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Reptigan Union's Star-Systems' Human Populations
Posted: Posted July 24th
Edited July 24th by chiarizio
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I finally figured out how to do the Yule-Simon distribution.
It involves the Beta function.
For some positive real parameter rho, the K-th planet’s population is proportional to
rho * Beta( K , rho + 1).

The Beta function is defined in terms of the Gamma function.
Beta(x,y) = (Gamma( x ) * Gamma( y ))/Gamma(x+y).

The Gamma function is related to the analytic continuation of the Factorial.
If x is a positive whole number, then Gamma(x) = (x-1)!
(That is, Gamma(x)=Factorial(x-1).)

So if x and y are positive whole numbers >= 1,
Beta(x,y) = ( (x-1)! * (y-1)! )/((x+y-1)!).

I use a constant-of-proportionality (10^10)/(rho*Beta(1,rho+1)) so that the 1st (ie most populous) star-system’s Human population will be 10 billion.

I tested several values of rho, to make the total human population of the first 4095 star-Systems be as close as possible to 92^6 which is about 606,355,000,000 (within 20,000).
I found rho=0.59 to be too small and rho=0.61 to be too big.
(The larger the value of rho, the faster the populations decline as rank increases, among the first several ranks.)
(So as rho increases, the total population of the first 4095 planets decreases, if the population of the 1st planet remains the same.)

So I use rho = 0.60, precise to within 0.01.

A Yule-Simon distribution results from a “preferential-attachment stochastic process”, in which the probability of immigration to a particular star-system, is proportional to some linear function of the population it already has.

This is also called the “rich get richer” effect; or “the Matthew effect” because of Matthew 25:14-30.

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It starts out like this:

1 10,000,000,000
2 6,250,000,000
3 4,807,692,308
4 4,006,410,256
5 3,483,835,006
6 3,110,566,969
7 2,827,788,154
8 2,604,541,721
9 2,422,829,508
10 2,271,402,663
11 2,142,832,701
12 2,031,996,527
13 1,935,234,788
14 1,849,856,782
15 1,773,835,271
16 1,705,610,837
17 1,643,962,253
18 1,587,918,085
19 1,536,694,921
20 1,489,653,240
21 1,446,265,281
22 1,406,091,246
23 1,368,761,390
24 1,333,962,371
25 1,301,426,704

And ends up like this:

4071 60,994,989
4072 60,986,001
4073 60,977,016
4074 60,968,035
4075 60,959,057
4076 60,950,083
4077 60,941,112
4078 60,932,145
4079 60,923,181
4080 60,914,221
4081 60,905,264
4082 60,896,311
4083 60,887,361
4084 60,878,415
4085 60,869,473
4086 60,860,533
4087 60,851,598
4088 60,842,666
4089 60,833,737
4090 60,824,812
4091 60,815,890
4092 60,806,972
4093 60,798,057
4094 60,789,146
4095 60,780,239

Total 607,237,692,169

I can’t figure out how to get the whole list from my spreadsheet into gtx0.

As you can see the total is off by slightly less than 1,000,000,000.



Edited July 24th by chiarizio
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Put it on a Google Sheet and provide us the link.

Posted July 24th by Black Yoshi
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https://docs.google.com/spreadsheets/d/1-IYRxceHGvsFqiSK-ZgVaLP8rlakEzRftAr3Six-hpI
It’s not the same.
I can only do the biggest 1000 planets, but already the 2nd planet is different.
And the 1000th planet’s population is under a quarter million.
One spreadsheet or the other contains a mistake.
I don’t know which.



Edited July 24th by chiarizio
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@Black Yoshi:
https://docs.google.com/spreadsheets/d/1-IYRxceHGvsFqiSK-ZgVaLP8rlakEzRftAr3Six-hpI

It’s not the same.
I can only do the biggest 1001 planets, but already the 2nd planet is different.
And the 1001st planet’s population is under a quarter million.
One spreadsheet or the other contains a mistake.
I don’t know which.



Posted July 24th by chiarizio
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@Xhin:
Why can’t I use apostrophes in the thread’s title?


Posted July 24th by chiarizio
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Why can’t I use apostrophes in the thread’s title?


Fixed. Try to use this apostrophe: ' rather than whatever the special one you were using was. ` works too apparently.

Posted July 24th by Xhin
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Xhin
Ground's what's around

Fixed. Try to use this apostrophe: ' rather than whatever the special one you were using was. ` works too apparently.


Hmm. I wasn’t aware there were three apostrophes on my keyboard. Still not sure where that straight-up-and-down one is.

י



Posted July 24th by chiarizio
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@Xhin:
Fixed. Try to use this apostrophe: ' rather than whatever the special one you were using was. ` works top apparently.


I just donˋt have that vertical apostrophe on any of my keyboards. And I have to go to the Deutsch keyboard to get the backwards one.

Posted July 25th by chiarizio
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@chiarizio: I'll need you to provide me access to the document; I've already sent an email.

Posted July 25th by Black Yoshi
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@Black Yoshi:
@chiarizio: I'll need you to provide me access to the document; I've already sent an email.

Hmm, I thought I had! I’ll look into it when I finish waking up.




Posted July 25th by chiarizio
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