Rank-Size Rule

They are closely linked. The rank-size rule is the general statement that city population is a power function of rank, P_r = P_1 r^{-q}, for some exponent q. Zipf's law is the special, much-cited case where q equals one, so the second city is half, the third a third, the tenth a tenth of the largest. In practice many countries have exponents near but not exactly one, so 'rank-size rule' refers to the family of relationships and 'Zipf's law' to the canonical unit-exponent benchmark.

They are complementary descriptions of the same phenomenon. The rank-size rule looks at the whole distribution and summarizes it with an exponent; the urban primacy index focuses on how dominant the single largest city is relative to the next ones. A rank-size exponent well above one corresponds to a primate distribution, where the leading city is far larger than the rule would predict, which a high two-city or four-city primacy index also flags. Using both gives a fuller picture than either alone.

No — it is a description, not a mechanism. It tells you that city sizes tend to follow a power law but not why. The most influential explanation is Gibrat's law of proportionate growth: if all cities grow at random rates independent of their size, the resulting steady-state size distribution is a power law with an exponent near one, reproducing Zipf's law. So the regularity is now understood as an emergent outcome of random urban growth, but the rank-size rule itself remains purely empirical.