Despite the ubiquitous emergence of skew distributions such as power law, log-normal, and Weibull distributions, there still lacks proper understanding of the mechanism as well as relations between them. It is studied how such distributions emerge in general evolving systems and what makes the difference between them. Beginning with a master equation for general evolving systems, we obtain the time evolution equation for the size distribution function. Obtained in the case of size changes proportional to the current size are the power law stationary distribution with an arbitrary exponent and the evolving distribution, which is of either log-normal or Weibull type asymptotically, depending on production and growth in the system. This master equation approach thus gives a unified description of those three types of skew distribution observed in a variety of systems, providing physical derivation of them and disclosing how they are related.

The wealth of a nation is changed by the internal economic growth of a nation and by the international trade among countries. Trade between countries are one of their most important interactions and thus expects to affect crucially the wealth distribution over countries. We reviewed the network properties of the international trade networks (ITN). We analyzed data sets of world trade. The data set include a total number of 190 countries from 1950 to 2000. We observed that the world trade network showed the uneven trading relationships which are measured by the disparity. The effective disparity followed a power law, < D(k) >∼ t δ , for the import and export network. We also construct the minimal spanning tree(MST) of international trade network, where each node is a country and directed links connecting them represent money flow from a source node to a target one. The topology of the MST shows the flow patterns of the international trades. From the MST we can identify the sub-economic zone if we delete the hub node. We observed that the cumulative degree distribution functions follow the power law, P>(k) ∼ k −α , with the average exponent α = 1.1(1)). We also calculated the betweenness centrality(BC) of the MST. The cumulative probability distribution of the betweenness centrality follows the power law, P>(BC) ∼ BC −β , with the average exponent β = 1.09(7).

We consider the criticality for firing structures of a simplified integrate-and-fire neural model on the regular network, small-world network, and random networks. We simplify an integrate-and-fire model suggested by Levina, Herrmann and Geisel (LHG). In our model we set up the synaptic strength as a constant value. We observed the power law behaviors of the probability distribution of the avalanche size and the life time of the avalanche. The critical exponents in the small-world network and the random network were the same as those in the fully connected network. However, in the regular one-dimensional ring, the model does not show the critical behaviors. In the simplified LHG model, the short-cuts are crucial role in the self-organized criticality. The simplified LHG model in three types of networks such as the fully connected network, the small-world network, and random network belong to the same universality class.

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