the Galton-Watson process
/ GAWL-ton WOT-son /
The Galton-Watson process is the simplest mathematical model of a reproducing population, and the historical birthplace of branching-process theory. It was introduced to answer a Victorian question — why do aristocratic family surnames die out? — and the answer turns out to depend on a single number, the mean number of children per individual. Start with one ancestor in generation 0. Each individual, independently of everyone else and of the past, produces a random number of children drawn from a fixed offspring distribution; those children form the next generation. The process Z_n is the total population size in generation n.
Formally, fix a probability distribution (p_k){k>=0} on the nonnegative integers, the offspring distribution, with mean m = sum_k k p_k. Let {X(n,i)} be independent copies of this law, X(n,i) being the number of children of the i-th individual in generation n. Then Z_0 = 1 and Z_(n+1) = sum_(i=1)^(Z_n) X(n,i), an empty sum (when Z_n = 0) being 0. This makes (Z_n) a time-homogeneous Markov chain on {0, 1, 2, ...} with the absorbing state 0 (extinction): once the population dies, it stays dead. The key structural facts follow by conditioning on Z_n: the mean grows geometrically, E[Z_n] = m^n, and the whole theory organizes itself around whether m is below, at, or above 1.
Why it matters: the Galton-Watson process is the prototype for every branching model that follows — multitype, continuous-time, continuous-state, spatial — and a workhorse heuristic far beyond demography. The local structure of a sparse random graph (the neighbourhood of a vertex in Erdos-Renyi G(n, c/n)) looks like a Galton-Watson tree with Poisson(c) offspring, which is exactly why the giant-component threshold sits at mean degree 1. The standing assumptions are real: offspring counts must be independent and identically distributed across individuals and generations, with no interaction, no carrying capacity, and no immigration. Drop independence or add density-dependence and you leave the Galton-Watson world.
Suppose each individual has 0 children with probability 1/2 and 2 children with probability 1/2, so m = 1 (critical). Starting from one ancestor, the population is a fair random walk in disguise: it either dies out quickly or grows, with E[Z_n] = 1 for all n yet extinction certain. By contrast m = 1.1 (each having on average 1.1 offspring) gives a positive chance of indefinite survival.
One ancestor, iid offspring, no interaction: survival is decided by the mean m alone.
The deterministic mean E[Z_n] = m^n is a treacherous guide: a critical process has E[Z_n] = 1 for all n yet dies out with probability one. Averages and almost-sure behaviour diverge sharply at criticality.