Demographics V: Urbanites

I've struggled for a while to come up with a reasonable mechanism to produce rural population numbers. The city numbers are determined by the growth system, but it doesn't account for those living outside the walls.

My first thoughts were that it should depend on the urbanization rate $u$ and infrastructure of the entire hegemony. Given $u$, the total urban population of the hegemony $P_u$, the total infrastructure of the hegemony $I_t$, and the local infrastructure $I_i$, I should be able to arrive at a reasonable rural population $P_r$: \[P_r = {I \over I_t} \left({P_u \over u} - P_u\right)\] So areas with high infrastructure will pull in a greater proportion of the total rural population. Makes sense at first.

But this can yield unreasonably high local densities. Considering the hegemony of Deribri. Deribri itself is quite small, and the more natural capital is the city of Guinrhyce (which is still in the Deribri hegemony). Guinrhyce has a population of $P_i = 44884$ and $I_i = 700$. For the Deribri kingdom/empire, $P_t = 939729$ and $I_t = 37787$. Assuming that $u$ is a function of $\bar{I}$, the average infrastructure of the hegemony, \[u\left(\bar{I}\right) = 0.96 \left(1-\exp\left(-\tau \bar{I}\right)\right) + 0.04\], where $\tau=0.00039$ is a constant which determines the speed at which the function varies, then $u\left(\bar{I}\right) = u\left(102\right) = 7.8\%$. This yields $P_r = 11108079$. Since the Kingdom of Deribri comprises 367 hexes, we can say that the population density of the entire kingdom is ${P_t + P_i \over 367 \cdot 347} = 94.605$ persons per square mile. This is pretty high. However, we can either 1) chalk it up to magic/it's my world, go kick rocks, 3) accept that the cities are too populous, or 3) increase the required distance between cities according to CPT.

Even if we accept a high density of 95 for the kingdom, there is still a really high local density based on the rural population. ${I_i \over I_t} = 1.852\%$, so Guinrhyce will support about $R_i = {I_i \over I_t} P_t = 206937$ people outside the city walls. This is a total hex density of ${P_i + R_i \over 347} = 706.409$ persons per square mile! Inside a city itself, this may be reasonable. But even leaving off the urban population still gives a density of around 600 pop/sqmi. Much too high.

However, it's still good to work through these things. These tools will be useful to approximate populations once I've reduced the overall city sizes appropriately.