J. T. Chang, Recent common ancestors of all present-day individuals. Chang considers a statistical model in which a population of constant size *n* goes through successive generations, and shows that (under suitable assumptions – infra) for *n* reasonably large (>~200 based on numerical simulations), there exists a single common ancestor of everyone in the population log_{2} *n* generations in the past – and if one goes approximately 1.77 log_{2} *n* generations back, *everyone* in the population is either a common ancestor of all people in the present generation, or of nobody. That is, either a given family line has died out completely, or just by statistical diffusion, they’ve become related to everyone living.

Applied to the population of Europe, this threshold seems to happen about 1000 years in the past. So it’s fairly likely that everyone with even a single European ancestor within the past 100 years or so can, in fact, claim descent from Charlemagne.

There are two technical assumptions in this paper. One is constant population size; it seems like it would be straightforward, although a technical pain in the ass, to relax this. The other more interesting one is that it assumes a random mating model; i.e., the probability that someone in generation *t* is a parent of someone in generation *t+1* is uniform. This obviously isn’t correct, but I can think of a good way to model something more realistic – consider a set of *k* populations of size *f*_{k}, each of which has random mating within it, and with a cross-mating probability distribution *p*_{k k’}. This could model the existence of disjoint social or geographic groups. I’m rather curious about whether this would substantially change the results. One interesting question is, given a total population size and a decomposition into subgroups, whether or not there’s a “critical size” for a subpopulation which will lead in finite time to that population dying out, becoming completely assimilated, or becoming ancestors of everybody.

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It seems like the cross-mating probability would definately have an effect. In the limit where the cross mating probability is much less than the probability of mating within the population, then each subpopulation will have it’s own common ancestor, distinct from the common ancestor of other subpopulations. So it seems like there must be some cross-mating probability above which the subpopulations merge (and below which they are separate). But there’s probably some interesting behavior when the cross-mating probability isn’t too high or too low, which I think is closer to reality.

That’s something that’s interesting me. It could also give interesting information about how genes propagate between mostly disjoint populations. Various limits could be very interesting – small populations that are either very exogamous or very endogamous, chains of populations each of which mates either with itself or with its nearest neighbor (as in geographically nearby groups), or medium-sized mostly-endogamous groups living very intertwined with a much larger group. (Like e.g. Jews in medieval Europe)

don’t they have something for you people?

like ritallin or something?

This

iswhat he’s like on Ritalin.Hmm? Nonsense. Never touch the stuff.