A theoretical approach for Pareto-Zipf law

Abstract: We suggest an analytical approach for Pareto-Zipf law, where we assume random multiplicative noise and fragmentation processes for the growth of the number of citizens of each city and the number of the cities, respectively.

1 Introduction: We show that the random multiplicative noise and the fragmentation processes [1] are conditionally similar and they give exponential variation within the related parameters with time. Hence, the probability of finding a city with a given population (size) decreases with this size and we have Pareto-Zipf law. The original theory is presented in the following section and the next one is devoted for discussion and conclusion.

2 Theory: We define a uniformly distributed random number (0≤ξ<1) which is utilized for several aims in the present simulation: For example, we use (ξI,t) at (t) for the city (I) with (0≤ξI,t<1), etc. Population of the cities (PI(t)) grow in time (t) with a random rate RI=RξI,t, where R is universal within a random multiplicative noise process,

PI(t) = (1 + RI)PI(t − 1) .

(1)

Please note that, as the initial cities grow in population (size) they fragment in the meantime. [2] Eq (1) may be written as

PI(t) =∏t=1 t (1 + RI)PI(1) ,

(2)

where PI(1) is the initial population of the city (I):

PI(1)=ξI,0Pmax ,

(3)

where (Pmax) is the maximum of initial population that the ancestor cities may have and (ξI,0) is as in the first paragraph here. Thus, Eq (2) may be written as

PI(t) =∏t=1 t (1 + RξI,t)ξI,0Pmax ,

(4)

where (R) is the maximum for the growth rate of the cities (and similarly Eq. (1) becomes PI(t) = (1 + RξI,t)PI(t − 1)). The number of the cities (M(t)) living at (t) increases with time in terms of fragmentation or emergence; and, the random fragmentation process can be conditionally related to the random multiplicative noise process.

2.1. Similarity between the random multiplicative noise and fragmentation processes We assume (M(t)) to vary as

M(t) = (1 + Eξt)M(t − 1) ,

(5)

where (E) is the (net, i.e., after subtracting the maximum rate for the random extinction) maximum for the random emergence rate of the cities. Eq. (5) may be written as

M(t) =∏t=1 t (1 + Eξ,t)ξ0M0 ,

(6)

where (M0) is the number of ancestor cities. Please note that, the random fragmentation is the same as random extinction of the given city which is replaced by two cities which emerge newly at t and the number of the current cities (M(t)) increases by 1, which is not important due to the present randomness. Suppose that the splitting ratio for the given fragmentation is S; this means that the offspring cities will have the following populations: PI(t)=SPI(t) and PM(t)+1(t)=(1-S)PI(t)) both of which are clearly less than the population of the fragmented city. It is clear that the offspring cities may be interchanged; i.e., (S)Æ(1-S) with (I)Æ (M(t)+1) and (1-S)Æ(S) with (M(t)+1)Æ(I). Thus a random fragmentation process may be considered as a random multiplicative noise process, where we have also a minimum for the population decay rate, since:

PI(t)=SPI(t)=S(1 + RξI,t)PI(t − 1))

(7)

which can be written as

PI(t)=(S + SRξI,t)PI(t − 1))=[1+(-1+S + SRξI,t)]PI(t − 1)
(8)

or as

PI(t)=[1-(1-S-SRξI,t)]PI(t − 1) ,

(9)

where the factor (1-S-SRξI,t) is the current (at (t)) random population decay rate and the maximum of which is (1-S) with ξI,t=0; and, similarly the mentioned maximum is (S) for (M(t)+1) with (S)Æ(1-S) or (1-S)Æ(S) for (I) as mentioned within the text before Eq. (7). Please note that, (S=1/2) may be taken as universal as well as a uniform random number, where average of (S) gives ½ in the long run. Secondly if (S≈0) or (S≈1) then it means that we do not have fragmentation practically, where one of the new cities goes extinction immediately.

2.2. Exponential growth in the population of each city and the number of the cities Eq. (1) reads

PI(t) − PI(t − 1)= RIPI(t − 1) ,

(10)

which may be written as

ΔPI(t) /Δt=RIPI(t)

(11)

or

ΔlnPI(t) = RIΔt

(12)

and similarly for (M(t)) in Eq. (6), where ln is the natural logarithm. Hence, the average of the logarithm of (PI(t)) in Eq. (12) increases with Rt/2 in time (t) since (RI=RξI,t) and the average of the uniform numbers (ξI,t) between 0 and 1 is ½. Therefore, whatever the populations of the cities (PI(t)) at a time t are, the probability of finding a city with a population P decreases exponentially with P which gives the power law with exponent -1 (Pareto-Zipf law) as we show in the next section. A different derivation of more general power laws was given by Levy and Solom

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