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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
Sky's the limit

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



@Black Yoshi:
I haven’t received your email! :-(
What address did you send it to?


Posted July 25th by chiarizio
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@Black Yoshi:
I figured out the gmail address you sent the gmail to.
I got your gmail.
GoogleSheets tells me I’ve added indiantrail09@gmail.com to the “Share” list for that sheet.
Did it work?

Posted July 25th by chiarizio
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Yes it did; I'll tale a look when I'm on my computer this afternoon.

Edited July 25th by Black Yoshi
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@chiarizio: Can you email me the actual excel spreadsheet? Or else upload the spreadsheet directly onto Google Sheets without copy/pasting anything?

Edited July 25th by Black Yoshi
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@Black Yoshi:
@chiarizio: Can you email me the actual excel spreadsheet?

Maybe. I’ll give it a try.

Or else upload the spreadsheet directly onto Google Sheets without copy/pasting anything?

I don’t know how. I can try to find out how.




Posted July 25th by chiarizio
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All you have to do is click and drag the Excel file onto your Google Drive.

Posted July 26th by Black Yoshi
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@Black Yoshi:

All you have to do is click and drag the Excel file onto your Google Drive.


It’s not Excel, it’s Numbers, but it might work anyway.

While looking at it I noticed an error.
I subtracted GammLn(k + rho) instead of GammaLn(k + 1 + rho).

Equivalently; Instead of calculating Beta(k, rho+1) as Gamma(k)*Gamma(rho+1)/Gamma(k+rho +1),
I used Gamma(k)*Gamma(rho +1)/Gamma(k+rho).
Dividing by Gamma(k+rho) instead of by Gamm(k+rho +1).



Anyway:
In trying to fix my error, I fucked up my spreadsheet to the point it’s no longer what you asked me to send you.

Do you still want me to try?

Posted July 27th by chiarizio
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Quite alright, I'd rather not trouble you any further than I have.

Posted July 27th by Black Yoshi
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Quite alright, I'd rather not trouble you any further than I have.


My worry is that I was troubling you! Or, rather, was just about to!

If I ever get it almost-figured out, I’ll make sure to give you access to everything I can give you access to.

I do not want to refuse help!

I also don’t want to scare help away.

I guess I’m not skilled at striking a balance.


Edited July 28th by chiarizio
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@Black Yoshi:
I have made a new census of the human populations of the 4095 star-systems in the Reptigan Union that have at least one permanent human inhabitant.
The way I did it was this:
First I made sequence of 4095 numbers, the first being 10^10, and each later one being (1-((10^10)/(92^6))) times the previous number in the sequence. (That’s between 0.98 and 0.99.)
Then I rounded each of those numbers off to the nearest whole number; unless it rounded off to 0, in which case I instead rounded it up to the next higher whole number (ie 1).

Past about the 1365th system, they all have just one permanent human inhabitant.

Reptigan Union space is going to contain some 200 million star-systems, of which probably 38 million are K or G stars, more than half of which will have a planet in the “Goldilocks Zone” where there’ll be liquid water somewhere on the planet’s surface year-round. I don’t know what fraction of those will actually be habitable.
I figure around at least 160,000 or 170,000 of them will actually be inhabited by intelligent species belonging to the Reptigan Union.
If a star-system’s human population is less than 10,000, I don’t expect them to sustain themselves, unless the total population of all Reptigan citizens (including allied species) there is at least 100,000.

—————

I have similar thoughts that I haven’t finished yet. I don’t want to commit them to writing until I’m pretty sure I won’t have to retract them later.

————-

I know I want the total human population to be 92^6; and I don’t want any planet’s population to be more than 10^10.

But do I really want 4095 human-inhabited star-systems? And do I really want the most populous planet to have all of 10^10 human inhabitants?


Edited July 29th by chiarizio
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It's your universe, you can set it up any way you wish.

Posted July 30th by Black Yoshi
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In the meantime, it is possible to export a Numbers worksheet in an Excel file:
https://www.nytimes.com/2016/08/18/technology/personaltech/converting-spreadsheets-in-apples-numbers-to-excel.html
Why not do that and send it to me so I can take a look?

Like I said though, at the end of the day it's your universe and you can build it however you want.

Posted July 30th by Black Yoshi
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It's your universe, you can set it up any way you wish.


Not if “the way I want” is mathematically inconsistent :-( ! And I’m afraid it might be!

—————

I’ll send you what I can, when I can, and thank you for any advice or instruction you give me.
Even if I decide not to use it right away, I know I’ll use it sometime!


Posted July 30th by chiarizio
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@Black Yoshi:
In the meantime, it is possible to export a Numbers worksheet in an Excel file:

https://www.nytimes.com/2016/08/18/technology/personaltech/converting-spreadsheets-in-apples-numbers-to-excel.html

Why not do that and send it to me so I can take a look?


They won’t let me read it.




Posted July 31st by chiarizio
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What happens when you try to read it?

Posted July 31st by Black Yoshi
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What happens when you try to read it?


It gets overlain by a page saying I can keep on if I subscribe for a dollar a week.

Posted July 31st by chiarizio
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Ah, right. On mobile I was able to make that go away, I thought you could do the same on desktop.

Anyway, here's another link that might help:
https://support.apple.com/kb/ph26057?locale=en_US

Posted July 31st by Black Yoshi
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@Black Yoshi:
Thanks!
It’s a little late right now, and I have to go to court tomorrow to get a “driving w/o proof of insurance” ticket waived/dismissed,
so I better go to bed and do it Thu or Fri or Sat or Sun..


Posted August 1st by chiarizio
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@Black Yoshi:
I made five more attempts in Sheets to come up with systematic censuses of the human populations of the 4095 star-systems in the late Reptigan Union that have the highest permanent human populations.
They are the different worksheets of this file:
https://docs.google.com/spreadsheets/d/1-zDS1bzgnaEpDFfpwcCdLTZolXrE2mufJM2QQIEYaio
The fifth, last, spreadsheet, is the one that’s currently my favorite.
The 25 most populous planets look like this:
K K^(-S) Population K as a fraction S
1 1.000000000000000E+00 10,000,000,000 9,999,999,999.92 6.18213545450000E-01 9.999999999921010E+09
2 6.514771369237740E-01 6,514,771,369 6,514,771,369.19 606,355,001,344 6.063550013487900E+01
3 5.070341107580490E-01 5,070,341,108 5,070,341,107.54 606,355,001,322
4 4.244224599343980E-01 4,244,224,599 4,244,224,599.31
5 3.697322004281710E-01 3,697,322,004 3,697,322,004.25
6 3.303211307993460E-01 3,303,211,308 3,303,211,307.97
7 3.002955831054370E-01 3,002,955,831 3,002,955,831.03
8 2.765015290442070E-01 2,765,015,290 2,765,015,290.42
9 2.570835894722050E-01 2,570,835,895 2,570,835,894.70
10 2.408720753634720E-01 2,408,720,754 2,408,720,753.62
11 2.270894382007190E-01 2,270,894,382 2,270,894,381.99
12 2.151966645585810E-01 2,151,966,646 2,151,966,645.57
13 2.048071578115140E-01 2,048,071,578 2,048,071,578.10
14 1.956357067123850E-01 1,956,357,067 1,956,357,067.11
15 1.874668374627150E-01 1,874,668,375 1,874,668,374.61
16 1.801344244967660E-01 1,801,344,245 1,801,344,244.95
17 1.735081222716340E-01 1,735,081,223 1,735,081,222.70
18 1.674840808194390E-01 1,674,840,808 1,674,840,808.18
19 1.619784364939160E-01 1,619,784,365 1,619,784,364.93
20 1.569226500226830E-01 1,569,226,500 1,569,226,500.21
21 1.522601039444350E-01 1,522,601,039 1,522,601,039.43
22 1.479435770246330E-01 1,479,435,770 1,479,435,770.23
23 1.439333410484500E-01 1,439,333,410 1,439,333,410.47
24 1.401957069021700E-01 1,401,957,069 1,401,957,069.01
25 1.367019000334580E-01 1,367,019,000 1,367,019,000.32

And the last 25 planets look like this:
4070 5.868131569910190E-03 58,681,316 58,681,315.70
4071 5.867240405932930E-03 58,672,404 58,672,404.06
4072 5.866349596121000E-03 58,663,496 58,663,495.96
4073 5.865459140246720E-03 58,654,591 58,654,591.40
4074 5.864569038082630E-03 58,645,690 58,645,690.38
4075 5.863679289401450E-03 58,636,793 58,636,792.89
4076 5.862789893976110E-03 58,627,899 58,627,898.94
4077 5.861900851579750E-03 58,619,009 58,619,008.52
4078 5.861012161985690E-03 58,610,122 58,610,121.62
4079 5.860123824967470E-03 58,601,238 58,601,238.25
4080 5.859235840298820E-03 58,592,358 58,592,358.40
4081 5.858348207753680E-03 58,583,482 58,583,482.08
4082 5.857460927106190E-03 58,574,609 58,574,609.27
4083 5.856573998130680E-03 58,565,740 58,565,739.98
4084 5.855687420601690E-03 58,556,874 58,556,874.21
4085 5.854801194293960E-03 58,548,012 58,548,011.94
4086 5.853915318982410E-03 58,539,153 58,539,153.19
4087 5.853029794442180E-03 58,530,298 58,530,297.94
4088 5.852144620448610E-03 58,521,446 58,521,446.20
4089 5.851259796777230E-03 58,512,598 58,512,597.97
4090 5.850375323203760E-03 58,503,753 58,503,753.23
4091 5.849491199504150E-03 58,494,912 58,494,911.99
4092 5.848607425454510E-03 58,486,074 58,486,074.25
4093 5.847724000831170E-03 58,477,240 58,477,240.01
4094 5.846840925410660E-03 58,468,409 58,468,409.25
4095 5.845958198969680E-03 58,459,582 58,459,581.99

6.063550013487900E+01 606,355,001,322






Edited August 3rd by chiarizio
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@elemtilas:
@linguistcat:
@Xhin:
@Black Yoshi:

I have to find a way to handle the fact that K-class stars will outnumber G-class stars at a ratio of approximately 12 to 7.
I want the Adpihi-descended humans to be one of the most influential species in the Reptigan Union.
But why wouldn’t the Space-Centipedes be the most influential, instead?
Indeed, why wouldn’t twelve of the nineteen most influential species, be native to K-star systems, rather than G-star systems?

———

M-star systems will outnumber K-star systems by about 6 to 1.
But I think planets orbiting a type-M star might have some trouble getting a stable biosphere and ecosystem going. At least for a sizeable fraction of such systems.
But even so, the number of intelligent species native to M stars might still outnumber the K stars’ species by 2 or 3 to 1.
However an M-star’s entire system — including but not limited to its habitable planet(s) — should be quite metal-poor.
That means they’ll have trouble inventing the kind of technology the Humans have.
And if they do achieve interplanetary space-travel, they’ll still be metal-poor.
So maybe M-native species won’t be very big on the interstellar scene until contacted by a species already in the Reptigan Union.

———

Maybe K-class stars’ systems are also metal-poor compared to G-stars’ systems?
So maybe, even if they’re just as quick as Adpihi humans to invent the technology, they can’t build their fleets or infrastructure as rapidly?

Anyone have any suggestions?
Or questions?


Posted August 12th by chiarizio
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