Our city size model is cast similarly to a Pareto inequality law, given as a cumulative distribution function (CDF). Define p( N = n ) as the expected number N in a probability distribution function (PDF) of cities having population n > Define P( N > n ) as the expected number (CDF) of cities having population of at least size n .
A power law for the exact number N of cities having population at size n is, p( N = n) ~ n


This implies for the CDF—the Pareto distribution—that P( N > n ) ~ n
 
= n
   
(1):
N cities have n or more inhabitants. This is equivalent to the Zipfian “the N th city has n inhabitants” (only the x and y axes are reversed – see Adamic 2002). Hence the rank r( N
= n ) with binned intervals ( n
´, inclusive of n ) is  r( N = n ) = ∫ n
∞ p( n ´) δ n ´ (the limit of a discrete sum as interval size → 0) = n
-1/
Zipf's law , however, is that α (alpha) ~ 1 ~ 1/ α , in which case β ~ 2 in a Pareto law.

.
Our model is a variant of a Pareto model in which we define pp( N = n ) as the expected number of people in cities having population n and PP( N > n ) as the expected number of people in cities having population of at least size n . We test for results in the same form as a Pareto inequality (a CDF version of Zipf’s law and of the power law), thus:
PP( N > n ) ~ n P( N > n ) = ∫ n
∞ pp( n ´) δ n ´= ∫ n
∞ n ´ p( n ´) δ n = 1/ n

. (2) ( ??
)
In the case where

~ 1,
PP( N > n ) ~ C – A log ( n ). (3) ( ??
)
The semilog relation in our graph, Y( n ) ~ C – A log ( n ), is close to what we find as in our empirical city size distributions, that is, we are seeing something close to Zipf’s law.
The alpha is not exactly unity so that a weak power or q -exponential may in some cases give a better fit.
Please check results (2) (??) and (3) (??). (2) cannot be = 1/ n

because if alpha is 1 then the exponent is zero. How does the term ∫ n
∞ n
´ p( n
´) δ n evaluate? This will go into the revision of our article with Natasa.