First, generally speaking, exponentiation of ancestry breaks down rapidly (pedigree collapse), otherwise we arrive at the absurd conclusion that any living person has a trillion great-to-the-thirtieth-grandparents. In reality, go back far enough and ancestors start occupying many positions in the tree.
Second, obvious counterexample: suppose Salazar Slytherin marries and has one child, who marries and has one child, etc...and fifty generations later, there is still only one descendant of Salazar Slytherin per generation. This counterexample can be broadened; suppose the descendants of Salazar Slytherin’s second child all died in the Black Plague. In short, it’s not the case that a distant ancestor is either everyone’s ancestor or no one’s...not until you get to mitochondrial Eve and y-chromosomal Adam, anyway, but that’s another story (and much older than 1000 years ago).
Actually, the exponentiation of ancestry proves that Salazar is the ancestor of everyone in Great Britain by the pigeonhole principle, except in your case where each of his descendants only has one child. That is an extreme outlier and not likely to happen, all things considered.
A recent paper shows that everyone who lived in Europe 1000 years ago is an ancestor of everyone living in Europe today (barring immigration of course):
We can furthermore conclude that pairs of individuals across Europe are reasonably likely to share common genetic ancestors within the last 1,000 years, and are certain to share many within the last 2,500 years. From our numerical results, the average number of genetic common ancestors from the last 1,000 years shared by individuals living at least 2,000 km apart is about 1⁄32 (and at least 1⁄80); between 1,000 and 2,000ya they share about one; and between 2,000 and 3,000 ya they share above 10. Since the chance is small that any genetic material has been transmitted along a particular genealogical path from ancestor to descendent more than eight generations deep [8]—about .008 at 240 ya, and 2.5×10−7 at 480 ya—this implies, conservatively, thousands of shared genealogical ancestors in only the last 1,000 years even between pairs of individuals separated by large geographic distances. At first sight this result seems counterintuitive. However, as 1,000 years is about 33 generations, and 2^33≈10^10 is far larger than the size of the European population, so long as populations have mixed sufficiently, by 1,000 years ago everyone (who left descendants) would be an ancestor of every present-day European.
Great Britain is a lot smaller than Europe as a whole, so it probably takes even less than 1000 years for this effect to work.
It’s true that a family tree either dies out or grows exponentially, but 2 is not necessarily the relevant exponent.
If the expected number of children an average descendant of Slytherin has is N, after 33 generations we should expect to see N^33 heirs of Slytherin. (You should think that this manipulation is suspicious as well, but it can be justified mathematically in, say, the Galton-Watson model.) Taking N=1.25 gives us only around 1500 descendants after 1000 years. And this is, if anything, an overestimate that does not take any intermarriage into account.
Powers of 2 only become relevant if you’re looking backwards from a specific person; for example, if you want to know whether two people have a common ancestor. In that respect, I (tentatively) believe the paper you link to.
One of us must be wrong; it can’t both be the case that everyone 1000 years ago is an ancestor of everyone living today, and that the average person 1000 years ago only had 1500 descendants.
I think N is closer to two or higher; assuming the average person has two children, they will have four grandchildren on average, eight great-grandchildren on average, and so on. So there really should be 2^33 heirs, though not 2^33 unique heirs; many of those heirs are just different genealogical paths to the same people.
I think if N were below two, it would be below the replacement rate and the population would shrink over time.
Indeed, I doubt that everyone 1000 years ago is an ancestor of everyone living today. I expect that everyone 1000 years ago is an ancestor of everyone [Edit: at least within a geographical region], of no-one, or is atypical in some way (for example, I expect a family that is well-off to have a number of children sampled from a different distribution, which has no reason to have mean greater than 2).
You are right, though, that across the board N has to be greater than 2 or else the global population would shrink over time. Moreover, if (when we look at Slytherin’s descendants specifically) N is 1 or less, we expect the Slytherin line to eventually die out. This leaves room for a line that neither dies out nor grows as quickly as population does overall.
True, bu 1) GB as a country has lots of immigration, 2) there are those Asian wizards sporadically mentioned before. Maybe the martial arts master whom Voldemort killed was another of Salazar’s grand(...)children, and the girl living ‘where they don’t get invitations to Hogwarts’ is yet another one?
(Or maybe she’s a resurrected, Obliviated and Time-turned Hermione. After all, Quirrell didn’t say he will restore her to Harry, andsuch a resolution would, from Harry’s point of view, equal the destruction of Hermione’s self and so a sacrifice large enough… Though that would violate restrictions on Time-turners).
On the other hand, it seems redundant of them to have discussed the possibility of a dying wizard making an Unspeakable Vow, and an opportunity for it not arising before the end).
The pigeonhole principle doesn’t say what you want it to. It guarantees that some ancestors will show up multiple times on everyone’s trees; it does not guarantee or even suggest that every ancestor present on anyone’s tree is present on everyone’s trees.
That aside, it’s not clear that the descendants of Salazar Slytherin would mix sufficiently with the rest of magical Britain in 1000 years of wizard generations (possibly longer than Muggle generations, given differing lifespans) for the paper’s findings to apply. Running with general experimental assumptions is not effective for specific and extraordinary cases.
That’s not how ancestry works.
First, generally speaking, exponentiation of ancestry breaks down rapidly (pedigree collapse), otherwise we arrive at the absurd conclusion that any living person has a trillion great-to-the-thirtieth-grandparents. In reality, go back far enough and ancestors start occupying many positions in the tree.
Second, obvious counterexample: suppose Salazar Slytherin marries and has one child, who marries and has one child, etc...and fifty generations later, there is still only one descendant of Salazar Slytherin per generation. This counterexample can be broadened; suppose the descendants of Salazar Slytherin’s second child all died in the Black Plague. In short, it’s not the case that a distant ancestor is either everyone’s ancestor or no one’s...not until you get to mitochondrial Eve and y-chromosomal Adam, anyway, but that’s another story (and much older than 1000 years ago).
Actually, the exponentiation of ancestry proves that Salazar is the ancestor of everyone in Great Britain by the pigeonhole principle, except in your case where each of his descendants only has one child. That is an extreme outlier and not likely to happen, all things considered.
A recent paper shows that everyone who lived in Europe 1000 years ago is an ancestor of everyone living in Europe today (barring immigration of course):
Great Britain is a lot smaller than Europe as a whole, so it probably takes even less than 1000 years for this effect to work.
It’s true that a family tree either dies out or grows exponentially, but 2 is not necessarily the relevant exponent.
If the expected number of children an average descendant of Slytherin has is N, after 33 generations we should expect to see N^33 heirs of Slytherin. (You should think that this manipulation is suspicious as well, but it can be justified mathematically in, say, the Galton-Watson model.) Taking N=1.25 gives us only around 1500 descendants after 1000 years. And this is, if anything, an overestimate that does not take any intermarriage into account.
Powers of 2 only become relevant if you’re looking backwards from a specific person; for example, if you want to know whether two people have a common ancestor. In that respect, I (tentatively) believe the paper you link to.
One of us must be wrong; it can’t both be the case that everyone 1000 years ago is an ancestor of everyone living today, and that the average person 1000 years ago only had 1500 descendants.
I think N is closer to two or higher; assuming the average person has two children, they will have four grandchildren on average, eight great-grandchildren on average, and so on. So there really should be 2^33 heirs, though not 2^33 unique heirs; many of those heirs are just different genealogical paths to the same people.
I think if N were below two, it would be below the replacement rate and the population would shrink over time.
Indeed, I doubt that everyone 1000 years ago is an ancestor of everyone living today. I expect that everyone 1000 years ago is an ancestor of everyone [Edit: at least within a geographical region], of no-one, or is atypical in some way (for example, I expect a family that is well-off to have a number of children sampled from a different distribution, which has no reason to have mean greater than 2).
You are right, though, that across the board N has to be greater than 2 or else the global population would shrink over time. Moreover, if (when we look at Slytherin’s descendants specifically) N is 1 or less, we expect the Slytherin line to eventually die out. This leaves room for a line that neither dies out nor grows as quickly as population does overall.
True, bu 1) GB as a country has lots of immigration, 2) there are those Asian wizards sporadically mentioned before. Maybe the martial arts master whom Voldemort killed was another of Salazar’s grand(...)children, and the girl living ‘where they don’t get invitations to Hogwarts’ is yet another one?
(Or maybe she’s a resurrected, Obliviated and Time-turned Hermione. After all, Quirrell didn’t say he will restore her to Harry, andsuch a resolution would, from Harry’s point of view, equal the destruction of Hermione’s self and so a sacrifice large enough… Though that would violate restrictions on Time-turners).
On the other hand, it seems redundant of them to have discussed the possibility of a dying wizard making an Unspeakable Vow, and an opportunity for it not arising before the end).
The pigeonhole principle doesn’t say what you want it to. It guarantees that some ancestors will show up multiple times on everyone’s trees; it does not guarantee or even suggest that every ancestor present on anyone’s tree is present on everyone’s trees.
That aside, it’s not clear that the descendants of Salazar Slytherin would mix sufficiently with the rest of magical Britain in 1000 years of wizard generations (possibly longer than Muggle generations, given differing lifespans) for the paper’s findings to apply. Running with general experimental assumptions is not effective for specific and extraordinary cases.