IEEE Computational Intelligence Magazine - May 2021 - 71

the consequence of changing C values
is minimal. The role of the inspection
cost is significant only when the game
has an intermediate level of difficulty
and, as expected, increasing inspection
cost C promotes higher cooperation.
This means that relying on the inspection cost to promote tax compliance is
more worthy when the population has
a mixture of cooperators and defectors.

trends of the BA scale-free networks
are similar. Networks with lower density (m = 2 and fitted network from
data) are able to better promote cooperation when a is increasing (the
game is harder). When the game is
easy (low a values), higher density is
better for achieving total cooperation
because it increases the speed of diffu-

sion through the network. These
results are in line with the well-mixed
population output, which jumps from
total defection to total cooperation
when the game is easy. This abrupt
shift in the model results is in agreement with the observation in Figure 1, where we have two extreme
cases (defection and harmony games)

B. Scale-free Networks Analysis

Final Frequency of Cooperators

1

0.8

Fitted Network From Data
Well-Mixed Population
SF (BA With m = 2)
SF (BA With m = 4)
SF (BA With m = 6)
SF (BA With m = 8)

0.6

0.4

0.2

0
0.7

0.6

0.5

0.4
0.3
α Values

Fitted Network From Data
Ass. SF (p = 0.5, m = 2)
Ass. SF (p = 1, m = 2)
Diss. SF (p = 0.5, m = 2)
Diss. SF (p = 1, m = 2)

1
Final Frequency of Cooperators

Here, we compare the dynamics of different scale-free networks with respect to
a well-mixed population. Apart from the
fitted real scale-free network, we considered four networks generated by the
Barabasi-Albert (BA) algorithm [57] for
different values of parameter m, controlling the average degree of the networks.
Additionally, we used the Xulvi-BrunetSokolov algorithm [58] to obtain
-assortative and disassortative scale-free
networks. The assortativity property of
networks denotes the preferences of
highly connected nodes to be connected
with other highly connected nodes [21].
On the other hand, the disassortativity
property denotes the preference of highly connected nodes to connect with less
connected nodes. Parameter p of the
Xulvi-Brunet-Sokolov algorithm is used
to control the degree of assortativity and
disassortativity of existing scale-free networks. In our case, we applied the algorithm to the most and least dense
networks generated by the BA algorithm
(i.e., m = 2 and m = 8). Thanks to these
network generation algorithms, we
employed 12 networks with diverse clustering coefficient (CC), diameter (D),
and density. Table III lists the features of
the networks.
Figure 8 shows the dynamics of the
model with the networks and a wellmixed population. We can see in the
upper plot of Figure 8 that cooperation is non-existent with a well-mixed
population, except when a is lower
than or equal to 0.2. Note that the
expected level of coexistence in the
well-mixed population can be analytically derived under some settings of
the tax fraud game. In the upper plot
of the figure, we can also see that the

0.2

0.1

0

Ass. SF (p = 0.5, m = 8)
Ass. SF (p = 1, m = 8)
Diss. SF (p = 0.5, m = 8)
Diss. SF (p = 1, m = 8)

0.8

0.6

0.4

0.2

0
0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

α Values
FIGURE 8 The upper plot shows the comparison among the well-mixed population, a network
obtained from real data, and scale-free networks generated by the BA algorithm for different
a values. The lower plot shows the comparison among different levels of assortativity and disassortativity in the scale-free networks. Density and assortativity impact the level of cooperation depending on the a values. A well-mixed population can only achieve cooperation when
the game is trivial (a # 0.2).

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