scaling in the Bak-Tang-Wiesenfeld sandpile and edge events. Phys Rev Lett 83:3952ADSCrossRefGoogle Scholar
  43. Vazquez A (2006) Polynomial growth in age-dependent branching processes with diverging reproductive number. Phys Rev Lett 96:038702ADSCrossRefGoogle Scholar
  44. Vazquez A, Balazs R, Andras L, Barabasi AL (2007) Impact of non-Poisson activity patterns on spreading processes. Phys Rev Lett 98:158702ADSCrossRefGoogle Scholar
  45. Vespignani A, Zapperi S, Pietronero L (1995) Renormalization approach to the self-organized critical behavior of sandpile models. Phys Rev E 51:1711ADSCrossRefGoogle Scholar
  46. Vespignani A, Dickman R, Munoz MA, Zapperi S (2000) Absorbing-state phase transitions in fixed-energy sandpiles. Phys Rev E 62:4564ADSCrossRefGoogle Scholar
  47. Yook SH, Jeong H, Barabasi AL, Tu Y (2001) Weighted evolving networks. Phys Rev Lett 86:5835ADSCrossRefGoogle Scholar
  48. Zapperi S, Lauritsen KB, Stanley HE (1995) Self-organized branching processes: mean-field theory for avalanches. Phys Rev Lett 75:4071ADSCrossRefGoogle Scholar
  49. Zapperi S, Castellano C, Colaiori F, Durin G (2005) Signature of effective mass in crackling-noise asymmetry. Nat Phys 1:46CrossRefGoogle Scholar

Books and Reviews

  1. Haccou P, Jagers P, Vatutin VA (2005) Branching processes: variation, growth, and extinction of populations. Cambridge University Press, CambridgeGoogle Scholar
  2. Harris TE (1989) The theory of branching processes. Dover, New YorkGoogle Scholar
  3. Jensen HJ (1998) Self-organized criticality. Cambridge University Press, CambridgeGoogle Scholar
  4. Kimmel M, Axelrod DE (2002) Branching processes in biology. Springer, New YorkGoogle Scholar
  5. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167Google Scholar
  6. Weiss GH (1994) Aspects and applications of the random walk. North-Holland, AmsterdamGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Applied PhysicsAalto UniversityEspooFinland
  2. 2.Research DepartmentDanish Meteorological InstituteCopenhagenDenmark