Power law in random multiplicative processes with spatio-temporal correlated multipliers

Abstract

It is well known that random multiplicative processes generate power-law probability distributions. We study how the spatio-temporal correlation of the multipliers influences the power-law exponent. We investigate two sources of the time correlation: the local environment and the global environment. In addition, we introduce two simple models through which we analytically and numerically show that the local and global environments yield different trends in the power-law exponent.Power-law distributions are ubiquitous not only in natural systems but also in social systems [1][2][3]. For instance, city sizes [4,5], firm sizes [6,7], stock returns [8,9], and personal incomes [10,11] follow the power-law(1) over large scales. This expression is widely known as Pareto's law [12] or Zipf's law [13], and it has been well investigated using various models. One well-known mechanism that generates power-law distribution is the random multiplicative process (RMP) [14][15][16][17][18][19]. This paper aims to clarify the influence of the spatiotemporal correlation of the multipliers on the power-law exponent γ. γ is known to decrease as the correlation time length increases [19]. However, there is no general formulation of RMPs with spatio-temporal correlation. To fulfill our aim, we consider two simple models, Model 1 and Model 2, in which the temporal correlation is led by the local environment and the global environment, respectively. We show analytically and numerically that the correlation time length influences the power-law exponent γ in Model 1, whereas it does not in Model 2. This paper is organized as follows. First, we introduce a binomial multiplicative process, in which the random multipliers can have only two values, and revisit the case where the multipliers have no correlation. Here, we propose a graphical method to estimate the exponent γ. Next, we separately analyze the cases of temporal and spatial correlation. We then analyze the effect of spatiotemporal correlation in Models 1 and 2. The results of the numerical simulations are compared with theoretical predictions. The paper concludes with a summary.Here, we consider a simple version of RMP,where t denotes a discrete time step and i specifies the elements (i = 1, 2, 3, . . . , N ). The variable x i (t) can represent any quantity such as population, firm size, or income, m i (t) is a random multiplier, and b is an additive * morita.satoru@shizuoka.ac.jp positive term. Although b may be a stochastic variable, we set b as a constant because it does not influence the following results. The initial condition is set to x i (0) = 1 so that x i (t) is always positive. For simplicity, we assume that the stochastic multiplier m i (t) is m + or m − , each with probability 1/2. Here, we set m + > 1 > m − > 0 and ln m + + ln m − < 0, so that eq. (2) has a stationary distribution. The cases of m + and m − are called good and bad environments, respectively. The mean of the multipliers is µ = (m + + m − )/2, and their variance is given by σ 2 = (m + − m − ) 2 /4. First,...

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