
Reptigan Union's StarSystems' Human Populations
Posted: Posted July 24thEdited July 24th by chiarizio
 I finally figured out how to do the YuleSimon distribution.
It involves the Beta function.
For some positive real parameter rho, the Kth 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) = (x1)!
(That is, Gamma(x)=Factorial(x1).)
So if x and y are positive whole numbers >= 1,
Beta(x,y) = ( (x1)! * (y1)! )/((x+y1)!).
I use a constantofproportionality (10^10)/(rho*Beta(1,rho+1)) so that the 1st (ie most populous) starsystem’s Human population will be 10 billion.
I tested several values of rho, to make the total human population of the first 4095 starSystems 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 YuleSimon distribution results from a “preferentialattachment stochastic process”, in which the probability of immigration to a particular starsystem, 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 25:1430.
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