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Ten circles meet each other

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Surname/tag: 100_circles
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Ten circles meet each other : a conjecture

The vocabulary and the context of this conjecture is defined in the 100 Circles page.

Abstract : Based on current state of connected profiles in WikiTree, and analysis of how the circles grow, we extrapolate figures that could be reached by systematic population of the ten first circles for an average profile of WikiTree (post-1600, Western world). We derive from those figures that the ten first circles for any two such profiles have necessarily a non-empty intersection, meaning that their relative distance is at most 20 degrees.

One million profiles in ten circles - Samuel Lothrop & al.

In the framework of the 100 Circles research, exploring various ways to assess centrality, we've found some profiles with a steep circles population growth. Our reference in this category is Samuel Lothrop (1622-1700), an Englishman who emigrated to America as part of the Puritan Great Migration and raised a large family in Connecticut.

The following table shows the cumulative population for his ten first circles at differentes dates.

  • By January 2022, the cumulative population even passed the million in the 9th circle!
  • By April 2023, CC10 passed 2 million.
Circle 2021-03-15 2021-04-16 2021-05-18 2021-07-04 2021-08-12 2022-01-21 2022-06-052022-09-062023-04-30
1 28 28 28 28 28 28 28 28 28
2 231 231 231 231 231 231 231231 232
3 1,176 1,174 1,176 1,179 1,182 1,188 1,193 1,201 1,222
4 5,222 5,243 5,269 5,310 5,316 5,396 5,455 5,514 5,690
5 20,935 21,081 21,211 21,415 21,494 21,974 22,346 22,686 23,874
6 72,280 72,814 73,419 74,398 74,850 77,355 79,318 80,860 86,001
7 204,638 206,322 208,491 212,111 214,040 222,697 230,299 235,686 254,184
8 474,440 479,186 485,350 494,182499,408 522,701 543,685 558,055 605,680
9 903,260913,384 925,691 941,829 952,479 1,004,103 1,048,790 1,078,696 1,175,442
10 1,520,7611,539,858 1,560,661 1,587,317 1,606,393 1,706,925 1,789,5681,842,016 2,014,650

Let's call "10C1M" such profiles for which cumulative population of the ten first circles passes the million threshold. Samuel Lothrop is certainly not the only one in this case, for example profiles in his first circle belong to this category also.

Mayflower passengers, such as Mary (Allerton) Cushman, Joseph Rogers, William Bradford, Love Brewster among others, are also 10C1M profiles.

More of them are presented by Eva Ekeblad in this page.

And there are certainly more of them all over the tree, difficult to identify, let alone count . And for many more, the cumulative population of the 10 first circles is well over the 100,000, it's just a question of time before it passes the million. Reaching the thousand range by the 4th circle is quite common for a profile in a reasonably "big family". From there, a geometric growth by a factor 10 every two circles is a reasonable ballpark estimation, leading to a 10th circle in the million range, when all is said and done, all possible branches explored and profiles added.

How the circles grow

Beyond the 10 first circles, it's interesting to see which proportion of globally connected profiles the cumulative population of Samuel Lothrop's circles represent, and how those figures evolve with time. In the following table, figures under CC10, CC15 ... are % of the total.

Date Total CC10 CC15 CC20 CC25CC30
2020-11-17 20,920,327 6.95 41.10 75.04 91.65 97.13
2021-07-0122,957,997 6.91 41.72 75.61 92.27 97.37
2022-07-0426,349,431 6.82 42.62 76.63 92.90 97.66
2022-09-0627,030,955 6.81 42.91 76.96 92.90 97.73

Those profiles are not exceptional

What makes Samuel Lothrop's singularity today is simply that he had the chance to live at the right place and time where a lot of WikiTreers have searched their ancestors. There was no conspiracy to populate his circles, it just happened, out of general WikiTree activity and growth. With the growth of WikiTree, we'll get more and more of those 10C1M profiles, and at some point in a distant future, it could be the majority of WikiTree "average" profiles, roughly any 19th century Westerner. For some profiles currently in focus in the 100 Circles project, we have indeed started a conspiracy to systematically populate the first circles. With time and patient work, many (if not every one) of them can become 10C1M, as a result of both systematic focused work and general background addition and reconnection of profiles in distant circles.

One million surely meets another one

Now let's take two 10C1M profiles A and B. Simple computation shows that the probability of the first ten circles of A and B having no common element is practically null, even if the overall population of WikiTree were to pass the billion range. Given two random subsets of one million people out of a hundred billion reference population (the estimated number of humans having lived in the time span covered by WikiTree), the probability to have no common element is 0.99999^1000000, which is less than 0.0001. The actual WikiTree population in any foreseeable future being well under this hundred billion figure, the said probability is actually stunningly smaller than that, and for all purposes can be considered as null. One can argue that the sample represented by the 10 circles of a given profile is not really random. But when you start to systematically populate circles, well before the 10th circle you see paths going in so many unexpected directions that the expansion really looks like a stochastic process.

Less than 20 degrees from each other

Whatever the state of growth of WikiTree, two 10C1M profiles A and B will have, with a probability close to 1, at least one common profile X at distance 10 (or less) of both A and B, providing a path from A to B which is less than 20 degrees.

If most average profiles (19th century Westerners) in due course of time and WikiTree growth can become 10C1M, a quite stunning consequence of all the above conjectures is that the distance between such profiles will eventually be brought under 20.

In the current size of The Tree (about 22 million connected), suppose the 10 circles of both A and B contain just one thousandth of the total, that is a mere 22,000. The same computing as above, for those sets to have no common element, gives a probability roughly equal to 0.999^22000 (about 2E-10). In other words those two samples, representing each only 0.1% of the total population, have almost certainly at least one common element. This counter-intuitive result is a variant of the Birthday Paradox.

The above result is based on the somehow bold assumption that the 10 circles for any profile represent a quasi-random sample of the global tree population, assumption which can be made if the population of those circles is large enough, the hard problem being how large is "large enough".

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