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Encyclopedia of Complexity and Systems Science pp 1–18Cite 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 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...

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Abbreviations

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 lose energy when it reaches the system boundary. In the neighborhood of a critical point, the avalanche distribution is described by a power-law distribution.

Critical properties and scaling:

The behavior of equilibrium and many nonequilibrium systems in steady states contains critical points where the systems display scale invariance and the correlation functions exhibit an algebraic behavior characterized by so-called critical exponents. A characteristic 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.

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.

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.

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.

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Author information

Authors and Affiliations

  1. Department of Applied Physics, Aalto University, 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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Correspondence to Mikko J. Alava .

Editor information

Editors and Affiliations

  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Alava, M.J., Lauritsen, K.B. (2015). Branching Processes. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_43-3

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_43-3

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