_{1}

The Zaklan model has become an excellent mechanism to control the tax evasion fluctuations (TEF) in a people- or agent-based community. Initially, the equilibrium Ising model (IM) had been used as a dynamic of temporal evolution of the Zaklan model near the critical point of the IM. On some complex network the IM presents no critical points or well-defined phase transitions. Then, through Monte Carlo simulations we study the recurring problem of the TEF control using the version of non-equilibrium Zaklan model as a control mechanism for TEF via agent-based non-equilibrium majority-vote model (MVM). Here we study the TEF on directed Barabási-Albert (BAD) and Apollonian (ANs) networks where the IM is not applied. We show that the Zaklan model can be also studied using non-equilibrium dynamics through of the non-equilibrium MVM on complex topologies cited above, giving the behavior of the TEF regardless of dynamic or topology used here.

The social and economic behavior of a community of people has been successfully studied using the Ising model close to its critical points [

According [

In Germany, the loss in taxes and social security by unreported “informal” work has been claimed to correspond to nearly two percent of the Gross Domestic Product.

Zaklan et al. [

In this work, we study the behavior of the tax evasion on an agent community of honest citizens and tax evaders, where the agents are positioned on sites of complex networks, but now using a version of non-equilibrium of the Zaklan model proposed by Lima [

The non-equilibrium model proposed by Lima [

R. Wintrobe and K. Gërxhani [

Here we briefly describe the complex networks used in this study as previously mentioned.

The Apollonian network is composed of

The undirected Barabási-Albert network [

In directed Barabási-Albert networks, the network itself is produced in the undirected Barabási-Albert networks way [

The traditional Zaklan model [

where

In this model an agent has an opinion and can assume the value

We further use a probability of an efficient audit p. Therefore, if tax evasion is detected by this audit, the agent must remain honest for a number k of time steps. Again, one time step is one sweep through the entire network.

In order to verify if there is a phase transition in MVM models on ANs Lima et al. [

The fraction of tax evaders is

where N is the total number and

For MVM it is known that for

Here, we follow the same steps we did in a previous work [

In

fluctuates near 0.082 (DBA). Case 5(b) with

In case 5(d) and 6(d) we also show that the tax evasion level decreases much more for an extreme set of punishment

To understand statistical errors, in

In this work we show that the Zaklan model of tax evasiom is very robust because we use the non-equilibrium dynamics of the MVM to simulate the Zaklan model, with results similar to equilibrium dynamics of the IM [

F. W. S. Lima thanks Dietrich Stauffer for many suggestions and fruitful discussions during the development of this work. We thank CNPq and FUNCAP for financial support. This work also was supported the system SGI Altix 1350 the computational park CENAPAD.UNICAMP-USP, SP-BRAZIL.

Francisco W. S.Lima, (2015) Tax Evasion Dynamics via Non-Equilibrium Model on Complex Networks. Theoretical Economics Letters,05,775-783. doi: 10.4236/tel.2015.56089