Draconis Combine: Probable World Populations

[Edit, 2019 Feb 28: I have refined my opinions on this topic. I no longer believe the Combine has room for more worlds of 500 million or more people; I now believe Ningxia's 23 million is average for the Combine's 379 "typical" stars, and the lowest 312 of those average 1 million people apiece while the highest 67 average 113 (or maybe 77.8) million apiece. /Edit]

Total Population: 90 billion (75% concentrated on administrative worlds)
Average Population per World: 260 million (30-40 million for non-capitals)
Minimum Population per "Inhabited" World: 1 million

Supposing that a single Behemoth DropShip can satisfy one day's food and water needs for 12.5 million people, and that a round trip takes 1 month, then just 700 Behemoths could supply all the food and water needs for 1/3rd of the Combine's 350 inhabited worlds. (Essentially every world with 5 million or fewer people.)

Ningxia, with 23 million inhabitants, would be something like the 120th most populous world in the Combine. There's about 25 worlds ranging from 100 million to 600 million inhabitants, an industrial world somewhere around 750 million, and another around 1.1 billion.


How Do I Figure?

The final chapter of House Kurita: The Draconis Combine (FASA1620, hereafter HK:DC) is an Atlas describing 33 worlds, 30 of which are capitals:

1 Luthien, which is 1.2167 as populous as the average district capital;
5 district capitals, which are about 1.2167*2 as populous as the average prefecture capital; and
24 (of the 25) prefecture capitals. 

Supposing these patterns continue, the next tier would have 125 worlds averaging 1.2167*3 fewer people than the prefecture capitals, and the following tier would have 625 worlds (some considered "inhabited," some not) averaging 1.2167*4 fewer people than the previous tier. Encouragingly, this forms an almost-straight line (yellow below).

counting Luthien's tier as T = 0
Y value shows how steeply population falls between tiers

I can't draw perfectly straight through Luthien (1,1) and the district capital tier (5, 1.2167) because that would force the population of the unlisted prefecture capital (Kagoshima) up to an impossible 14 billion. Assuming instead that Kagoshima is at all similar to the other 24 prefecture capitals, and assuming that the sample in the Atlas actually bears extrapolation, then the average population at each tier has to fall faster than [1.2167^tier * factorial of tier]--or in other words, the tier of 125 worlds must average less than 481 million people apiece, and the tier of 625 worlds must average less than 99 million.

If that sounds like a lot of people per world, well, it is and it isn't. Remember that's the upward limit, and that population isn't distributed evenly. Here's the district capitals (in blue) and the prefecture capitals (in red) graphed as fractions of their tier's average:

value of missing capital assumes prefectures average exactly 1.2167*2 as much as district capitals

The district capitals are more or less level, which is what we'd expect if they were near the maximum possible population. The least populous district capital, Galedon, falls in line with the prefecture capitals instead; its fraction of the districts' average (last blue dot in graph above) marks the foot of the prefectures' curve, and its population of 1.6 billion sits squarely in the prefectures' median. This is interesting because the least populous prefecture capital, Ningxia, also seems to fall down a tier; I think it's reasonable to use it as a signpost for the next tier's median and foot.

Three worlds in the Atlas are not capitals: the industrial worlds of Altair and Hachiman, both 3.6 billion; and the resort world of Thessalonika, 1.4 billion. Looking at my graph above, I see that the top two prefecture capitals bracket the top of the prefecture line, so I figure Altair and Hachiman will likewise bracket the top of this next tier's line. The prefectures' 1/3 - 2/3 * ln(x) curve looks realistic enough, so I'll try to model the next tier with a realistic curve too.

Some hours of algebra and Wolfram|Alpha later, I've discovered that no ln(x) curve will fit (they all raise the tier average over 481 million people). My other go-to for natural looking curves, 1/x, looks okay but a little on the low side, so I'll hedge my bets by combining them.

Max: 1.45796 / x  +  0.87315 * ln(x)  -  1.42641, scaled to Hachiman
Min: 0.799736 / x  -  0.44901 * ln(x)  -  0.813028, scaled to Altair
Best Guess: 0.368905 / x  -  0.00573755 * ln(x)  -  0.362388, scaled to Altair + Hachiman

That gives this tier (of 125 worlds) an average population of 91 million to 177 million per world. My best guess is around 128 million. (At first these numbers seemed low to me, but then I remembered what 2013 Earth looks like.)


The Combine has about 350 "inhabited" worlds-

-but the Lyran book says for every "inhabited" world there are as many as ten more with populations too small to count, and I can't help but notice that 350+350*10 (=3850) is marvelously close to 1+5+25+125+625+3125 (=3906). Plus, I count roughly 412 star systems on the HK:DC map, and I've got to wonder whether the 60 extra systems are empty, or have space stations, or have been exhausted by Drac economics, or what, and I'm hoping to find a clue as to how many "inhabited" Drac worlds are doubled or tripled up around a single star.

I think I can take Thessalonika as marking the top of the next tier, the tier of 625 worlds. I'll estimate the median as Ningxia's population^2 / Galedon's population, and the foot as Ningxia's fraction^2 / Galedon's fraction. 

For the following tier of 3125 uninhabited worlds, I estimate (very roughly, from the progression of Dieron to Irurzun to Altair to Thessalonika) 216 million as the top population, and Ningxia^3 / Galedon^2 for the median and foot.

Here's where it breaks down.

Once I get beyond halfway in the 625 tier, world values depend more on the equation I picked than on the initial values; it swings too much to make any conclusions about "uninhabited" planets, or how frequently "inhabited" planets double up around a single star.

More rigorous methods and actual logistic/population equations might yield better results.