Mathematical Statistics · Random Branching Processes
The Galton–Watson Process
Reference
The Galton–Watson process is a probabilistic model of a branching population.
Each particle independently produces a random number of offspring according to
the same distribution. The key parameter is the expected number of offspring
\(m = \mathbb{E}[\xi]\). For \(m < 1\) the population is guaranteed to die out,
while for \(m > 1\), with probability \(1-q > 0\), it can grow without bound.
Alive in generation T
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Extinct trees
Extinct tree
A tree that did not survive to the last generation \(T\).
In the subcritical regime, all trees become extinct with probability 1.
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Share of the dominant tree
Giant component
The giant-component effect is that, as the generation number grows, the share of descendants coming from some single particle tends to 100%.
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Theoretical q
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Distribution of the offspring number ξ
Generating function
\[f(s) = \mathbb{E}[s^\xi] = e^{\lambda(s-1)}\]
The extinction probability is the smallest root of \(f(q)=q\) in \([0,1]\).
Distribution of tree sizes in the process (generation T)
Tree of a starting particle
All descendants of one starting particle form its tree.
The histogram shows the size of each tree in generation \(T\).
Forest of the process — trees colored by starting particle