Power-law exponent of the Bouchaud-M\'ezard model on regular random network

We study the Bouchaud-M\'ezard model on a regular random network. By assuming adiabaticity and independency, and utilizing the generalized central limit theorem and the Tauberian theorem, we derive an equation that determines the exponent of the probability distribution function of the wealth as $x\rightarrow \infty$. The analysis shows that the exponent can be smaller than 2, while a mean-field analysis always gives the exponent as being larger than 2. The results of our analysis are shown to be good agreement with those of the numerical simulations.