Maths question (I think) about families

[Rogin the Armchair fan]

Assume a population where everyone follows the British/American norm of taking their father's surname as their own.

That then gets passed on down the generations. When girl children with surname X marry, they take the boy's surname Y instead, and then so do all their children (boys or girls). And so on.

Maths question bit Given that some families will only have daughters, and some no children at all, does this naturally tend towards a position where, given enough time, everyone will end up with the same surname?

I was musing on this earlier putting Xmas cards in the bin, thinking that out of my Grandad and his brother, they had three sons between them (my Dad and my uncles), but between the three of them they had only two boys, of whom I was one (my other cousins on that side are all girls). And I've had two daughters, so it's only my brother (who is as yet childless) who is ever (potentially) going to pass our family surname into the "next" generation. If you follow me.

My ex-wife's family line went similarly - her Dad had no male siblings, he had two daughters, so now none of his grandchildren or great-nieces/nephews bear his surname.

This must happen to a lot of family names, I thought. So how long before we're all called the same?

[Wyatt Earp]

Maths question (I think) about families

Broadly speaking, very rare surnames are in danger of winking out of existence like that, but more common surnames don't, because for even though your branch of the Smith family have no sons, you can rely on other branches to do so.

[Rogin the Armchair fan]

Maths question (I think) about families

So extrapolating ... assume there are (I don't know) 1,000 distinct surnames, and in a given population 20% are already "Smith, Brown and Evans", how many generations, do you reckon, before everyone's "Smith"?

Even if the answer's something ridiculous like "23,000AD", I still think it's interesting if it's going to happen one day.

[Ginger Yellow]

Maths question (I think) about families

The thing is, that assumes no immigration and no newly minted surnames. Neither of which holds. I suppose it could still be an interesting exercise, similar to the maths exercises which "show" that the most recent common ancestor of (nearly) every living human lived only a few thousand years ago (Australia, especially Tasmania fucks things up a bit). But it has no real predictive power.

[Wyatt Earp]

Maths question (I think) about families

No, what I'm saying is that the frequencies stay roughly constant.

Look at it this way. Suppose Smiths form 1% of a population of 60m (as they do, pretty much). Suppose for simplicity that everybody marries. Now, each fresh generation of Smiths consists of roughly 300 thousand men, who stay as Smith, and 300 thousand women, who change their names (apart from the 1% who happen to marry Smiths, which we'll ignore for simplicity; they make no difference to the argument anyway).

That sounds like the number of Smiths halves. But hang on a minute: who do those 300 thousand Smith men marry? They marry women called something other than Smith.

So 300 thousand women change their name from Smith. But 300 thousand women change their name to Smith. So the number of Smiths stays the same.

There's a bit of random drift, of course, which means frequencies gradually alter over time. And because of that, there's some "culling" of rare surnames (because once the frequency randomly hits zero, it obviously never rises again).

I would imagine the rate of "culling" would, in the absence of things like immigration, reach a rough equilibrium with the rate of new coinages, transcription errors from generation to generation, and so on.

[Guest]

Maths question (I think) about families

Wow. I just posted the stupidest thing. Move along, nothing to see, etc.

[Rogin the Armchair fan]

Maths question (I think) about families

No, I think I've just done the same.

[Wyatt Earp]

Maths question (I think) about families

I think you could, without doing violence to this, treat the population of each surname as a random walk, based on a Poisson distribution for each individual. If so, I think I've worked out how to model the "drift" effect numerically, but the model would take ages to run. My probability theory's not up to the theoretical calculation of the "time to extinction" (even for a closed population with no fresh coinages and prefect transcription).

There seems to have been a landmark paper, on "time to extinction" in complicated random walks like this, by Lotka, who as well as driving a New York taxi and talking in a funny voice, pretty much invented population dynamics. This also seems to be a pretty active field, too, with a paper published, aiming to deal with the complexities that enter the model when you're dealing with sexually reproducing couples instead of individuals, as recently as 2001.

[hobbes]

Maths question (I think) about families

Of course, no-one has factored in people who marry but keep their surname.

[Wyatt Earp]

Maths question (I think) about families

Doesn't affect the "What happens on average?" argument, unless women with certain surnames are more likely to do that than others. Probably does affect the "What happens if you allow for random drift?" calculation, which already looks fiendish-ish.

[Wyatt Earp]

Maths question (I think) about families

Oh hang on: I've just remembered the sum of Poisson distributions is a Poisson distribution. (I hate stats.)

With my noddy model, the numerical calculation doesn't look too bad after all. Hang on.

[Wyatt Earp]

Maths question (I think) about families

It is actually taking a while. I've got to go to a lecture; will leave it running, and see if it's finished when I get back.

[Sean of the Shed]

Maths question (I think) about families

What about people who double barrel their surnames? Does that become a new name or does it count as two?

[WOM]

Maths question (I think) about families

As a Smith, I'm tired of my name being used as an example of a common one. And I'm sure there are millions of others who feel as I do.

[Wyatt Earp]

Maths question (I think) about families

OK, so I ran my simplistic model for a thousand generations, using the most popular 250 surnames in the UK and their frequencies, assuming that there's no overall change in population size, that there's no immigration or new coinages and that all surnames are equally fecund (probably a false assumption, that; some surnames will very likely be associated with larger families).

On average, frequencies changed by about 10%. The biggest frequency change was only around 50%. None of the top 250 went extinct in that time. So there's a drift, as predicted, but it's pretty stable really, as guessed.

I'll knock together a diagram when I get time. But broadly, the time taken for one of the top 250 surnames to go extinct would certainly have to be ten thousand generations or more, I'd have thought, though I haven't had time to run the model that far.

The time taken for all surnames bar one to go extinct is going to be mind-bogglingly immense, though there's a theorem that says that under these very artificial assumptions, it must eventually happen.

[ursus arctos]

Maths question (I think) about families

Keeping the same artificial assumptions, it would happen more quickly (but still likely require hundreds of generations) in some place like Korea (where three surnames already account for roughly half the native population).

[Wyatt Earp]

Maths question (I think) about families

It would happen more quickly, yes, though I'd say you're still underestimating the time needed there by at least a couple of orders of magnitude!

[ursus arctos]

Maths question (I think) about families

My guessestimate was for the time needed for the first common surname to go extinct.

That of course depends on how you define "common". Wiki suggests that there are only 250 Korean surnames now, so you would need to pick a lower threshhold.

And yeah, it would take a very, very long time to get down to nothing but "Kim", even in Korea.

[Guest]

Maths question (I think) about families

I think probably one of the most annoying things about being a Smith is the look on hotel receptionist faces when you check in.

"I have a reservation for Smith"

...and then they give you the raised eyebrows look, the look which says "Yeah, I know your game pal. And I suppose you will inevitably try and pass off the blindingly obvious prostitute as your wife later on. Disgrace."

All that because of a name.....

[Stumpy Pepys]

Maths question (I think) about families

I'd have loved to have gone on a dirty weekend and booked ourselves in as Mr & Mrs Smith. It must have been terribly exciting.

[Ginger Yellow]

Maths question (I think) about families

I think probably one of the most annoying things about being a Smith is the look on hotel receptionist faces when you check in.

Is that still the case? Most hotels I've been to recently have asked for a credit card at check-in. Then again, I only really go to hotels abroad, so maybe it's just for foreigners.

[Wyatt Earp]

Maths question (I think) about families



Here's an animation to illustrate what happens over 1000 generations according to my dumbass model. The top 250 surnames start off in order of frequency, and then there's a bit of a random drift effect, as you see. The animation has 101 frames: one every ten generations from zero to 1000.

(In passing, I think this is possibly a reasonably good example of a worthwhile use of animation, which is something we discussed a while ago.)

Edit: hang on, I'll make it bigger.

Edit 2: there it is, a bit bigger.