_{1}

Based on people or agent community, we use the Zaklan model as a mechanism to control the tax evasion fluctuations. Here, we use the non-equilibrium Sánchez-López-Rodríguez model (SLR), i.e . directed Watts-Strogatz networks, as a dynamics of the temporal evolution of the Zaklan model. We simulate the response of tax cheaters to punishment by an auditing authority as well as to the behavior of their neighbors. The higher the punishment is, the smaller is the simulated probability to cheat. This reasonable result shows that our model is qualitatively good.

Over the past three decades, the Ising model (IM) was used successfully in the study of biological, social, behavioral, opinion dynamic and economic systems [

Zaklan et al. [

Lima [

Many social and economic networks in the real world exhibit directed relations. This may be modeled by including directed links in the corresponding complex network model. The motivation of this work is to study tax evasion on directed SW networks (DSW) via a non-equilibrium dynamics model (SLR) with the objective to make this model as realistic as possible.

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 DSW networks, but now using a non-equilibrium SLR model proposed by Sánchez, López and Rodríguez (SLR) [

Small-world (SW) networks have typical distance between two randomly selected nodes which increases only logarithmically with increasing number of nodes. SW networks are intermediate between the regular local networks and the random networks. They have two interesting features as high clustering which is the characteristic of regular networks, and the other is short path length that is characteristic of random networks. The combination these two features suggests that SW networks can be used to describe the behavior of various real systems that present interactions between nodes, agents or people as social and economic systems. Here we briefly describe the DSW and undirected SW (USW) networks used in this study as previously mentioned.

・ DSW networks

In the SW networks, we can introduce an asymmetric disorder, in such a way that we redirect a fraction p of the links. This redirecting results in a directed network, preserving the outgoing node of the redirected link but changing the incoming node, i.e., when A is tied to B, B may not be linked to A but to someone else instead. When

・ USW networks

In the USW networks, different from DSW networks, there exists a reciprocity or symmetry of redirected links, i.e., if node A selects node B as incoming neighbor then A is also an incoming neighbor of B. The neighbor relations were such that if A has B as a neighbor; B has A as a neighbor.

The original Zaklan model [

where the sum is carried out over the l mates of agent i. The external noise or social temperature T is included allowing some degree of randomness in the time evolution. Then, for a given value of the external social tem- perature, the update of the model is performed as follows: at each step, an agent (network site) is randomly chosen and its corresponding

・ 1): If

・ 2): If

which depends on temperature, i.e., an unfavorable change. Therefore, this model is a non-equilibrium model, since detailed balance is not satisfied, due to the directedness of the links.

Phase transition theory distinguishes between first-order or discontinuous transitions and second-order or continuous transitions. For first-order transitions, the quantity of main interest, like the density of liquids in equilibrium with their vapour at fixed pressure, jumps discontinuously at some boiling temperature if the liquid is heated. Ferromagnets if heated show a second-order transition: at a sharp Curie temperature their magneti- sation goes to zero continuously though with infinite slope. Tricritical points in more complicated systems separate continuous transitions, at one side of the tricritical point, from discontinuous transitions at the other side. For our DSW networks, the tricritical point lies at a rewiring probability

In order to model tax evasion, we further use for all agents one probability of an efficient audit

The fraction of tax evaders is

where N is the total number and

For SLR it is known that for

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

In

we plot the baseline case

To understand statistical errors, in

In this work, we used the SLR model as a temporal evolution dynamics of the Zaklan model. This model explores the effect of a directed small-world topology. On DSW networks, the SLR model presents two types of phase transition dependent rewiring probability p. And

What is needed in the future are more quantitative empirical data how tax cheaters react to punishment. For example, some governments bought bank account data stolen from Swiss banks, and as a result, lots of people admitted to the tax authorities that they did not report the income from the investments on these account to them. But details on amount of income, amount of punishment, and behavior in later years are hidden behind tax secrecy.

The author thanks Dietrich Stauffer for many suggestions and fruitful discussions during the development of this work. He thanks CNPq and FUNCAP for financial support. This work also was supported by the system SGI Altix 1350, the computational park CENAPAD, UNICAMP-USP, SP-BRAZIL.

Francisco W. S. Lima, (2016) Tax Evasion Dynamics via Non-Equilibrium Model on Directed Small-World Networks. Theoretical Economics Letters,06,819-826. doi: 10.4236/tel.2016.64086