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Encyclopedia of Complexity and Systems Science pp 644–657Cite as

Branching Processes

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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 mostobvious question, of importance in particular for the inhabitants of the island, is whether this number will go to zero. Humans die and reproduce in stepsof one, and therefore one can try to analyze this fate mathematically by writing down what is called master equations, to describe the dynamics asa “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 “branchingnumber”, or the number of offspring λ that a human being will have [1,2,3,4].

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

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Abbreviations

Markov process:

A  process characterized by a set of probabilities to go from a certain state at time t to another state at time \( { t+1 } \). These transition probabilities are independent of the history of the process and only depend on a fixed probability assigned to the transition.

Critical properties and scaling:

The behavior of equilibrium and many non‐equilibrium systems in steady states contain critical points where the systems display scale invariance and the correlation functions exhibit an algebraic behavior characterized by so‐called critical exponents. A characteristics of this type of behavior is the lack of finite length and time scales (also reminiscent of fractals). The behavior near the critical points can be described by scaling functions that are universal and that do not depend on the detailed microscopic dynamics.

Avalanches:

When a system is perturbed in such a way that a disturbance propagates throughout the system one speak of an avalanche. The local avalanche dynamics may either conserve energy (particles) or dissipate energy. The avalanche may also loose energy when it reaches the system boundary. In the neighborhood of a critical point the avalanche distribution is described by a power‐law distribution.

Self‐organized criticality (SOC):

SOC is the surprising “critical” state in which many systems from physics to biology to social ones find themselves. In physics jargon, they exhibit scale‐invariance, which means that the dynamics – consisting of avalanches – has no typical scale in time or space. The really necessary ingredient is that there is a hidden, fine‐tuned balance between how such systems are driven to create the dynamic response, and how they dissipate the input (“energy”) to still remain in balance.

Networks:

These are descriptions of interacting systems, where in graph theoretical language nodes or vertices are connected by links or edges. The interesting thing can be the structure of the network, and its dynamics, or that of a process on the top of it like the spreading of computer viruses on the Internet.

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    Google Scholar 

  2. Asmussen S, Hering H (1883) Branching Processes. Birkhäuser, Boston

    Google Scholar 

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    Google Scholar 

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    MATH  Google Scholar 

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    Google Scholar 

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    Google Scholar 

  7. Jensen HJ (1998) Self‐Organized Criticality. Cambridge University Press, Cambridge

    MATH  Google Scholar 

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Acknowledgments

We are grateful to our colleague Stefano Zapperi with whom we have collaborated on topics related to networks, avalanches, and branchingprocesses. This work was supported by the Academy of Finland through the Center of Excellence program (M.J.A.) and EUMETSAT's GRAS Satellite ApplicationFacility (K.B.L.).

Author information

Authors and Affiliations

  1. Department of Engineering Physics, Espoo University of Technology, Espoo, Finland

    Mikko J. Alava

  2. Research Department, Danish Meteorological Institute, Copenhagen, Denmark

    Kent Bækgaard Lauritsen

Authors
  1. Mikko J. Alava
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  2. Kent Bækgaard Lauritsen
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2009 Springer-Verlag

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Alava, M.J., Lauritsen, K. (2009). Branching Processes. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_43

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