Complex network model of the phase transition on the wealth distributions - from Pareto to the society without middle class

A model of distribution of the wealth in a society based on the properties of complex networks has been proposed. The wealth is interpreted as a consequence of communication possibilities and proportional to the number of connections possessed by a person (as a vertex of the social network). Numerical simulation of wealth distribution shows a transition from the Pareto law to distribution with a gap demonstrating the absence of the «middle class». Such a transition has been described as a second‐order phase transition, the order parameter has been introduced and the value of the critical exponent has been found. The distribution of wealth among the people in many countries is described by the Pareto law [1‐2], i.e. the number of persons   N k depends upon their wealth k according to   N k k    , where  is a constant. Pareto distributions (scale free distributions) occur in many fields, for instance, in energy distribution of earthquakes (the Gutenberg‐‐Richter law [3]), word frequency distribution in texts (the Zipf law [4]), size of fragments during material breaking [5], etc. The universality of such power distributions is caused by the fact that they are characteristic for the systems, which formation involves (is related to) clusterization, when the union of the «rich» with the «rich» is more preferable as compared with that of the «rich» with the «poor» or the «poor» with the «poor». The effect of priority of such clusterization was named by R. Merton as the Matthew effect [6], according to the biblical quote – «For unto every one that hath shall be given, and he shall have abundance…». However, there are economies with wealth distribution that do not follow Pareto law. In such cases the function   N k has a step (a gap) which can be interpreted as the absence of the middle class [7, 8]. Such an economy seems to be much less acceptable than that one that follows Pareto law. 2 In present work we propose an approach based on theory of complex networks [9‐11]. Introduction of additional rules for wealth allocation triggers Pareto distribution to distribution without middle class. We described such a transition in terms of the theory of phase transitions [12,13] where we revealed the presence of a threshold, introduced the order parameter, and the critical exponent was calculated. Here we consider the society as the graph and …