Consider the amount of communication a congressman has to do with both his constituency and with his fellow legislators.
Some constant fraction of the constituency probably wants to communicate with the legislator, which is c·P/L communication with him where c is a constant, P is the population of the country, and L is the cardinality of the legislature.
Meanwhile each legislator needs to communicate with all the others (or anyhow a constant fraction of them) to get things done (e.g. convince them to enact something he wants). That's about k·L communication for each legislator per thing he wants to do (where k is another constant).
If we now minimize
Suppose the communication with the constituents is by mail or e-email or telephone; but the communication with fellow legislators is face-to-face 1-on-1 meetings in random order. Further, all the legislators are along one long corridor. Then Joe Legislator typically must walk distance L to reach a random target legislator. Then the difficulty of communication with the L-1 others is then not proportional to L, but rather to its square. In that case we need instead to optimize by minimizing
Similarly if we had assumed a three-dimensional grid we would have found L∝P3/7. If we had assumed his constituents had to travel distances proportional to (P/L)1/2 to meet with the legislator, then...
So it seems that, no matter which assumption we make, some power law is the "right answer" – albeit it perhaps now is less clear what the correct power is! The best power seems fairly clearly to lie within the real interval [1/4, 1/2].
[I do not see any good argument for L∝log(P).]
ACE claims they did a survey of cross-country data, finding a good fit of L to the cube root of 2·P·W·A where P=population, W=fraction of it of "working age", and A=literacy fraction. "Very few countries have assemblies that are larger than twice the size predicted by this equation and only a few have assemblies that are smaller than half [of it]."
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