Somebody check my #math (and #thinking), please.

Following recent discussion about centralization of #fediverse I am trying to establish the "desirable" sizes for instances, following certain assumptions I list below (with links to their sources).

A. The #instance is a #community (social network), thus its functionality follows the dynamics of such.

B. The close-knitted community (village) size (I know you) I derive conservatively from Dunbar's Number = 150

C. The medium-sized community I calculate based on 2 degrees of separation (I know someone who knows you). Which is a typical dynamics for a small town community.

D. The large community size (I know someone who knows someone who knows you) comes from the assumption that the limit is at the "influence dissipation" size, thus three degrees of separation. This one corresponds with a bigger town or a city district, being the largest population that still maintains natural mechanisms of cohesion, control & communication.

Of course all specific numbers are approximate. The key thing, I believe, is to give people a way to assess what community they want to create or join.

The math I use comes from here and here.

So, I start from
"An average path length between two nodes in a random network (D) is equal to ln N / ln K, where N = total nodes and K = acquaintances per node."

In my case I know K=150 and D being either 2 or 3. I need to calculate N

N = e^(ln K * D)

1. Small community (I know you) will then be 150 persons.

2. Medium community (I know someone who knows you) will then be

D = 2
K = 150
ln K = 5.010 635
ln N = 10.02127
N = 22 500 (and I am baffled )

3. Large community (I know someone who knows someone who knows you) will then be:

D = 3
K = 150
ln K = 5.010 635
ln N = 15.031905
N = 3 374 997 (and I am stupefied )

I am aware of the power of the exponential growth, but still, results #2 and #3 are orders of magnitude above what I expected. And I do not understand why.

Would someone point me towards understanding (and better model), please?