Small-World Effects in Wealth Distribution

We construct a model of wealth distribution, based on an interactive multiplicative stochastic process on static complex networks. Through numerical simulations we show that a decrease in the number of links discourages equality in wealth distribution, while the rewiring of links in small-world networks encourages it. Inequal distributions obey log-normal distributions, which are produced by wealth clustering. The rewiring of links breaks the wealth clustering and makes wealth obey the mean field type (power law) distributions. A mechanism that explains the appearance of log-normal distributions with a power law tail is proposed. e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] 1 With the advance of the theory of complex networks [1], it has become clear that many social networks can be understood as small-world networks [2]. On the other hand, recent high-precision studies of economic phenomena, helped by the availability of high-quality data in digital format [3], is starting to reveal the basic characterics of wealth distribution, which form a basis and at the same time a final product of economic activities. In view of these developments, we propose a stochastic model of wealth distribution built on a complex network and present the results of our numerical simulation in this paper. Several models for wealth distribution have been proposed with some reality and success [4, 5, 6, 7]. They, however, lack one feature we consider important: In some models, each agent (economic body) goes through a stochastic process of increasing and decreasing wealth, quite independently from other agents. In some others, each agent interacts with either randomly-selected agents or neighboring agents on a fixed lattice. Apparently, such a selection of economic partnership/competition is far from reality, as is evident from complex network studies. This is one reason we propose our model, which features both the stochastic nature of economic activities and underlying network structures that are expected to be close to those of the relevant social activities. We also note that studies of economic phenomena that build upon the underlying complex networks are novel. In considering the mechanisms behind wealth distribution, it is important to include at least two basic natures of the wealth. One is a random change of the value of wealth and the other is the liquidity of it. As a minimal model that contains these features we chose an interactive multiplicative stochastic process [6], which is defined by the following evolution equation: Wi(t+ 1) = ai(t)Wi(t) + σ Wi(t) + ∑ j(6=i) Jij(t)Wj(t)− ∑ j(6=i) Jji(t)Wi(t), (1) where Wi(t) is the wealth of the i-th agent at the time t, ai(t) is a Gaussian random variable with mean m and variance 2σ, which describes the spontaneous increase or decrease of the wealth. The rest of the terms in the r.h.s. of Eq. (1) describe transactions on business networks. We note that in Ref. [6] the process corresponding to Eq. (1) is described by a stochastic differential equation in the Stratonovich sense, which generates the second term proportional to σ when discretized. In addition, although naive dis2 cretization brings in a time interval ∆t in Eq.(1), it has been absorbed by the rescaling of the parameters. The transaction matrix Jij is chosen based on the underlying network structure: If the i-th agent is directly connected to the j-th agent by a link, we take Jij = J/Zi, where Zi is the number of agents connected to the i-th agent (“neighbors”). Otherwise, Jij = 0. The parameter J is a positive number, which represents the ratio of transactions and the wealth. Thus, the third term in the r.h.s. of Eq. (1) describes the incoming wealth, and the fourth term describes the outgoing wealth. Under this selection of transaction matrix, Eq. (1) can be written as follows: