An Interacting Agent Model of Economic Crisis

Yuichi Ikeda
arXiv:2001.11843v1 [physics.soc-ph] 30 Jan 2020

Abstract Most national economies are linked by international trade. Consequently,

economic globalization forms a massive and complex economic network with strong
links, that is, interactions arising from increasing trade. Various interesting collective
motions are expected to emerge from strong economic interactions in a global econ-
omy under trade liberalization. Among the various economic collective motions,
economic crises are our most intriguing problem. In our previous studies, we have
revealed that the Kuramoto’s coupled limit-cycle oscillator model and the Ising-like
spin model on networks are invaluable tools for characterizing the economic crises.
In this study, we develop a mathematical theory to describe an interacting agent
model that derives the coupled limit-cycle oscillator model and the Ising-like spin
model by using appropriate approximations. Our interacting agent model suggests
phase synchronization and spin ordering during economic crises. We confirm the
emergence of the phase synchronization and spin ordering during economic crises
by analyzing various economic time series data. We also develop a network recon-
struction model based on entropy maximization that considers the sparsity of the
network. Here network reconstruction means estimating a network’s adjacency ma-
trix from a node’s local information. The interbank network is reconstructed using
the developed model, and a comparison is made of the reconstructed network with
the actual data. We successfully reproduce the interbank network and the known
stylized facts. In addition, the exogenous shock acting on an industry community in
a supply chain network and financial sector are estimated. Estimation of exogenous
shocks acting on communities of in the real economy in the supply chain network
provide evidence of the channels of distress propagating from the financial sector to
the real economy through the supply chain network.

Yuichi Ikeda
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University, e-mail:
ikeda.yuichi.2w@kyoto-u.ac.jp

1
2 Yuichi Ikeda

1 Introduction

Most national economies are linked by international trade. Consequently, economic

globalization forms a massive and complex economic network with strong links,
that is, interactions due to increasing trade. In Japan, several small and medium
enterprises would achieve higher economic growth by free trade based on the estab-
lishment of economic partnership agreement, such as the Trans-Pacific Partnership.
Various collective motions exist in natural phenomena. For instance, a heavy nucleus
that consists of a few hundred nucleons is largely deformed in a highly excited state
and subsequently proceeds to nuclear fission. This phenomenon is a well-known
example of quantum mechanical collective motion due to strong nuclear force be-
tween nucleons. From the analogy with the collective motions in natural phenomena,
various interesting collective motions are expected to emerge because of strong eco-
nomic interactions in a global economy under trade liberalization. Among the various
economic collective motions, economic crises are our most intriguing problem.
Business Cycle There have been several theoretical studies on the concept of the
“business cycle” [1, 2, 3]. In recent times, the synchronization [4] of interna-
tional business cycle as an example of the economic collective motion has attracted
economists and physicists [5]. Synchronization of business cycles across countries
has been discussed using correlation coefficients between GDP time series [6]. How-
ever, this method remains only a primitive first step, and a more definitive analysis
using a suitable quantity describing business cycles is needed. In an analysis of busi-
ness cycles, an important question is the significance of individual (micro) versus
aggregate (macro) shocks. Foerster et al. [7] used factor analysis to show that the
volatility of U.S. industrial production was largely explained by aggregate shocks
and partly by cross-sectoral correlations from the individual shocks transformed
through the trade linkage. The interdependent relationship of the global economy
has become stronger because of the increase in international trade and investments
[8, 9, 10, 11].
We took a different approach to analyze the shocks to explain the synchronization
in international business cycles. We analyzed the quarterly GDP time series for
Australia, Canada, France, Italy, the United Kingdom, and the United States from
Q2 1960 to Q1 2010 to determine the synchronization in international business
cycles [13]. The followings results were obtained.
(1) The angular frequencies ωi estimated using the Hilbert transform are al-
most identical for the six countries. Therefore, frequency entrainment is observed.
Moreover, the phase locking indicator σ(t) shows that partial phase locking is ob-
served for the analyzed countries, representing direct evidence of synchronization in
international business cycles.
(2) A coupled limit-cycle oscillator model was developed to explain the synchro-
nization mechanism. A regression analysis showed that the model is a very good fit
for the phase time series of the GDP growth rate. The validity of the model implies
that the origin of the synchronization is the interaction resulting from international
trade.
An Interacting Agent Model of Economic Crisis 3

(3) We also showed that information from economic shocks is carried by phase
time series θ i (t). The comovement and individual shocks are separated using the
random matrix theory. A natural interpretation of the individual shocks is that they
are “technological shocks”. The present analysis demonstrates that average phase
fluctuations well explain business cycles, particularly recessions. Because it is highly
unlikely that all of the countries are subject to common negative technological shocks,
the results obtained suggest that pure “technological shocks” cannot explain business
cycles.
(4) Finally, the obtained results suggest that business cycles may be understood
as comovement dynamics described by the coupled limit-cycle oscillators exposed
to random individual shocks. The interaction strength in the model became large in
parallel with the increase in the amounts of exports and imports relative to GDP.
Therefore, a significant part of comovements comes from international trade.
We observed various tpes of collective motions for economic dynamics, such as
synchronization of business cycles [13, 14, 15], on the massive complex economic
network. The linkages among national economies play important roles in economic
crises and during normal economic states. Once an economic crisis occurs in a certain
country, the influence propagates instantaneously toward the rest of the world. For
instance, the global economic crisis initiated by the bankruptcy of Lehman Brothers
in 2008 is still fresh in our minds. The massive and complex global economic network
might show characteristic collective motion during economic crises.
Economic Crisis Numerous preceding studies attempted to explain the character-
istics of stock market crash using spin variables in the econophysics literature. First,
we note some content and mathematical descriptions from previous studies [16],
[17]. In particular, we note studies by Kaizoji and Sornette [18], [19], [20], [21],
[22], [23], [24], [25] in which investor strategies (buy or sell) are modeled as spin
variables, with stock prices varying depending on the differences in the number of
spin-ups. In addition, the feedback effect on an investor’s decision making through
a neighbor’s strategies can explain bubble formations and crashes. For instance, the
temporal evolution is simulated by adding random components in Sornette and Zhou
[21]. Most papers adopted two-state spin variables; however, the study by Vikram
and Sinha [23] adopted three-state spin variables. Note that the purpose of these
studies was to reproduce the scaling law and not explain phase transitions.
In contrast, economics journals aim to explain the optimality of investors’ decision
making [22]. In Nadal et al. [25], phase transition is discussed, starting with discrete
choice theory. Many papers have similar discussions on phase transitions, with slight
variations in optimization and profit maximization. Although empirical studies using
real data are relatively few, Wall Street market crash in 1929, 1962, and 1987 and
the Hong Kong Stock Exchange crash in 1997 were studied in Johansen, Ledoit, and
Sornette [24]. Note that elaborate theoretical studies exist on phase transition effects
on networks and the thermodynamics of networks [27], [28], [29], [30]. Furthermore,
preceding studies on macroprudential policy exist that mainly focus on time series
analyses of macroeconomic variables [31], [32], [33].
4 Yuichi Ikeda

Although the market crash is an important part of an economic crisis, our main
interest is that the real economy consists of many industries in various countries. We
analyzed industry-sector-specific international trade data to clarify the structure and
dynamics of communities that consist of industry sectors in various countries linked
by international trade [34]. We applied conventional community analysis to each time
slice of the international trade network data: the World Input Output Database. This
database contains industry-sector-specific international trade data on 41 countries
and 35 industry sectors from 1995 to 2011. Once the community structure was
obtained for each year, the links between communities in adjoining years were
identified using the Jaccard index as a similarity measure between communities in
adjoining years.
The identified linked communities show that six backbone structures exist in the
international trade network. The largest linked community is the Financial Interme-
diation sector and the Renting of Machines and Equipments sector in the United
States and the United Kingdom. The second is the Mining and Quarrying sector in
the rest of the world, Russia, Canada, and Australia. The third is the Basic Metals
and Fabricated Metal sector in the rest of the world, Germany, Japan, and the United
States. These community structures indicate that international trade is actively trans-
acted among the same or similar industry sectors. Furthermore, the robustness of
the observed community structure was confirmed by quantifying the variations in
the information for perturbed network structure. The theoretical study conducted
using a coupled limit-cycle oscillator model suggests that the interaction terms from
international trade can be viewed as the origin of the synchronization.
The economic crisis of 2008 showed that the conventional microprudential pol-
icy to ensure the soundness of individual banks was not sufficient, and prudential
regulations to cover the entire financial sector were desired. Such regulations attract
increasing attention, and policies related to such regulations are called a macro-
prudential policy that aims to reduce systemic risk in the entire financial sector by
regulating the relationship between the financial sector and the real economy. We
studied channels of distress propagation from the financial sector to the real economy
through the supply chain network in Japan from 1980 to 2015 using a Ising-like spin
model on networks [35]. An estimation of exogenous shocks acting on communities
of the real economy in the supply chain network provided evidence of channels of
distress propagation from the financial sector to the real economy through the sup-
ply chain network. Causal networks between exogenous shocks and macroeconomic
variables clarified the characteristics of the lead lag relationship between exogenous
shocks and macroeconomic variables when the bubble bursts.

2 Interacting Agent Models

The coupled limit-cycle oscillator model and the Ising-like spin model on networks
are invaluable tools to characterize an economic crisis, as described in Section 1. In
An Interacting Agent Model of Economic Crisis 5

this section, we develop a mathematical theory to describe an interacting agent model

that derives the aforementioned two models using appropriate approximations.

2.1 Interacting Agent Model on Complex Network

Hamiltonian Dynamics Our system consists of N company agents and M bank

agents. The states of the agents are specified by multi-dimensional state vectors qi
and q j for companies and banks, respectively. If we consider (1) security indicators:
(1-1) total common equity divided by total assets and (1-2) fixed assets divided by
total common equity; (2) profitability indicators: (2-1) operating income divided by
total assets and (2-2) operating income divided by total revenue; (3) capital efficiency
indicator: total revenue divided by total assets, and (4) growth indicator: operating
income at time t divided by operating income at time t − 1 as variables to the
soundness of companies, state vectors qi are expressed in six dimensional space.
The agents interact in the following way:
Õ Õ Õ Õ
Hint (q) = − HC,i qi − HB, j q j −JC ai j qi q j −JC B bi j qi q j , (1)
i ∈C j ∈B i ∈C, j ∈C i ∈C, j ∈B

where HC , HB , JC , JC B , ai j , and bi j represent the exogenous shock acting on

companies, the exogenous shock acting on banks, the strength of the inter-company
interactions, the strength of company-bank interactions, the adjacent matrix of the
supply chain network, and the adjacent matrix of the bank to company lending
network, respectively. The Hamiltonian H(q, p) of the system is the sum of the
kinetic energy of th e companies (the first term), the kinetic energy of the banks (the
second term), and the interaction potential Hint (q):

Õ p 2 Õ p 2j
i
H(q, p) = + + Hint (q), (2)
i ∈C
2m j ∈B
2m

where pi and m are the multi-dimensional momentum vector and the mass of agent
i. Here, we set m = 1 without loss of generality. We obtain the canonical equations
of motion for agent i:
∂H(q, p)
= qÛ i, (3)
∂ pi
∂H(q, p)
= − pÛ i . (4)
∂ qi
From Eq. (3), we obtain
pi
= qÛ i . (5)
m
By substituting Eq. (5) into Eq. (4), we obtain the equation of motion of the company
agent i:
6 Yuichi Ikeda
Õ Õ
qÜi = HC,i + JC ai j + a ji q j + JC B bi j q j ,


(6)
j ∈C j ∈B

and the equation of motion of bank agent j:

Õ
qÜ j = HB, j + JC B bi j qi . (7)
i ∈C

Langevin Dynamics Hamiltonian dynamics are applicable to a system in which its

total energy is conserved. However, to be noted is that the economic system is open
and no quantity is conserved accurately. Considering this point, we add a term for
constant energy inflow Pi and a term for energy dissipation outside the system −α qÛ i
to the equation of motion for company agent i in Eq. (6):
Õ Õ
qÜi = Pi − α qÛ i + HC,i + JC ai j + a ji q j + JC B bi j q j ,


(8)
j ∈C j ∈B

where we assume that energy dissipation is proportional to the velocity of agent qÛ i .

The stochastic differential equation in Eq. (8) is called the Langevin equation.
Similarly, way, we obtain the Langevin equation for bank agent j:
Õ
qÜ j = P j0 − α 0 qÛ j + HB, j + JC B bi j qi, (9)
i ∈C

If a system’s inertia is very large and thus qÜi ' 0, we obtain the following first
order stochastic differential equation for company agent i:

Pi HC,i JC Õ JC B Õ
qÛ i = + + ai j + a ji q j + bi j q j .


(10)
α α α j ∈C α j ∈B

Similarly, we obtain the first order stochastic differential equation for bank agent j:

P j0 HB, j JC B Õ
qÛ j = + + 0 bi j qi . (11)
α0 α0 α i ∈C

2.2 Ising Model with Exogenous Shock

Underlying Approximated Picture Suppose that qi is a one dimensional variable,

and we assume that magnitude |qi | varies slowly compared with orientation si . We
approximate the magnitude as |qi | ≈ const. and obtain:
qi
si = sgn , (12)
|qi |
∂ ∂ ∂si 1 ∂
= = . (13)
∂qi ∂si ∂qi |qi | ∂si
An Interacting Agent Model of Economic Crisis 7

Derived Model Stock price xi,t (i = 1, · · · , N(or M), t = 1, · · · , T) is assumed to

be a surrogate variable to indicate the soundness of companies or banks. The one-
dimensional spin variable si,t was estimated from the log return of daily stock prices,
ri,t :
si,t = +1 qi,t = log xi,t − log xi,t−1 ≥ 0


(14)
si,t = −1 qi,t = log xi,t − log xi,t−1 < 0


(15)
Here spin-up: si,t = +1 indicates that company i is in good condition; on spin-down:
si,t = −1 indicates that company i is in bad condition. The macroscopic order pa-
rameter Mt = i si,t is an indicator of the soundness of the macro economy, which
Í
is regarded as an extreme simplification to capture the soundness of the economy. In
addition to this simplification, spin variables include noise information because we
have various distortions in the stock market caused by irrational investor decision
making. The spin variables of companies interact with the spins of other compa-
nies through the supply chain network and interact with banks through the lending
network. Those interactions between companies and banks are mathematically ex-
pressed as Hamiltonian. A Hamiltonian is written as follows:
Õ Õ Õ Õ
Hint (s) = −HC si,t − HB si,t − JC ai j si,t s j,t − JC B ai j si,t s j,t
i ∈C i ∈B i ∈C, j ∈C i ∈C, j ∈B
(16)
where HC and HB are the exogenous shocks acting on companies and banks, respec-
tively. ai j represents the elements of adjacency matrix A of the supply chain network
that is treated as a binary directed network.
When spins are exposed for exogenous shock Hext , an effective shock of

He f f = Hext + Hint (17)

acts on each spin. By calculating interaction Hint of the Hamiltonian of Eq. (16),
exogenous shock Hext was estimated by considering the nearest neighbor companies
in the supply chain network

µHext J i j (ai j + a ji )si,t

 Í 
Mt
= tanh + µ (18)
Nµ kT kT N

where T represents temperature as a measure of the activeness of the economy, which

is considered proportional to GDP per capita. We note that supply chain network
data are a prerequisite condition for estimating exogenous shock Hext .
In the current model, the interactions between banks
Õ
JBB ti j si,t s j,t (19)
i ∈B, j ∈B

were ignored because of a lack of data for interbank network ti j . This lack of data
caused by the central bank not making public the data on transactions between banks.
A method to reconstruct the interbank network is described in Section 3.
8 Yuichi Ikeda

2.3 Kuramoto Model with Exogenous Shock

Underlying Approximated Picture When qi is a two-dimensional variable, we

treat this quantity as if it is a complex variable. We assume that amplitude | qi | varies
slowly compared with phase θ i . We approximate the amplitude as | qi | ≈ const. and,
thus, obtain:

qi = | qi |eiθi , (20)
∂ ∂ ∂θ i
= . (21)
∂ qi ∂θ i ∂ qi
Derived Model The business cycle is observed in most industrialized economies.
Economists have studied this phenomenon by means of mathematical models, in-
cluding various types of linear, nonlinear, and coupled oscillator models.
Interdependence, or coupling, between industries in the business cycle has been
studied for more than half a century. A study of the linkages between markets
and industries using nonlinear difference equations suggests a dynamical coupling
among industries [40]. A nonlinear oscillator model of the business cycle was then
developed using a nonlinear accelerator as the generation mechanism [41]. We stress
the necessity of nonlinearity because linear models are unable to reproduce sustained
cyclical behavior, and tend to either die out or diverge to infinity.
However, a simple linear economic model, based on ordinary economic princi-
ples, optimization behavior, and rational expectations, can produce cyclical behavior
much like that found in business cycles [42]. An important question aside from
synchronization in the business cycle is whether sectoral or aggregate shocks are
responsible for the observed cycle. This question was empirically examined; it was
found that business cycle fluctuations are caused by small sectoral shocks, rather
than by large common shocks [43].
As the third model category, coupled oscillators were developed to study noisy
oscillating processes such as national economies [44] [45]. Simulations and empir-
ical analyses showed that synchronization between the business cycles of different
countries is consistent with such mode-locking behavior. Along this line of ap-
proach, a nonlinear mode-locking mechanism was further studied that described a
synchronized business cycle between different industrial sectors [46].
Many collective synchronization phenomena are known in physical and biological
systems [48]. Physical examples include clocks hanging on a wall, an array of lasers,
microwave oscillators, and Josephson junctions. Biological examples include syn-
chronously flashing fireflies, networks of pacemaker cells in the heart, and metabolic
synchrony in yeast cell suspensions.
Kuramoto established the coupled limit-cycle oscillator model to explain this
wide variety of synchronization phenomena [47] [48] [49]. In the Kuramoto model,
the dynamics of the oscillators are governed by
An Interacting Agent Model of Economic Crisis 9

N
Õ
θÛi = ωi + k ji sin(θ j − θ i ), (22)
j=1

where θ i , ωi , and k ji are the oscillator phase, the natural frequency, and the coupling
strength, respectively. If the coupling strength ki j exceeds a certain threshold that
equals the natural frequency ωi , the system exhibits synchronization.
By explicitly writing amplitude |qi | and phase θ i , the third term of the R.H.S. in
Eq. (1) is rewritten as follows:
Õ Õ
JC ai j qi q j = JC ai j | q j || qi | cos(θ j − θ i ). (23)
i ∈C, j ∈C i ∈C, j ∈C

The spatial derivative of Eq. (23) is obtained:

∂ Õ ∂θ i ∂ Õ
JC ai j qi q j = JC ai j | q j ||qi | cos(θ j − θ i )
∂ qi i ∈C, j ∈C ∂ qi ∂θ i i ∈C, j ∈C
(24)
∂θ i Õ
= ai j + a ji | q j || qi | sin(θ j − θ i ).


JC
∂ qi j ∈C

By substituting Eq. (24) into the stochastic differential equation for company agent
i of Eq. (10), we obtain the following equation:

∂ qi dθ i Pi HC,i
= +
∂θ i dt α α
JC ∂θ i Õ
+ ai j + a ji | q j ||qi | sin(θ j − θ i )


α ∂ qi j ∈C (25)
JC B ∂θ i Õ
+ bi j |q j ||qi | sin(θ j − θ i ).
α ∂ qi j ∈B

Consequently, we obtain the stochastic differential equation, which is equivalent to

the Kuramoto model of Eq. (22) with an additional exogenous shock term:

Pi HC,i ∂ qi
 
dθ i 1
= +
dt | qi | 2 α α ∂θ i
JC Õ | q j|
+ ai j + a ji sin(θ j − θ i )


α j ∈C | qi | (26)

JC B Õ |q j |
+ bi j sin(θ j − θ i ).
α j ∈B | qi |
10 Yuichi Ikeda

3 Network Reconstruction

Network reconstruction estimates a network’s adjacency matrix from a node’s lo-

cal information. We developed a network reconstruction model based on entropy
maximization and considering network sparsity.

3.1 Existing Models

MaxEnt algorithm The MaxEnt algorithm maximizes entropy S by changing ti j

under the given total lending siout and the total borrowing siin for bank i [36], [37].
The analytical solution of this algorithm is easily obtained as

siout sin
j
tiMj E = , (27)
G
Õ Õ
G= siout = j .
sin (28)
i j

However, noted is that the solution to Eq. (27) provides a fully connected network,
although real-world networks are often known as sparse networks.
Iterative proportional fitting Iterative proportional fitting (IPF) has been intro-
duced to correct the dense property of tiMj E at least partially. By minimizing the
Kullback-Leibler divergence between a generic nonnegative ti j with null diagonal
entries and the MaxEnt solution tiMj E in Eq. (27), we obtain tiIjPF [38]:

© Õ ti j ª Õ tiIjPF
min ­ ti j ln M E ® = tiIjPF ln M E . (29)
ti j ti j
«i j(i,j) ¬ i j(i,j)
The solution tiIjPF has null diagonal elements, but does show the sparsity equivalent
to real-world networks.
Drehmann and Tarashev approach Starting from the MaxEnt matrix tiMj E , a
sparse network is obtained in the following three steps [39]: First, choose a random
set of off-diagonal elements to be zero. Second, treat the remaining nonzero elements
as random variables distributed uniformly between zero and twice their MaxEnt
estimated value tiDT ME
j ∼ U(0, 2ti j ). Therefore, the expected value of weights under
this distribution coincides with the MaxEnt matrix tiMj E . Third, the IPF algorithm is
run to correctly restore the value of the total lending siout and the total borrowing
siin . However, note that we need to specify the set of off-diagonal nonzero elements.
Therefore, accurate sparsity does not emerge spontaneously in this approach.
An Interacting Agent Model of Economic Crisis 11

3.2 Ridge Entropy Maximization Model

Convex Optimization We develop a reconstruction model for the economic net-

work and apply it to the interbank network in which nodes and links are banks
and lending or borrowing amounts, respectively. First, we maximize configuration
entropy S under the given total lending siout and total borrowing siin for bank i.
Configuration entropy S is written using bilateral transaction ti j between bank i
and j as follows,
Í 
t !
! !
ij i j Õ Õ Õ
S = log Î ≈ ti j log ti j − ti j log ti j . (30)
i j ti j ! ij ij ij

Here, an approximation is applied to the factorial ! using Stirling’s formula. The first
Í of the R.H.S. of Eq.(30) does not change the value of S by changing ti j because
term
i j ti j is constant. Consequently, we have a convex objective function:
Õ
S=− ti j log ti j . (31)
ij

Entropy S is to be maximized with the following constraints:

Õ
siout = ti j , (32)
j
Õ
j =
sin ti j , (33)
i
Õ
G= ti j . (34)
ij

Here, constraints Eq. (32) and Eq. (33) correspond to local information about each
node.
Sparse Modeling The accuracy of the reconstruction will be improved using the
sparsity of the interbank network. We have two different types of sparsity here. The
first is characterized by the skewness of the observed bilateral transaction distribu-
tions. The second type of sparsity is characterized by the skewness of the observed
in-degree and out-degree distributions. Therefore, a limited fraction of nodes have a
large number of links, and most nodes have a small number of links. Consequently,
the adjacency matrix of international trade is sparse.
To take into account the first type of sparsity, the objective function of Eq. (31)
is modified by applying the concept of Lasso (least absolute shrinkage and selection
operator) [51] [52] [53] to our convex optimization problem.
By considering this fact, our problem is reformulated as the maximization of
objective function z:
12 Yuichi Ikeda
Õ Õ Õ
z(ti j ) = S − ti2j = − ti j log ti j − β ti2j (35)
ij ij ij

with local constraints. Here the second term of R.H.S. of Eq. (35) is L2 regularization.
Ridge Entropy Maximization Model In the theory of thermodynamics, a system’s
equilibrium is obtained by minimizing thermodynamic potential F:

F = E − TS (36)

where E, T, and S are internal energy, temperature, and entropy, respectively. Eq.
(36) is rewritten as a maximization problem as follows:
1 1
z ≡ − F = S − E. (37)
T T
We note that Eq. (37) has the same structure as Eq. (35). Thus, we interpret the
meaning of control parameter β and L2 regularization as inverse temperature and
internal energy, respectively. In summary, we have a ridge entropy maximization
model [55] [56]:

maximize z(pi j ) = − pi j log pi j − β p2i j

Í Í
ij ij

subject to G =
Í
i j ti j

siou t ti j (38)
= =
Í Í
G j G j pi j

s ij n ti j
= =
Í Í
G i G i pi j

ti j ≥ 0

4 Empirical Validation of Models

The model described in Section 2 suggests the phase synchronization and the spin
ordering during an economic crisis. In this section, we confirm the phase synchro-
nization and the spin ordering by analyizing varilous economic time series data. In
addition, the exogenous shock acting on an industry community in a supply chain
network and the financial sector are estimated. An estimation of exogenous shocks
acting on communities of the real economy in the supply chain network provided
evidence of the channels of distress propagation from the financial sector to the real
economy through the supply chain network. Finally, we point out that the interactions
between banks were ignored in the interacting agent model explained in Section 2.2
given a lack of transaction data ti j in an interbank network. This lack of data is caused
An Interacting Agent Model of Economic Crisis 13

by the central bank not making public the data on transactions between banks. In
this section, the interbank network is reconstructed and the reconstructed network is
compared with the actual data and the known stylized facts.

4.1 Phase Synchronization and Spin Ordering during Economic

Crises

1
0.9
0.8
0.7
Order Parameter

0.6
all sector
0.5
linked comm1
0.4 linked comm2
linked comm3
0.3
linked comm4
0.2 linked comm5
linked comm6
0.1
0

Fig. 1 Temporal change in amplitude for the order parameters r(t) of phase synchonization for each
community: We applied a conventional community analysis to each time slice of the international
trade network data: the World Input Output Database. This database contains the industry-sector-
specific international trade data on 41 countries and 35 industry sectors from 1995 to 2011. Once
the community structure was obtained for each year, the links between communities in adjoining
years was identified by using the Jaccard index as a similarity measure between communities in
adjoining years.

We evaluated the phase time series of the growth rate of value added for 1435
nodes (41 countries and 35 industry sectors) from 1995 to 2011in the World Input
Output Database) using the Hilbert transform and then estimated the order parameters
of the phase synchronization for communities [34]. The order parameter of the phase
synchronization is defined by
N
1 Õ iθ j (t)
u(t) = r(t)eiφ(t) = e . (39)
N j=1

The respective amplitude for the order parameter of each community was observed to
be greater than the amplitude for all sectors. Therefore, active trade produces higher
phase coherence within each community. The temporal change in amplitude for the
14 Yuichi Ikeda

n1
b1
c1
n2
c2

b2
c3
c

b3

Fig. 2 Temporal changes of the order parameter of spin ordering for the real economy (companies)
MC (t): The order parameter shows the high spin ordering MC, t ≈ 1 during the bubble periods,
and MC, t ≈ −1 during the crisis periods.
An Interacting Agent Model of Economic Crisis 15

n1
b1
c1
n2
c2

b2
c3 c

b3

Fig. 3 Temporal changes of the order parameter of spin ordering for financial sector (banks)
M B (t): The order parameter shows the high spin ordering M B, t ≈ 1 during the bubble periods,
16 Yuichi Ikeda

order parameters for each community are shown in Fig. 1 from 1996 to 2011. Phase
coherence decreased gradually in the late 1990s but increased sharply in 2002. From
2002, the amplitudes for the order parameter remained relatively high. In particular,
from 2002 to 2004, and from 2006 to 2008, we observe high phase coherence. The
first period was right after the dot-com crash and lasted from March 2000 to October
2002. The second period corresponds to the subprime mortgage crisis that occurred
between December 2007 and early 2009. These results are consistent with the results
obtained in the previous study [13].
The stock price is the daily time series for the period from January 1, 1980 to
December 31, 2015. The spin variable si,t was estimated using Eqs. (14) and (15).
The order parameters of spin ordering are defined by:
Õ
MC,t = si,t . (40)
i ∈C
Õ
MB,t = si,t . (41)
i ∈B

for the real economy and the financial sector, respectively. Temporal changes in the
order parameter of spin ordering for the real economy (companies) and the financial
sector (banks) are shown in Figs. 2 and 3, respectively. The symbols in Figs. n1,
b1, c1, n2, c2, b2, c3, and b3 show “Normal period: 1980 - 1985”, “Bubble period:
1985 - 1989”, “Asset bubble crisis: 1989 - 1993”, “Normal period: 1993 - 1997”,
“Financial crisis: 1997 - 2003”, “Bubble period: 2003 - 2006”, “US subprime loan
crisis and the Great East Japan Earthquake: 2006 - 2012”, and “BOJ monetary
easing: 2013 - present”, respectively. The order parameters shows high spin ordering
MC,t ≈ MB,t ≈ 1 during the bubble periods, and MC,t ≈ MB,t ≈ −1 during the
crisis periods.
In Fig. 1, we note that the phase synchronization was observed between 2002 and
2004, and between 2006 and 2008. For these periods, the high spin ordering was
observed in Figs. 2 and 3. The phase synchronization and high spin ordering are
explained by the Kuramoto and Ising models, respectively, and are are interpreted as
the collective motions in an economy. This observation of the phase synchronization
and high spin ordering in the same period supports the validity of the interacting
agent models explained in Section 2.

4.2 Estimation of Exogenous Shock

Exogenous shocks were estimated using Eq. (18), and the major mode of an exoge-
nous shock was extracted by eliminating shocks smaller than 90 % of the maximal
or the minimal shock. The major mode of exogenous shock acting on the financial
sector is shown in Fig. 4. The obtained exogenous shock acting on the financial
sector indicates large negative shocks at the beginnings of c1 (1989) and c2 (1997)
but no large negative shock during period c3 (2008). Therefore, the effect of the U.S.
An Interacting Agent Model of Economic Crisis 17

subprime loan crisis on the Japanese economy was introduced through shocks to the
real economy (e.g., the sudden decrease of exports to the United States), not through
direct shocks in the financial sector.
The major mode of an exogenous shock acting on the community, which consists
of construction, transportation equipment, and precision machinery sectors, is shown
in Fig. 5. For this community, no large negative shock was obtained at the beginnings
of c1 (1989) and c2 (1997). In Fig. 2, we note that MC,t ≈ 1 is observed for this real
economy community at the beginnings of c1 (1989) and c2 (1997). This observation
is interpreted as the existence of channels of distress propagation from the financial
sector to the real economy through the supply chain network in Japan. We observe a
negative but insignificant exogenous shock on the real economy at the beginning of
the U.S. subprime loan crisis (c3).
1.0
Exogenous Shcok

b1 b2 b3
n1 n2
0.0

c1 c2 c3
−1.0

1980 1984 1988 1992 1996 2000 2004 2008 2012

Year

Fig. 4 Major mode of exogenous shock acting on the financial sector: The obtained exogenous
shock acting on the financial sector indicates large negative shocks at the beginnings of c1 (1989)
and c2 (1997), but no large negative shock during period c3 (2008).

4.3 Reconstruction of Interbank Network

The interbank network in Japan was reconstructed using the ridge entropy maxi-
mization model in Eq. (38). The number of banks in each category is 5, 59, 3, and
31 for major commercial bank, leading regional bank, trust bank, and second-tier
regional bank, respectively. Call loan siout of bank i and call money sin
j of bank j are
taken from each bank’s balance sheet and are provided as constraints of the model.
In addition to the banks, a slack variable is incorporated into the model to balance
the aggregated call loan and the aggregated call money. In the objective function in
Eqs. (38), we assumed β = 15.
The distribution of transaction ti j for the reconstructed interbank network in
2005 is shown in the left panel of Fig. 6. The leftmost peak in the distribution is
regarded as zero and, thus, corresponds to spurious links. Transactions ti j for the
18 Yuichi Ikeda

Exogenous Shock
b1 b2 b3
0.5
−0.5 n1 n2
c1 c2 c3
−1.5

1980 1984 1988 1992 1996 2000 2004 2008 2012

Year

Fig. 5 Major mode of exogenous shock acting on the community, which consists of the construction,
transportation equipment, and precision machinery sectors: The obtained exogenous shock acting
on these sectors indicates no large negative shock at the beginnings of c1 (1989) and c2 (1997), but
show a large negative shock during period c3 (2008).

log(Reconstructed Transactions)

−3 −2 −1 0 1 2 3 4
1500

−3
−2
1000

−1
log(Actual Transactions)
Frequency

0
1
500

2
3
0

−5 0 5 10
4

log(Reconstructed Transaction)

Fig. 6 Distribution of transactions ti j for the reconstructed interbank network in 2005 is shown in
the left panel. A comparison of transactions between four categories of banks for the reconstructed
interbank network and the actual values is shown in the right panel.

reconstructed interbank network were used to calculate the transaction between four
bank categories and compared with the actual values taken from Table 4 in [57].
In the right panel of Fig. 6, a comparison is shown of transactions among four
categories of banks for the reconstructed interbank network and the actual values.
This comparison confirms that the accuracy of the reconstruction model is acceptably
good.
For the reconstructed interbank network, we obtain the following characteristics,
which are consistent with the previously known stylized facts: the short path length,
the small clustering coefficient, the disassortative property, and the core and periph-
eral structure. Community analysis shows that the number of communities is two to
An Interacting Agent Model of Economic Crisis 19

three in a normal period and one during an economic crisis (2003, 2008 - 2013).
The major nodes in each community have been the major commercial banks. Since
2013, the major commercial banks have lost the average PageRank, and the leading
regional banks have obtained both the average degree and the average PageRank.
This observed changing role of banks is considered to be the result of the quantitative
and qualitative monetary easing policy started by the Bank of Japan in April 2013.

5 Conclusions

Most national economies are linked by international trade. Consequently, economic

globalization forms a massive and complex economic network with strong links,
that is, interactions resulting from increasing trade. From the analogy of collective
motions in natural phenomena, various interesting collective motions are expected
to emerge from strong economic interactions in the global economy under trade
liberalization. Among various economic collective motions, the economic crisis is
our most intriguing problem.
We revealed in our previous studies that Kuramoto’s coupled limit-cycle oscil-
lator model and the Ising-like spin model on networks were invaluable tools for
characterizing economic crises. In this study, we developed a mathematical theory
to describe an interacting agent model that derives these two models using appro-
priate approximations. We have a clear understanding of the theoretical relationship
between the Kuramoto model and the Ising-like spin model on networks. The model
describes a system that has company and bank agents interacting with each other
under exogenous shocks using coupled stochastic differential equations. Our interact-
ing agent model suggests the emergence of phase synchronization and spin ordering
during an economic crisis. We also developed a network reconstruction model based
on entropy maximization considering the sparsity of network. Here, network re-
construction means estimating a network’s adjacency matrix from a node’s local
information taken from the financial statement data. This reconstruction model is
needed because the central bank has yet to provide transaction data among banks to
the public.
We confirmed the emergence of phase synchronization and spin ordering during
an economic crisis by analyzing various economic time series data. In addition, the
exogenous shocks acting on an industry community in a supply chain network and
the financial sector were estimated. The major mode of exogenous shocks acting on a
community, which consists of construction, transportation equipment, and precision
machinery sectors was estimated. For this community, no large negative shock was
obtained during the crises beginning in 1989 and 1997. However, negative spin
ordering is observed for this real economy community during crises beginning in
1989 and 1997. Estimation of exogenous shocks acting on communities of the real
economy in the supply chain network provided evidence of channels of distress
propagation from the financial sector to the real economy through the supply chain
network.
20 Yuichi Ikeda

Finally, we pointed out that, in our interacting agent model, interactions among
banks were ignored because of the lack of transaction data in the interbank network.
The interbank network was reconstructed using the developed model, and the recon-
structed network and the actual data were compared. We successfully reproduce the
interbank network and the known stylized facts.

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