Branching Processes

04 Dec 2014 18:23

A class of stochastic process important as models in genetics and population biology, chemical kinetics, and filtering. The basic idea is that there are a number of objects, often called particles, which, in some random fashion, reproduce ("branch") and die out; they can be of multiple types and occupy differing spatial locations. They can pursue their trajectories and their biographies either independently, or with some kind of statistical dependence across particles.

The most basic version has one type of particle, and no spatial considerations. At each time step, each parrticle gives rise to a random number of offspring; the distribution of offspring is fixed, and the number is independent across time-steps and across lineages (IID). This is the so-called Galton-Watson branching process. Galton introduced it as a model of the survival of (patrilneal) family names, so that only male offspring counted; he required the distribution of time until a given lineage went extinct. This was provided almost immediately by Watson, in a very elegant use of the method of generating functions, which is, itself, reproduced in probability textbooks down to the present day. (However, when I first encoutnered the problem, in a probability class, the teacher presented it as one about the survival of matrilineal lineages, defined by inheritance of mitochondrial DNA. Whether this was conscious subversion of the patriarchy, or just a reflection of the changing scientific interests between the 1890s and the 1990s, I couldn't say.)

See also: Epidemiology; Social Contagion

To read:

  • David Assaf, Larry Goldstein and Ester Samuel-Cahn, "An unexpected connection between branching processes and optimal stopping", math.PR/0510587 = Journal of Applied Probability 37 (2000): 613--6 [This sounds like a nice pedagogical topic for a course in stochastic processes. I teach a course in stochastic processes....]
  • Michael Assaf and Baruch Meerson, "Spectral Theory of Metastability and Extinction in Birth-Death Systems", Physical Review Letters 97 (2006): 200602 = cond-mat/0610415
  • Krishna B. Athreya, Branching Processes
  • K. B. Athreya, A.P. Ghosh, S. Sethuraman, "Growth of preferential attachment random graphs via continuous-time branching processes", math.PR/0701649
  • Ellen Baake, Hans-Otto Georgii, "Mutation, selection, and ancestry in branching models: a variational approach", q-bio.PE/0611018
  • Romulus Breban, Raffaele Vardavas and Sally Blower, "Linking population-level models with growing networks: A class of epidemic models", Physical Review E 72 (2005): 046110
  • Nicolas Champagnat, Régis Ferrière, Sylvie Méléar, "Individual-based probabilistic models of adaptive evolution and various scaling approximations", math.PR/0510453
  • Charles R. Doering, Khachik V. Sargsyan and Leonard M. Sander, "Extinction times for birth-death processes: exact results, continuum asymptotics, and the failure of the Fokker-Planck approximation", q-bio/0401016
  • Pierre Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems [This looks really, really cool]
  • Janos Englander, "Branching diffusions, superdiffusions and random media", Probability Surveys 4 (2007): 303--364, arxiv:0710.0236
  • Benjamin Golub and Matthew O. Jackson, "Using selection bias to explain the observed structure of Internet diffusions", Proceedings of the National Academy of Sciences (USA) 107 (2010): 10833--10836
  • Vicenc Gomez, Hilbert J. Kappen and Andreas Kaltenbrunner, "Modeling the structure and evolution of discussion cascades", arxiv:1011.0673
  • P. Haccou et al., Branching Processes: Variation, Growth, and Extinction of Populations
  • Jose Luis Iribarren and Esteban Moro, "Branching Dynamics of Viral Information Spreading", <cite>Physical Review E 84 (2011): 046116
  • Predrag R. Jelenkovic, Jian Tan, "Modulated Branching Processes, Origins of Power Laws and Queueing Duality", 0709.4297
  • Junghyo Jo, Jean-Yves Fortin, M. Y. Choi, "Weibull-type limiting distribution for replicative systems", Physical Review E 83 (2011): 031123, arxiv:1103.3038
  • Jean-Francois Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations
  • Brendan P. M. McCabe1, Gael M. Martin2, David Harris3, "Efficient probabilistic forecasts for counts", Journal of the Royal Statistical Society B 73 (2011): 253--272
  • Sebastian Müller, "Strong recurrence for branching Markov chains", arxiv:0710.4651
  • Fabricio Murai, Bruno Ribeiro, Don Towsley, Krista Gile, "Characterizing Branching Processes from Sampled Data", arxiv:1302.5847
  • Victor M. Panaretos, "Partially observed branching processes for stochastic epidemics", Journal of Mathematical Biology 54 (2007): 645--668
  • David Sankoff, "Branching Processes with Terminal Types: Application to Context-Free Grammars", Journal of Applied Probability 8 (1971): 233--240 [JSTOR]
  • D. Sornette and S. Utkin, "Limits of declustering methods for disentangling exogenous from endogenous events in time series with foreshocks, main shocks, and aftershocks", Physical Review E 79 (2009): 061110, arxiv:0903.3217

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