The social and economic behavior of a community of people has been successfully studied using the Ising model close to its critical points [1] -[5] , see also [6] -[9] .

According [10] [12] -[14] , the tax evasion in a community of people remains a major cause of concern for governments. Second [15] , the higher levels of tax evasion generally occur in less developed countries that present a lower amount of trust that people have in governmental institutions. Gächter [16] , Frey and Torgler [17] have provided empirical evidences that indicate that tax payers tend to condition their decision regarding whether to pay taxes or not on the tax evasion decision of the members of their group or neighborhood as also provide experimental evidence on the relevance of conditional cooperation for tax morale. Within the context Zaklan et al. [7] [8] developed an economics model to study the problem of tax evasion dynamics on a people community. However, this model deals with illegal tax evasion only. They used the equilibrium Ising model on a square lattice (SL) and Monte-Carlo simulations with the Glauber algorithms to study the proposed model.

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. [8] also have studied the implications of conditional cooperation in a multi-agent-based framework. They considered a large number of agents (people) who interact locally with each other and base their decision whether to evade taxes or not on the behavior of their neighbors. They used the Ising model on a SL to study the behavior of tax evasion and furthermore add a policy maker’s tax enforcement mechanism. In their model the enforcement mechanism consists of two components: a probability (p) of an audit each person is subject to in every period, and a length (k) of time detected tax evaders remain honest.

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 [18] [19] . Here, each agent does have an opinion in the presence of a constant social noise (q) as in the traditional Majority-Vote model (MVM) [20] .

The non-equilibrium model proposed by Lima [18] [19] is based on the knowledge that we do not live in a social equilibrium and any rumor or gossip can lead to a government or market chaos. Then, a non-equilibrium model (MVM) explains better events of non-equilibrium and makes this model more realistic to explain social and economic behavior of a community of people. We ask if also this more realistic model reproduces the main results of the earlier versions and the answer is yes. Other motivation to use Lima model is that this presents a well-defined phase transition on directed Barabási-Albert and Apollonian networks. In these complex networks the equilibrium IM does not present a phase transition making it impossible to use the equilibrium Zaklan model via IM because this has no “social temperature” on these complex networks.

R. Wintrobe and K. Gërxhani [15] argued that less developed countries may have high tax evasion because of less trust in government. Zaklan et al. [7] [8] proposed a model to study this problem , called here the Zaklan model, using Monte Carlo simulations and a equilibrium IM dynamics on square lattices. Their results are in good agreement with analytical and experimental results obtained by [15] . On some topologies like ANs and DBA the IM does not present a phase transittion making this model inappropriate to study TEF. However, the MVM does not present this limitation on these topologies. The source of this distinction is due to the different behavior of noise in each of these models [27] . In the IM, the probability of switching a highly connected spin against the local majority is smaller than a less connected one; since the energy variation is larger for a more connected spin. In the MVM, the probability of a spin switching against the local majority is always given by q, independent on the number of neighbors of this spin. The motivation of this work is to study tax evasion on complex networks via a nonequilibrium dynamics model (MVM) with the objective to make this model as realistic as possible.

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

The traditional Zaklan model [7] consists of a number of homogeneous agents located on a regular or irregular structure. In every time period each network site is inhabited by an individual, spin, who can either be an honest tax payer or a cheater. It is assumed that initially everybody is honest. Each period individuals can rethink their behaviour and have the opportunity om become the opposite type of agent they were in the previous period. The network neighborhood of every individual is composed of z people, agents to network nodes. Each agent’s social network may either prefer tax evasion or reject it. Individual decision making depends on two factors: First, the type of network every agent is connected with, exerts influence on what type of citizen she becomes in the respective period. On the other hand, people’s decisions are partly autonomous, i.e. they are not only influenced by the constitution of their vicinity. The autonomous part of individual decision making is responsible for the emergence of the tax evasion problem, because some initially honest tax payers decide to evade taxes and then exert influence on others to do so as well. Applied to tax evasion we can interpret the model as follows: Tax evaders have the greatest influence to turn honest citizens into tax evaders if they constitute a majority in the respective neighborhood. If the majority evades, one is likely also to evade. On the other hand, if most people in the vicinity are honest, the respective individual is likely to become a tax payer if she was a tax evader before. The model also presents an enforcement mechanism that consists of two components: a probability of an efficient audit p; and if tax evasion is detected and punished, the individual remains honest for a number k of periods. One time unit is one sweep through the entire system. The temporal evolution this model can be performed by using an equilibrium or non-equilibrium dynamics. In the MVM on network, the system dynamics traditionally is as follows. We assign a spin variable with values to each node of the network. At each step we try to spin flip a node. The flip is accepted with probability

where is the sign of x if, if. To calculate our sum runs over the z nearest neighbors j of spin i on the network. Equation (1) means that with probability the spin will adopt the same state as the majority of its neighbors. The noise parameter plays a role similar to the temperature in equilibrium systems: the smaller q, the greater the probability of parallel aligning with the local majority.

In this model an agent has an opinion and can assume the value depending on the opinion of the majority of its neighbors.

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. [27] measured the relaxation time as a funtion of the noise parameter q, independent of our tax question. They started the system with all spins up and a number N of spins equal to 7,174,456 (). They determined the time after which the magnetisation has flipped its sign for the first time, and then took the median value of nine samples. As one can see in Figure 1, the relaxation time goes to infinity at some positive q value near 0.176 (), indicating a second order phase transition. In contrast, the Ising model on ANs [21] [22] and directed BA networks has no phase transition and agrees with the modified Arrhenius law for relaxation time [23] .

The fraction of tax evaders is

where N is the total number and the honest number of agents. The tax evasion is calculated at every time step t of system evolution.

For MVM it is known that for, half of the people are honest and the other half cheat, while for either one opinion or the other opinion dominates. Because of this behavior we set a fixed noise (q) to some values slightly below, where the case that agents distribute in equal proportions onto the two alternatives is excluded. We set with (ANs) and (DBA) such that we see flips of the whole system in the baseline cases. Then we vary the degrees of punishment (, 10 and 50) and audit probability rate (, 10% and 90%). Therefore, if tax evasion is detected, the enforcement mechanism p and the time of punishment k are triggered in order to control the tax evasion level. The punished individuals remain honest for a certain number k of periods, as explained before in Section 3.

Here, we follow the same steps we did in a previous work [18] . Therefore, we first will present the baseline case and, i.e., no use of enforcement, at and with sites for ANs and DBA. All simulations are performed over 20,000 time steps, as shown in Figure 2. For very low noise the part of autonomous decisions almost completely disappears. The individuals then base their decision solely on what most of their neighbours do. A rising noise has the opposite effect. Individuals then decide more autonomously. Therefore, Figure 2 was expanded to two examples (Figure 2(b) and Figure 2(d)), in order to show how much the results change if one uses various random numbers, average on 20 different seeds. Error bars cannot describe this randomness properly. (For the later figures the error bars are visible from the fluctuations in time which show a band of fractions.) Although everybody is honest initially, it is impossible to predict roughly which level of tax compliance will be reached at some time step in the future.

Figure 3 and Figure 4 illustrate different simulation settings for ANs and DBA, for each considered combination of degree of punishment (, 10 and 50) and audit probability (, 10% and 90%), where the tax evasion is plotted over 20,000 time steps. Both a rise in audit probability (greater p) and a higher penalty (greater k) work to flatten the time series of tax evasion and to shift the band of possible non-compliance values towards more compliance. However, the simulations show that even extreme enforcement measures (and) cannot fully solve the problem of tax evasion.

In Figure 5 and Figure 6 we plot tax evasion for ANs and DBA, but now with, again for different enforcement k and audit probability p. Now the fluctuations are much smaller since the network is nearly nine times larger. For case Figure 5(a) and Figure 5(b) we plot the baseline case and, i.e., no use of enforcement for ANs and DBA and parameters as in Figure 3 and Figure 4. The probable error for part 5(c) fluctuates near 0.0031 (ANs) and is much smaller than the symbols (circle), but the one in 6(c)

fluctuates near 0.082 (DBA). Case 5(b) with shows already a strong reduction of tax evasion on ANs, the same does not occur for DBA 6(b). In case 5(c) and 6(c) we show the tax evasion level decreases, on ANs and DBA, for a more realistic set of possible values for the degree of punishment and audit probability [7] [15] .

In case 5(d) and 6(d) we also show that the tax evasion level decreases much more for an extreme set of punishment and audit probability [7] for both networks. Therefore, our model also works for large networks.

To understand statistical errors, in Figure 5 and Figure 6 we plot tax evasion for ANs and DBA with now for the case and. We found from 20 samples in part (c) that the tax evasion remains at around 20% (ANs) and 30% (DBA), but with fluctuations in time larger than from sample to sample: The probable errors are much smaller than the fluctuations seen in part (c).

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 [7] [8] , and also on various topologies [18] . Therefore, as the model of Zaklan incorporates concepts from both sociophysics and econophysics, we argue here that the best framework for simulating this kind of a model is the one of complex networks like ANs and DBA networks, where citizens always make a prior consultation with their nearest neighbors before making any final decisions. Also here we found the plausible result that tax evasion is diminished by higher audit probability p and stronger punishment k on both ANs and DBA networks.

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