Webs, trees and branches - Complex Webs: Anticipating the Improbable - page 83

to be found in a variety of phenomena. This imbalance is ultimately interpretable as the
implicit unfairness found in complex webs.
Let us set the words of a text in order of frequency of appearance, and rank them
assigning the rank r

= 1 to the most frequently used word, the rank r = 2totheword

( )

with the highest frequency after the first, and so on. The function W r denotes the

number of times the word of rank r appears. Zipf found

K
r η ,

W ( r ) = (2.58)

where 1 and K is simply determined from ( 2.54 ).

Let us imagine that the number of times a word appears, W , can be interpreted as the
wealth of that word. This makes it possible to define the probability

η ≈

, namely the
probability that a wealth larger than W exists. According to Pareto the distribution of
wealth (actually Pareto's data were on the distribution of income, but we shall not dwell
on the distinction between income and wealth here) is

( W )

A
W k .

( W ) = (2.59)

The distribution density ψ( W ) is given by the derivative of the probability with respect

to the web variable W ,

d
dW (

ψ( W ) =− W ), (2.60)

which yields

B
W a ,

ψ( W ) = (2.61)

with the normalization constant given by

B = kA (2.62)

and the new power-law index related to the old by

a = k + 1 . (2.63)

Now let us take ( 2.58 ) into account in the distribution of wealth. Imagine that we
randomly select, with probability density p a word of rank r from the collection of

words with distribution number given by ( 2.58 ). A relation between the wealth variable
and a continuous version of the rank variable is established using the equality between
the probability of realizing wealth in the interval ( W

( r ),

, W + dW ) and having the rank in

the interval ( r , r + dr ),

ψ( W ) dW = p ( r ) dr . (2.64)

We are exploring the asymptotic condition r 1, and this makes it possible for us to

move from the discrete to the continuous representation. The equality ( 2.64 ) generates
a relation between the two distribution densities in terms of the Jacobian between the
two variates,