Branching Processes

A branching process, in its simplest Galton-Watson form, starts with one individual in generation 0. That individual produces a random number of offspring, drawn from some fixed distribution, forming generation 1; each of those offspring independently produces its own random number of offspring forming generation 2, and so on. The question the model was originally built to answer — do aristocratic family surnames die out? — generalizes to any setting where something spreads by each "unit" independently spawning a random count of successors: an epidemic's chain of infections, a nuclear chain reaction, or a viral social-media share cascade.

The whole long-run behavior is governed by one number: $m$, the mean number of offspring per individual. If $m < 1$, the expected population shrinks each generation and the process dies out with probability 1. If $m > 1$, there is a positive probability the population survives forever and, conditional on survival, grows geometrically. If $m = 1$ exactly (the "critical" case), the process still dies out with probability 1, but takes much longer on average to do so than the $m<1$ case — a knife-edge that shows up whenever a system sits exactly at a reproduction threshold.

The exact extinction probability, when $m>1$, is the smallest root in $[0,1]$ of $s = G(s)$, where $G$ is the offspring-count's probability generating function — a fixed-point equation that appears throughout branching-process theory and its applications to queueing and epidemic modeling.