Branching Processes
27 Jul 2023 12:02
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:
Compartment Models;
Epidemic Models;
Social Contagion
Recommended (introductory):
- Geoffrey Grimmett and David Stirzaker, Probability and Random
Processes [This is my favorite probability textbook, and returns to
branching processes in many places.]
Recommended (forbiddingly technical):
- P. Del Moral and L. Miclo, "Branching and Interacting Particle
Systems Approximations of Feynman-Kac Formulae with Applications to Nonlinear
Filtering", in J. Azema, M. Emery, M. Ledoux and M. Yor
(eds)., Semainaire de Probabilites XXXIV (Springer-Verlag, 2000),
pp. 1--145 [Postscript
preprint. Looks like a trial run for Del Moral's book, below, which I've
yet to read.]
To read:
David Assaf, Larry Goldstein and Ester Samuel-Cahn, "An unexpected
connection between branching processes and optimal stopping", Journal of
Applied Probability 37 (2000):
613--6, math.PR/0510587
[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
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. Martin, David Harris, "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
Su-Chan Park, Joachim Krug, Léo Touzo & Peter Mörters, "Branching with Selection and Mutation I: Mutant Fitness of Fréchet Type",
Journal of Statistical Physics 190 (2023): 115
David Sankoff, "Branching Processes with Terminal Types:
Application to Context-Free Grammars", Journal of Applied
Probability 8 (1971): 233--240
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