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Topological Properties of the Bornhold Model
R. Borrasca Netoa, B. J. Beckera,b, C. R. da Cunhaa
aInstituto de F́ısica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS,
91501-970 Brazil
bWarren Corretora de Tı́tulos e Valores Mobiliários e Câmbio Ltda, Porto Alegre, RS,
90035-190 Brazil
Abstract
We study an extension of the Bornholdt spin-market model using config-
urable network topology. The distribution of returns was studied using the
probability plot correlation coefficient and indicates three different behaviors
for the tails of distribution of returns. The price volatility was studied by
fitting the magnetization with a Wiener process and a power-law behavior
was found for the volatility as a function of the levels of randomness and
connectivity of the network. Both parameters have opposing effects on the
risk as inferred from the Shannon entropy of the magnetization. Also, we
show that trading can spatially auto-organize depending on the conditions
of a control space. Finally, we show that there is a range of values of the
control space that renders a model that reproduces real-market data.
Keywords: Econophysics, Ising model, Watts-Strogatz network,
small-world network, Bornholdt model
1. Introduction
MELHORAR
The Ising model1 is a well stablished tool in statistical mechanics mostly
used to study some magnetization properties of materials.2,3 More recently,
many adaptations of this model have been proposed to study financial sys-5
tems.4,5, 6, 7 A particularly interesting adaptation is the Bornholdt model7
wherein spins are mapped to buying/selling behavior.8,9, 10,11
Email address: creq@if.ufrgs.br (C. R. da Cunha)
Preprint submitted to Physica A July 12, 2021
The Bornholdt approach yields some properties of a real-market such as
heavy tails and volatility clustering.7,9, 11 On the other hand, it is still a non-
realistic model, since, for instance, it only considers interactions on a regular10
grid. Here we extend Bornholdt model to a more realistic topology given by
the small-world model of Watts and Strogatz12 (WS).
The goal of this paper is to propose a model with small-world network
topology that yields real financial market characteristics, and compare cer-
tain aspects of the artificial market with real data.15
2. Bornholdt Model
In the instantaneous Bornholdt model,7 the dynamics of each spin Si ∈
[−1,+1], i = 1, . . . , N2 in a N × N square lattice is updated in accordance
with a heat-bath dynamics given by:
Si(t+ 1) =
{
+1 with probability p =
[
1 + e−2βhi(t)
]−1
,
−1 otherwise.
(1)
In this equation, β is the inverse temperature, and hi is a local magnetization20
given by:
hi(t) =
N∑
j∈N (i)
JijSj − αSi
∣∣∣∣∣∣ 1|N (i)|
∑
j∈N (i)
Sj
∣∣∣∣∣∣ , (2)
whereN (i) indicates the neighborhood of the ith site (usually a von Neumann
neighborhood on a regular 2D grid), J is a constant matrix with elements
related to the coupling to each nearest neighbor. This corresponds to a ferrog-
magnetic order where spins tend to align in acoordance with their neighbors.25
The constant α > 0 links the local magnetization to a demagnetizing field
that seeks global anti-ferrogmagnetic order through dipole interaction. This
dynamics produces metastable phases as shown in Fig. 1.
For the study of a particular asset, the spin state maps to two distinct
market behaviors: seller (−1) and buyer (+1). Therefore, the total excess30
demand, and consequently the price of such asset, is mapped to the lattice
magnetization, given by:
M(t) = 〈S〉 = 1
N
N∑
i=1
Si. (3)
2
Lisete
Sticky Note
No segundo termo, dentro do módulo, deve haver divisão por N e não N cursivo. Assim, o segundo termo confere acoplamento a magnetização.
Lisete
Sticky Note
h_i is a local field
Lisete
Sticky Note
A descrição de h_i está carente. Para o primeiro termo, é bom mencionar que é uma contribuição igual a do modelo de Ising. E que o spin_i tende a ocupar a configuração de menor energia, alinhando-se a seus vizinhos, i.e contribuição ferromagnética. Podemos comentar aqui sobre a natureza de um investidor de imitar seus pares, uma vez que isso é uma prática válida, fácil e que não requer procura de informação.

Quanto ao segundo termo, a descrição está bem sucinta. Falta por no contexto do trader. Esse demagnetizing field estaria constantemente impulsionando o trader a mudar de estado S_i = +-1, o quao maior for |M|. Mas por que? Na prática, esse termo faz o trader alinhado a M desalinhar-se, tanto como o contrário. Ou seja, reflete tanto o comportamento de querer comprar na baixa e vender na alta (e desejar estar na minoria, desalinhado a M), tanto como o comportamento de trend following do preço e movimento de manada/rebanho (portanto, desejando estar na maioria, alinhado-se a M).
t = 0 t = 1000 t = 2000
t = 4000t = 3000
Figure 1: Snapshots of a 32x32 lattice with T = 1.5, α = 4 and J = 1. Monte Carlo simulation steps.
Through the time steps, there are metastable phases, defined by clusters of specific behaviors (black
squares represent buyers, whereas whites squares represent sellers), such as in t = 2000 and t = 3000.
After a while, there is a sharp transition where information about the past structure of the market is
quickly lost, as in t = 1000 and t = 4000.
Under this approach, the log-returns are given by:
r(t) = ln
[
M(t)
M(t− 1)
]
. (4)
One of the major strenghts of this model is that it exhibits some stylized
facts such as non-Gaussian fluctuations and volatility clustering.7,13 On the35
other hand, the simplicity of the model does not allow it to be finely tuned
to show exactly the same behavior found in real markets.
3. Proposed Model
In order to improve Bornhold’s model, we keep the same dynamics de-
scribed by Eqs. 1 and 2, but instead of using a N−regular lattice, we impose40
this dynamics on a Watts-Strogatz12,14,15 network described by a control
space C = (K, β) | K ∈ N∗, β ∈ [0, 1]. In order to construct such network,
we start with a lattice wherein each of its N nodes are connected to their first
K neighbors. Nodes are then randomly rewired with a probability β that
yields small-world properties.12 This network model is convenient to study45
the behavior of real populations, since its topology can be finely adjusted to
reflect realistic social structures.14,15 On the other hand, the network model
3
Lisete
Sticky Note
Essa descrição é problemática. Preço não é igual a M. Isso dá preços negativos e retornos indefinidos.

P = exp(M), assim o log retorno será linear em M: ret(t) = M(t) - M(t-1).

Essa descrição está atualizada no segundo paper: Kaizoji et al 2002. 
alone does not say anything about the economic interactions between its
nodes.
M
(t
)x
1
0
-2
−1
0
−2
−2
0
−4
2
0
4
2
4
−2
−4
−2
−4
β = 0 β = 0.01 β = 0.5
K = 2 K = 8 K = 240
0 250 500
Time - t
0 250 500 0 250 500
Figure 2: Top - Simulations with 4 different values of β for a Bornholdt model applied to a Watts
Strogatz network, with 2401 agents and K = 4. The gray line represents a brownian motion fit, while
the black line represents the magnetization of the system. Bottom - Simulations with 4 different values
of K for a Bornholdt model applied to a Watts Strogatz network, with 2401 agents and β = 0.01. It is
notable that for K = 240 ( 10%N), the magnetization presents a almost-linear behavior, that does not
fit well with real data. The gray line represents a brownian motion fit, while the black line represents the
magnetization of the system.
Figure 2 shows that the improved model can produce time series for the50
magnetization with different behaviors depending on the control space C.
Next we detail some properties of the new model.
4. Properties of the Model
In order to produce a realistic econophysical model, not only it is impor-
tant for it to reproduce relevant stylized facts but is equally important that55
it can be tuned to reproduce specific bahavior found in real market systems.
In this section we investigate the distribution of returns and their volatility,comparing them with those of the S&P500 index. Later we explore the eco-
nomic risks associated with the model by studying the entropy of system.
At least 1024 nodes were used in all simulations. The first XX simulations60
steps were used for thermalizing the system and the remaining XXX steps
were used to compute the statistics. The magnetization of the network was
calculated using Eq. 3 and the corresponding log-returns were calculated
using Eq. 4.
The S&P500 index was particularly chosen because, instead of a single65
asset, it captures the information of a group of stocks. Figure 3 shows a
4
time series for the S&P 500 index between XX/2020 and XX/2021 with data
obtained from XXXX. The returns for this experiment data set were also
computed using Eq. 4.
β [%]
0 25 50 75 100
0
0.22
0.44
0.66
Figure 3: Stock prices for the S&P index from 2020 to 2021, alongside with a fitted Brownian Motion
time-series. The inset shows a map with optimized values for the pair of parameters (β,K). Inside the
curve, in the colored area, all combinations of β and K yields a good aproximation for the volatility of
the S&P index, with tolerance of O(10−7).
4.1. Distribution of Returns70
One of the most important features of the Bornholdt model is the presence
of non-Gaussian fluctuations in the distribution of returns.7 These fluctua-
tions can be studied using a probability plot correlation coefficient (PPCC)
analysis using the Tuckey-Lambda16 (TL) as a theoretical reference.13 This
is a distribution whose probability density function does not have a closed75
form, but its quantile function is given by:
Q(p;λ) =
 λ
−1
[
pλ − (1− p)]λ
]
for λ 6= 0,
ln
(
p
1−p
)
otherwise.
(5)
In this last equation, λ is a parameter that dictates the shape of the distri-
bution. For instance, λ = −1 approximates a Cauchy distribution, λ = 1
approximates a uniform distribution, and λ = 0.14 approximates a normal
distribution.80
In the PPCC procedure, the correlation between the quantile of a data
set and that of the TL distribution is calculated for different values of λ.
The parameter that yields the highest correlation is taken as an estimator
for the distribution of the data set under investigation. Values of λ smaller
than 0.14 indicate heavy tails, whereas λ bigger than this value indicate a85
5
Lisete
Sticky Note
2020 - 2021 é um periodo de alta vol, vale comentar. Uma amostra maior é aconselhável.

Existe também a a questão de escala. Que justificativa existe que podemos comparar os dados diarios de sp500 com os da simulações?
β [%]
1.0 17.5 34.0 50.5 67.0 83.5 100
λ x 102
0
2
4
6
8
10
0.30
0.69
1.07
1.46
K
 [
%
N
]
Figure 4: 2D map with the corresponding fitted λ for each combination of K and β, for a WS system
with 1024 agents.
light tailed distribution. A map the λ obtained from the magnetization as a
function of the control space is shown in Fig. 4.
As shown in the figure, small values of K and β (colocar valores) delin-
eate a boundary that separates light and heavy tail distributions. The bigger
the value of β, the more random the network becomes, and according to our90
analysis, the more uniform the distribution of returns becomes. High values
of λ (λ > 1) indicate a localized distribution. This corresponds to a situ-
ation where the magnetization becomes binary producing sharp transitions
between saturating values.
On the other hand, low levels of randomization (conversely, high values of95
regularity) of the network as well as a small connectivity levels make it more
difficult for the magnetization to saturate. Nonetheless, the regularity of the
network tends to make the transitions of the magnetization more similar,
producing a heavy tail distribution.
It is possible, though, to fine tune the distribution of returns by modifying100
the topolgy of the network through the values of the control space. The
estimated λ for the S&P500 varies between XXX and YYY and this can be
obtained with a control space C = ZZZ.
4.2. Price Volatility
As previously shown in Fig. 2, it is visible that the volatility of the returns105
vary depending on the values of the control space. In order to obtain more
information about this volatility, the magnetization was fitted with a Wiener
process described by Eq. 6:
dS(t) = µdt+ σdWt, (6)
where µ is a drift parameter and σ is the volatility. The latter as a function of
the control space parameters is shown in Fig. 5. In both cases, the volatility110
6
fits well with a power-law function of the type Axb. Whereas the fitting
gives a power exponent of ∼ −0.37 for the dependency with K, it gives an
exponent of ∼ 0.24 for the dependency with β. Consequently, the volatility
shows a self-similar behavior as a function of either parameter.
−4
−3
−3
1 2 3 −2 −1 0
Figure 5: Volatility σ as a function of WS parameters K (left) and β (right) for a simulation performed
with 2401 nodes.
The inset of Fig. 3 shows the locus of values of the control space (β,K)115
that produce the same volatility of the S&P500 index within an O(10−7)
error. ESTES VALORES BATEM COM OS DA SUBSECAO ANTERIOR?
4.3. Accumulated risk
For a discrete random variable X with a defined set of possible outcomes
[x1, x2, . . . , xN ] the Shannon entropy
17 (Ssh) of the system can be interpreted120
as a measure of uncertainty, defined as:
Ssh(X) = −
N∑
i=1
P (xi) logP (xi), (7)
where P (xi) is the probability of finding X = xi.
If we consider the price of an asset to be a random variable, its Shannon
entropy estimates the risk associated with this asset.
For processes with high entropy, the probability of a set of possible out-125
comes is approximately the same for every xi ∈ [x1, . . . , xN ] (e.g. flipping a
coin), and the risk is assumed to be high. On the other side, for processes
with low entropy, there is a state xi that is more likely than the others and
the risk is assumed to be low since this state can possibly be either pursued
or avoided.130
The entropy of the magnetization as a function of the control space is
shown in Fig. 6. The randomization of the network is reflected on the entropy
7
Lisete
Sticky Note
No gráfico, à esquerda quais eram os valores de beta? E à direita quais eram os K? Não ficou claro para mim.
of the magnetization as S increases with β. On the other hand, there is a
slight tendendy for the entropy to be reduced as the number of connections is
increased. It is possible to tune the network to obtain entropy values similar135
to those of the S&P500 index. Although, β and K have opposite effects,
values between XXX and XXX produce a network with a more realistic
behavior.
QUE TAL UM MAPA AQUI TAMBEM??
1
0
.(
S
(t
) 
−
 1
0
)
0.8
1.0
1.2
1.4
1.6
1.8
2.0
S
(t)
8.0
8.5
9.0
9.5
10.0
β [%]
K [%N]
20 40 60 800
20 40 60 80
Figure 6: The entropy of grid magnetization as a function of the WS parameters β (left) and K (right),
for a system with 1024 agents. The dashed line indicates the entropy of the S&P 500 time series.
4.4. Spatial Organization140
We define a configuration entropy Sc as a measure of the spatial orga-
nization of the network. For a generic node i with |N (〉)| neighbors, its
neighborhood has 2n possible states in which it can be organized. Therefore,
the configurational entropy for this node can be calculated using:
Sc(i) =
2n∑
j=0
pj ln pj, (8)
where pj represents the probability that the neighborhood is in a state j. The145
latter can be computed by first producing a histogram of neighborhoods. The
configuration entropy of the network is just the average entropy for all nodes
Sc = 〈Sc(i)〉.
8
We investigate the impact of the spatial entropy on the magnetization of
the network through the Pearson correlation coefficient between them. This150
is shown in Fig. 7 for different values of the control space.
Figure 7: Module of correlation between the prices and configuration entropy of the system for different
parameters of the WS model, in a 1024 agents system. For each column, there is a differentvalue of β
with fixed K, and for each row there is a different value of K with fixed β.
An erratic behavior is observed for small values of randomization β and
high connectivity levels K. On the other hand, as the network is made
more random the correlation shows an exponential decay. The stationary
correlation, though tends to be higher for low connectivity levels. Therefore,155
the spatial organization of the agents correlates with the magnetization of
the network for random networks. This suggests that, at these conditions,
trading self-organizes in the network.
COMO ISTO SE COMPARA COM OS VALORES DE BETA E K QUE
MELHOR REPRESENTAM O SP500?160
5. Conclusions
We presented a spin-based model for trading based on the Bornhold dy-
namics on a Watts-Strogatz network. By fine-adjusting the parameters of a
control space, it is possible to obtain several properties found in real-market
data.165
The distribution of returns was studied using the probability plot corre-
lation coefficient (PPCC) analysis with the Tuckey-Lambda as a reference
9
Lisete
Sticky Note
alpha não está em control space. Qua valores de alpha tem usado? A razzão entra J e alpha em Bornholdt dá a razão entre interações locais e globais e isso impacta muito o que vamos observar quando aumentamos K. Não ficou claro para mim como isso ficou tratado nos resultados.
distribution. Our analysis indicate the existence of a boundary between ligh
and heavy tail distributions. Moreover, high values of randomization of the
network yields binary transitions of the magnetization.170
The price was fitted with a Wiener process. The volatility shows a power-
law behavior as a function of the connectivity and randomness levels of the
networks. The control parameters have opposing impacts on the Shannon
entropy of the magnetization. An increase in the network randomness leads
to an entropy increase, whereas an increase in connectivity leads to a reduc-175
tion in entropy. Finally, we observed that the spatial distribution of trade
self-organizes and correlates with the magnetization for specific values of the
control space.
By comparing all theses measures of the theoretical model with those of
the S&P500 index, we found that values of network randomness around XXX180
and network connectiveness around YYY yield a model that can be used to
study real-market data.
Acknowledgements
This material is based upon work supported by the Air Force Office of
Scientific Research under award number FA9550-20-1-0377.185
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	Introduction
	Bornholdt Model
	Proposed Model
	Properties of the Model
	Distribution of Returns
	Price Volatility
	Accumulated risk
	Spatial Organization
	Conclusions