Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004, pp. 672∼676
Effects of Smartness, Preferential Attachment and Variable Number
of Agents on Herd Behavior in Financial Markets
Sungmin Lee
Department of Physics and Research Institute of Basic Sciences, Kyung Hee University, Seoul 130-701
Yup Kim∗
Department of Physics and Research Institute of Basic Sciences, Kyung Hee University, Seoul 130-701 and
Korea Institute for Advanced Study, Seoul 130-722
(Received 8 October 2003)
We study the effects of various factors on stochastic formation of opinion clusters, modelled by
an evolving network, and herd behaviors in financial markets. The first factor considered is the
smartness of an agent, such that the agent does not choose a linking partner from among members
of the same cluster to which the agent belongs. The second factor is preferential attachment, such
that a larger linking probability is assigned to a partner with more assigned links. The third factor
considered is the variation of the number of agents in the market. It is found that smartness and
preferential attachment enhance the herd behavior in a market with a fixed number of agents. In
a market with a variable number of agents, we find two characteristic herding behaviors. In a
market with a relatively low average number of agents, we find a relative increase in the probability
of relatively small returns. In a market with a relatively large average number of agents, we find
nearly the same behavior as with a fixed number of agents.
PACS numbers: 05.65.+b, 05.45.Tp, 87.23.Ge, 02.50.Le
Keywords: Herd behavior, Financial crashes, Financial markets, Price return
posed a model for opinion cluster formation and information dispersal by agents in a random network. In this
model, they considered a random dispersion of information, agents sharing the same information from a group
that makes decisions as a whole, and, whenever a group
performs an action, the network necessarily adapts to
this change. When the information dispersion is much
faster than trading activity in the model, the distribution of the number of agents sharing the same information shows a power law in which the distribution of return shows a power-law decay and an exponential cutoff.
On the other hand, when the dispersion of information
becomes slower, a smooth transition to truncated exponential tails and a relative increase in the probability of
extremely high returns are proved. The relative increase
in the probability of return is argued to be the signal for
‘financial crashes’ [11].
Even though the EZ model explained many important characteristics of financial markets rather well, the
model missed some key factors in the real markets. In
their model, the number of agents is fixed. In contrast,
the number of agents is variable, because participating
agents can leave the market or new agents can participate in the market. When the opinion cluster is formed
in the EZ model, the randomly selected agent is linked
to other randomly chosen agent without considering the
I. INTRODUCTION
Recently, there have been various studies in order to
understand economical and social phenomena based on
physical methods [1–5]. Similar characteristics have
been found between financial markets and physical systems with a large number of interacting units, such as
the existence of long-term volatility correlation [2, 3, 6],
the fast decay of linear correlation [7], fat tails in the
distribution of price change [2,3,7,8], switching from one
strategy group to another [9], and mutual interactions
among market participants through herding and imitation behaviors [8, 10]. Among such studies, there have
been many interesting studies on the analysis of price
change [8, 10–13]. Especially, the phenomena that the
distribution of return R and the logarithmic change of
the market price have enhanced tails larger than in a
Gaussian distribution have drawn much attention. For
the fat tail distribution, several microscopic models have
been suggested, such as the dynamic multiagent model
[14,15], the herd behavior model [11–13,16], a static percolation model [10] and a self-organized model [17].
Recently, Eguiluz and Zimmermann [EZ] [11] pro∗ Corresponding
Author Email: [email protected]
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Effects of Smartness, Preferential Attachment· · · – Sungmin Lee and Yup Kim
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specific properties of the agent. However, an agent in the
real market should have an inclination not to be linked to
any other member in the same opinion cluster to which
the agent belongs. Furthermore, an agent should want to
link with an other agent with greater information power
(or more links). In this paper, we therefore make modified EZ models in which the key factors missed in the
EZ model are included. We also aim to see how differently these modified models explain the characteristics
of herding behaviors, as compared with the original EZ
model.
II. MODIFIED MODELS WITH THE
NUMBER OF AGENTS FIXED
In this section, we introduce modified models in which
the specific properties of the agent mentioned in Section
I are considered. Here, we think only models with the
number of agents fixed. The specific properties considered are the following two factors. The first is that an
agent selects the linking partner only from among those
who are not a member of the same opinion cluster to
which the selecting agent belongs. From now on, we will
call such an agent a smart agent. In contrast, we call
an agent who selects the linking partner without considering the clustering properties a normal agent. In the
EZ model, only normal agents are considered. The second is related to the probability assignment of a selecting
linking partner among those possible. The simplest one
is random selection, i.e., to give equal probability to each
possible partner, as in the EZ model. The other possibility, which considers the information power or number
of links, is to assign the probability in proportion to the
links which the partner already has. Let us assume a
number of possible linking partners Np (t) and a number
of links of some partner i Li (t) at a certain time t. Then,
the linking probability Pi (t) of an agent to the partner i
can be assigned as
Pi (t) = Li (t)/
Np
X
Lj (t).
(1)
j=1
The assignment in Eq. (1) is called the preferential
attachment in the recently-developed complex network
theories [18–20].
Now, we will describe our modified model with a fixed
number of agents in detail. A market is composed of N
agents. There are three states Φl of each agent: an inactive state is Φl = 0 (waiting) and an active state is either
Φl = +1 (buying) or Φl = −1 (selling). Initially, all the
agents are waiting and have no links [11]. A network of
links among agents evolves dynamically in the following
way: at each time step ti (I) first an agent k is randomly
selected. (II) Next, with probability (1 − a), the agent k
chooses a linking partner and makes a link between them.
If k is a smart agent, then the possible partners are the
Fig. 1. Log-log plot of the distribution P (R) against return R for a modified EZ model with only smart agents and
preferential selection. The plot is for N = 1000 and a = 0.01.
The inset is the same plot for a = 0.20. The solid lines represent the relation P (R) ' R−α with α = 1.6.
agents who are not members of the cluster to which the
agent k belongs. If k is a normal agent, possible partners are all the other N − 1 agents in the market. The
probability of choice of a partner from among possible
ones can be assigned either preferentially or randomly.
During the linking process, the chosen agent remains inactive (Φk = 0). (III) With probability a, the state of
k becomes active by randomly choosing the state +1 or
−1, and all other agents who are in the same cluster as k
follow the same action.
The aggregate state of the marP
ket si = s(ti ) = j=1,N Φj is measured. Then all the
agents in the active cluster become isolated with no links
and inactive.
From s(ti ) we get the time series data of prices similar to price index data in real markets. We follow the
evolution of price P (ti ) introduced in Ref. 21 as
P (ti+1 ) = P (ti ) exp(si /λ) ,
(2)
where λ is a parameter which controls the size of the
updates and provides a measure of the liquidity of the
market [11]. We set λ = 50000 as in Ref. 11. The price
return R(ti ), the logarithmic change of the market price,
is defined as
R(ti ) = ln[P (ti+1 )] − ln[P (ti )] .
(3)
To see the effects of smartness and preferential attachment, we obtain the time series of s(ti ) and R(ti ) for our
modified models by simulations. The simulations are
performed for the number of agents N = 100, 1000, 10000
and for various values of a. In Fig. 1 we show the distribution P (R) of price return R for the modified model
with only smart agents and preferential selection. At
a = 0.01 this shows ‘financial crashes’: a relative increase
in the probability of high returns is observed. There is a
critical value a∗ which varies with the kind of agent and
the method of selection. For a < a∗ , ‘financial crashes’
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Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004
Fig. 2. Plot of critical value a∗ against the number
N of agents. NR means the model with normal agents
and random selection. NP means the model with normal
agents and preferential selection. SR means that with smart
agents and random selection. SP means that with smart
agents and preferential selection. a∗ ’s are estimated for N
= 100, 1000, 10000. The doted line represents the relation
a∗ (N ) = a∗o + A exp(−N/No ) with a∗o = 0.042, A = 0.259
and No = 4.488 ∗ 102 . The solid lines between data points
are obtained by simple interpolations. This figure can also
be the estimated phase diagram for the crash-possible phase
and the no-crash phase.
occur in P (R) as in Fig. 1. For a > a∗ , P (R) shows a
power-law decay and an exponential cutoff as in the inset
of Fig. 1. The solid lines in Fig. 2 show a power law
R−α with exponent α = 1.6. Note that one can observe
power-law decay in a range of returns, regardless of the
relative increase in P (R). This result is consistent with
Ref. 11. The exponent α does not change with or without preferential attachment. The distribution of returns
is related to the distribution of clusters. If P (s), the
probability distribution of clusters of size s, satisfies the
power law P (s) ' s−β , then P (R) is equal to P (s) times
the probability of choosing a given cluster that is proportional to s: P (R) ' R−α ' s ∗ s−β [11], but the effect
of the preferential attachment is not enough to change
the distribution of clusters, because all the agents in the
active cluster become isolated with no links and inactive after trading. Only preferential attachment makes
clusters grow effectively and rapidly. So the distribution
of return shows the power law and crashes at relatively
large a with preferential attachment.
We estimate the critical value a∗ that is dependent
upon the kind of agents and the method of selection for
N = 100, 1000, 10000 (Fig. 2). The solid lines between
data points are from simple interpolations. From this, we
can estimate the phase diagram for the crash-possible
phase (below the lines) and no-crash phase (above the
lines). In Fig. 2 it is clearly seen that the effects of
the difference between preferential selection and random
selection exist, even for large N . In contrast, the effects
of the difference between smart agents and normal agents
become relatively weak as N increases. However, we can
see from Fig. 2 that preferential selection and smartness
enhance the herd behavior for financial crashes.
For the estimation of a∗ in the limit N → ∞, we
have tried to fit the best functional form to the data
for the model with normal agents and random selection. The best functional form we obtained is a∗ (N ) =
a∗o + A exp(−N/No ) with a∗o = 0.042, A = 0.259 and
No = 4.488 ∗ 102 . From this functional form we can estimate a∗ (N → ∞) ' a∗o (6= 0). This result means that
the crash-possible phase still exists in the limit N → ∞
for the model with normal agents and random selection.
This kind of estimation for a∗ (N → ∞) is possible for
other models, and thus the crash-possible phase should
exist in the limit N → ∞.
III. MODIFIED MODELS WITH A
VARIABLE NUMBER OF AGENTS
In this section, we consider the effects of variation of
the number of agents. For the sake of simplicity, it is
assumed that one new agent at a time can participate
in the market and all active agents leave the market after trading. The details of the modified model with a
variable number of agents are as follows: at each time
step ti (I) with probability q, a new agent participates
in the market. (II) With (1 − q), an agent k in the
market is selected at random. (III) With probability
(1 − a), the agent k chooses a linking partner and makes
a link between them. Of course, k can be either a smart
agent or a normal agent. The probability of choice of a
partner among possible ones can be assigned either preferentially or randomly as in the models in Section II.
During the linking process, the chosen agent remains inactive (Φk = 0). (IV) With a, the state of k becomes
active by randomly choosing the state +1 or −1, and
all the agents who are in the same cluster as k follow
the same action.
P The aggregate state of the market
si = s(ti ) = j=1,N Φj is measured. Then, all active
agents leave the market. By the process (I), the number of agents in the market increases. In contrast, the
number decreases by the process (IV).
For small q, we can expect that the volume of trading
or si is small. The reason is that the expected average number of agents in the market is small, and then
the size of the opinion cluster is also small. The typical
herding behavior for small q (q = 0.1) is shown in Fig. 3.
For small q and large a, any opinion cluster among the
relatively small average number of agents cannot grow
enough to show herding behavior. That is why P (R) for
a = 0.3 decreases monotonically with R (See the inset
of Fig. 3). Furthermore, P (R) for small q and large a
manifests exponential decay as P (R) = A exp(−R/Ro )
as shown in the inset of Fig. 3. The exponential decay
behavior for small q is somewhat peculiar, compared to
Effects of Smartness, Preferential Attachment· · · – Sungmin Lee and Yup Kim
Fig. 3. Log-log plot of P (R) against R for q = 0.1 and
a = 0.0005. Symbols for the data points are the same as those
in Fig. 2. The inset is the semi-log plot of P (R) against R
for q = 0.1 and a = 0.3. The solid line in the inset represents
the relation P (R) = A exp(−R/R0 ) with A = 0.4 and R0 =
1.4 × 10−5 .
Fig. 4. Log-log plot of P (R) against R for q = 0.5 and
a = 0.01. The inset is the same plot for a = 0.3. The solid
line of the inset represents the relation P (R) ' R−α with
α = 1.5.
the power-law behavior (Fig. 1) for the models with a
fixed number of agents. In contrast, the herding behavior
for small q and small a is completely different from that
for small q and large a. The typical behavior for small q
and small a (a = 0.0005) is shown in Fig. 3. The relative increase of P (R) occurs for relatively small R as in
Fig. 3, which should be some kind of signal of a deeplycorrelated herding behavior in a small group of agents.
This kind of behavior can be expected from the fact that
information sharing occurs frequently for the small average number of agents and the opinion cluster can grow
almost maximally to show the deeply-correlated herding
behavior. As can be seen in Fig. 3, the effects of the
smartness and the preferential attachment on P (R) are
negligible for large R, even though the effects can be seen
for small R.
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For large q, we can expect the average number of
agents in the market to be large enough to form a large
opinion cluster. The volume of trading, or si , is also expected to be large. Typical behavior of P (R) for large q
(q = 0.5) is shown in Fig. 4. The results for P (R) for
large q are generally quite similar to those for the models with a fixed number of agents in Fig. 1. For small
a (a = 0.01), this shows the relative increase in P (R)
at high R. For large a (a = 0.3), P (R) shows a powerlaw decay as P (R) = R−α with α = 1.5. (See the inset
of Fig. 4). The same power-law decay of P (R) exists
for small a before the relative increase of P (R) appears.
(See the main part of Fig. 4). In the main part of Fig.
4, we can also see that smartness enhances the relative
increase of P (R) at high R, but preferential attachment
does not seem to perform this crucial role.
IV. SUMMARY AND DISCUSSION
In the modified models with a fixed number of agents,
it is found that preferential selection and smartness enhance the herd behavior, but the effects of the difference
between smart agents and normal agents become relatively weak as N increases. In the model with a variable
number of agents, we find two regimes. For small q, we
find the behavior of exponential decay P (R) for large a
and a relative increase in P (R) at relatively small R for
small a. For large q, P (R) is nearly the same as that for
the model with a fixed number of agents.
As can be inferred from Figs. 3 and 4, the effects of
smartness and preferential attachment are not so clear
for the models with a variable number of agents. For
clarity of the effects, we must estimate the phase diagram
of the crash-possible phase and the no-crash phase in
the q − a parameter space as in Fig. 2. The estimate of
the phase diagram needs an enormously large computing
time. So, we leave the estimate of the phase diagram for
future studies.
ACKNOWLEDGMENTS
This research was supported in part by Grant No.
R01-2001-000-00025-0 from the Basic Research Program
of KOSEF.
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Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004
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