IEEE Computational Intelligence Magazine - May 2021 - 68

worse fitness). Mathematically, agent i
with strategy X adopts strategy Y of agent
j (a randomly selected direct contact of i)
with a probability given by Equation 1,
where b is the intensity of selection
parameter, set to 0.5 in the model.

prob tij =

1
t-1
t - 1 . (1)
1 + e -b( f j - f i )

IV. Data Analysis and Parameters
of the Model
A. Data Description

Most of the parameters in the model
were set using real data. The real data
used for our study includes VAT declarations by businesses registered in the
Canary Islands (Spain) in 2002 about
transactions, with persons or firms,

exceeding 3,005.06 euros. The anonymized data was accessible only within the
tax administration under confidentiality
agreement of not revealing any information that could be used to identify either
the buyers or sellers, and under the commitment that the data was not to be
passed on to third parties in any format.
Every taxpayer must independently
declare purchases and sales. Therefore, all
transactions between two firms should be
declared twice-once each by the seller
and the buyer. The original information is
split into two files: one with the total sales
in the economy and the other with total
purchases. A database was built by merging both files and taking the transactions
(sales and purchases in 2002) declared by
the same firms in the two files into
account. Transactions without a counter-

Probability (Cumulative)

100
10-1
10-2
10-3
10-4
100

100.5

101

101.5
Degree

102

102.5

103

FIGURE 3 Degree distribution of the VAT declaration network.

Probability (Cumulative)

100
10-1
10-2
10-3
10-4
10-5
106

107

108
Weight

FIGURE 4 Edge-weight distribution in the real VAT declaration network.

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2021

109

part were eliminated, since in some cases
(e.g., individual buyers) a counterpart is
not obliged to declare. Transactions where
both counterparts match were also
removed. Finally, the number of firms in
the database is N = 32, 886 including
E = 197,791 operations.
B. Generation of the Real Social
Network of Transactions

In order to examine the structure of VAT
declarations, we set up a network of
firms, where two firms are linked when
they declare transactions between them.
The network is undirected, since we do
not differentiate whether the firm acts as
a buyer or a seller, although the amount
declared by the seller is included as a link
weight. Specifically, we use a scale-free
network starting with the degree distribution of the VAT declaration network
(Figure 3), which is a long-tailed distribution. Following the methodology in
Clauset et al. [55], the network fits to a
power law k -c with x min = 88 and
c = 3.04. According to the taxonomy in
Broido and Clauset [24], the VAT declaration network is a weak scale-free network, since its power law cannot be
rejected and it includes more than 50
nodes. For the model, we have built a
scale-free network of 10,000 nodes, with
the same exponent c and x min.
C. Feeding Transaction
Parameters from Data

The values and distribution of the quantities to be paid " d L, d H , were set from
the empirical data. Figure 4 shows the
edge-weight distribution by considering
sellers' declarations (note that we obtained
a similar distribution when we looked at
buyers' declarations). More than 95% of
the transaction amounts declared are
under 1 million euros, whereas 0.01% are
over 100 million euros. On the other
hand, according to official records, the
percentage of large firms (those with
more than 20 employees) in the Canary
Islands is about 2% [56].Therefore, we set
the probability of an edge to have a high
transaction to 0.02 (i.e., prob dH = 0.02).
Additionally, we obtained a ratio
r = 45.7589 between the average values
of the total volume of transactions of

IEEE Computational Intelligence Magazine - May 2021

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