Galton–Watson process


The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. The process models family names as patrilineal, while offspring are randomly either male or female, and names become extinct if the family name line dies out. This is an accurate description of Y chromosome transmission in genetics, and the model is thus useful for understanding human Y-chromosome DNA haplogroups, and is also of use in understanding other processes ; but its application to actual extinction of family names is fraught. In practice, family names change for many other reasons, and dying out of name line is only one factor, as discussed in [|examples], below; the Galton–Watson process is thus of limited applicability in understanding actual family name distributions.
There was concern amongst the Victorians that aristocratic surnames were becoming extinct. Galton originally posed a mathematical question regarding the distribution of surnames in an idealized population in an 1873 issue of The Educational Times, and the Reverend Henry William Watson replied with a solution. Together, they then wrote an 1874 paper titled "On the probability of the extinction of families" in the Journal of the Anthropological Institute of Great Britain and Ireland. Galton and Watson appear to have derived their process independently of the earlier work by I. J. Bienaymé; see Heyde and Seneta 1977. For a detailed history see Kendall.

Concepts

Assume, for the sake of the model, that surnames are passed on to all male children by their father. Suppose the number of a man's sons to be a random variable distributed on the set . Further suppose the numbers of different men's sons to be independent random variables, all having the same distribution.
Then the simplest substantial mathematical conclusion is that if the average number of a man's sons is 1 or less, then their surname will almost surely die out, and if it is more than 1, then there is more than zero probability that it will survive for any given number of generations.
Modern applications include the survival probabilities for a new mutant gene, or the initiation of a nuclear chain reaction, or the dynamics of disease outbreaks in their first generations of spread, or the chances of extinction of small population of organisms; as well as explaining why only a handful of males in the deep past of humanity now have any surviving male-line descendants, reflected in a rather small number of distinctive human Y-chromosome DNA haplogroups.
A corollary of high extinction probabilities is that if a lineage has survived, it is likely to have experienced, purely by chance, an unusually high growth rate in its early generations at least when compared to the rest of the population.

Mathematical definition

A Galton–Watson process is a stochastic process which evolves according to the recurrence formula X0 = 1 and
where is a set of independent and identically-distributed natural number-valued random variables.
In the analogy with family names, Xn can be thought of as the number of descendants in the nth generation, and can be thought of as the number of children of the jth of these descendants. The recurrence relation states that the number of descendants in the n+1st generation is the sum, over all nth generation descendants, of the number of children of that descendant.
The extinction probability is given by
This is clearly equal to zero if each member of the population has exactly one descendant. Excluding this case there exists
a simple necessary and sufficient condition, which is given in the next section.

Extinction criterion for Galton–Watson process

In the non-trivial case the probability of final extinction is equal to one if E ≤ 1 and strictly less than one if E > 1.
The process can be treated analytically using the method of probability generating functions.
If the number of children ξ j at each node follows a Poisson distribution with parameter λ, a particularly simple recurrence can be found for the total extinction probability xn for a process starting with a single individual at time n = 0:
giving the above curves.

Bisexual Galton–Watson process

In the classical Galton-Watson process described above, only men are considered, effectively modeling reproduction as asexual. A model more closely following actual sexual reproduction is the so-called "bisexual Galton-Watson process", where only couples reproduce. In this process, each child is supposed as male or female, independently of each other, with a specified probability, and a so-called "mating function" determines how many couples will form in a given generation. As before, reproduction of different couples are considered to be independent of each other. Now the analogue of the trivial case corresponds to the case of each male and female reproducing in exactly one couple, having one male and one female descendant, and that the mating function takes the value of the minimum of the number of males and females.
Since the total reproduction within a generation depends now strongly on the mating function, there exists in general no simple necessary and sufficient condition for final extinction as is the case in the classical Galton-Watson process. However, excluding the non-trivial case, the concept of the averaged reproduction mean allows for a general sufficient condition for final extinction, treated in the next section.

Extinction criterion

If in the non-trivial case the averaged reproduction mean per couple stays bounded over all generations and will not exceed 1 for a sufficiently large population size, then the probability of final extinction is always 1.

Examples

Citing historical examples of Galton–Watson process is complicated due to the history of family names often deviating significantly from the theoretical model. Notably, new names can be created, existing names can be changed over a person's lifetime, and people historically have often assumed names of unrelated persons, particularly nobility. Thus, a small number of family names at present is not in itself evidence for names having become extinct over time, or that they did so due to dying out of family name lines – that requires that there were more names in the past and that they die out due to the line dying out, rather than the name changing for other reasons, such as vassals assuming the name of their lord.
Chinese names are a well-studied example of surname extinction: there are currently only about 3,100 surnames in use in China, compared with close to 12,000 recorded in the past, with 22% of the population sharing the names Li, Wang and Zhang, and the top 200 names covering 96% of the population. Names have changed or become extinct for various reasons such as people taking the names of their rulers, orthographic simplifications, taboos against using characters from an emperor's name, among others. While family name lines dying out may be a factor in the surname extinction, it is by no means the only or even a significant factor. Indeed, the most significant factor affecting the surname frequency is other ethnic groups identifying as Han and adopting Han names. Further, while new names have arisen for various reasons, this has been outweighed by old names disappearing.
By contrast, some nations have adopted family names only recently. This means both that they have not experienced surname extinction for an extended period, and that the names were adopted when the nation had a relatively large population, rather than the smaller populations of ancient times. Further, these names have often been chosen creatively and are very diverse. Examples include:
On the other hand, some examples of high concentration of family names is not primarily due to the Galton–Watson process: