Transcript PowerPoint 프레젠테이션 - Complex Systems and Statistical Physics

복잡계 네트워크와 경제학
Complex Networks and Economics
정하웅
KAIST, 물리학과
복잡계 및 통계물리 연구실
http://stat.kaist.ac.kr
http://complex.kaist.ac.kr
복잡계 네트워크??
?
• 복잡계 네트워크 = 복잡계 + 네트워크
• 네트워크??
방송망, 통신망, 피라미드-판매망 etc…
You know what it means… 
• 복잡계??
복잡계(complex system)란 여러 구성 요소로 이루어진 집단에서
각 요소가 다른 요소와 끊임없이 상호작용을 하는 체계다.
복잡계는 한마디로 지금까지 '불가사의'로 여겨졌던 부분에 대해 새
롭게 접근하는 패러다임이다. 과학자들은 복잡계를 해석하기 위해
△선형이 아닌 비선형적 수학해석 △절대작용이 아닌 상호작용 △
연속성이 아닌 불연속성 △환원이 아닌 종합을 기본 '법칙'으로 삼
는다. (from Yahoo 시사정보)
Society
Internet
Human body :
NETWORKS!
chemical network
Society
Nodes: individuals
Links: social relationship
(family/work/friendship/etc.)
S. Milgram (1967)
John Guare
“Six Degrees of Separation”
Social networks: Many individuals with diverse
social interactions between them.
9-11 Terror Hijacker’s Network
Econo-network:
network of economic agents
Communication networks
The Earth is developing an electronic nervous system,
a network with diverse nodes and links are
-computers
-phone lines
-routers
-TV cables
-satellites
-EM waves
Communication
networks: Many
non-identical
components
with diverse
connections
between them.
GENOME
protein-gene
interactions
PROTEOME
protein-protein
interactions
METABOLISM
Bio-chemical
reactions
Citrate Cycle
Complex systems
Made of
many non-identical elements
connected by diverse interactions.
NETWORK
Erdös-Rényi model
(1960)
Connect with
probability p
p=1/6
N=10
k ~ 1.5
Pál Erdös
(1913-1996)
Degree Distribution
P(k) : prob. that a certain
node will have k links
Poisson distribution
- Random
- Democratic
ARE COMPLEX NETWORKS
REALLY RANDOM?
To test this:
We need to pragmatically investigate
the topology of large real networks.
World Wide Web
Nodes: WWW documents
Links: URL links
800 million documents
(S. Lawrence, Nature,1999)
ROBOT:
collects all
URL’s found in a
document and follows
them recursively
R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999)
What did we expect?
k ~ 6
P(k=500) ~ 10-99
NWWW ~ 109
 N(k=500)~10-90
We find:
out= 2.45
 in = 2.1
P(k=500) ~ 10-6
NWWW ~ 109
 N(k=500) ~ 103
Pout(k) ~ k-out
Pin(k) ~ k- in
What does it mean?
Poisson distribution
Exponential Network
Power-law distribution
Scale-free Network
INTERNET BACKBONE
Nodes: computers, routers
Links: physical lines
(Faloutsos, Faloutsos and Faloutsos, 1999)
SEX-Web
Nodes: people (females; males)
Links: sexual relationships
4781 Swedes; 18-74;
59% response rate.
(Liljeros et al. Nature 2001)
ACTOR CONNECTIVITIES
Nodes: actors
Links: cast jointly
Days of Thunder (1990)
Far and Away
(1992)
Eyes Wide Shut (1999)
N = 212,250 actors
k = 28.78
P(k) ~k-
=2.3
GENOME
protein-gene
interactions
PROTEOME
protein-protein
interactions
METABOLISM
Bio-chemical
reactions
Citrate Cycle
protein-protein
network
(yeast)
Jeong et al.
Nature 411, 41 (2001)
p53 network
(mammals)
metabolic
network
(E. coli)
Jeong et al.
Nature 407, 651 (2000).
Other Examples of Scale-Free Network
Email network
Nodes: individual email address
Links: email communication
Phone-call networks
Nodes: phone-number
Links: completed phone call
(Abello et al, 1999)
Networks in linguistics
Nodes: words
Links: appear next or one word apart from each other
(Ferrer et al, 2001)
Networks in Electronic auction (eBay)
Nodes: agents, individuals
Links: bids for the same item
(H. Jeong et al, 2001)
Network from eBay bidding history data
Simple rule: Connect two agents
when they are bidding to the same item
1
2
2
1
3
4
Item 1
5
3
6
Item 2
7
9
4
5
8
6
7
8
9
Degree distribution P(k)
More relevant Econo network
Nodes: individual, company, country...
Links: economic activities
Nodes: Company (stock)
Links: connect two company
when they are correlated!
Pi(t) : stock price at time t ,
Return : Yi  ln Pi (t )  ln Pi (t  1)
Example
4
Return
T1
T2 T3 T4
X
Y
Z
W
1
3
-1
2
0
2
0
0
-1
1
1
-2
0
2
0
0
3
2
X
1
Y
0
Z
1
2
3
4
W
-1
-2
-3
Correlation (X,Y) = 1
cf) Distance (X,Y) = 4
Correlation (X,Z) = -1
Distance (X,Z) = 2.83
Correlation (X,W) = 1
Distance (X,W) = 1.41
Distance(X , Y) 
p
2
(
X
[
j
]

Y
[
j
]
)

j 1
Connect
highly correlated
companies!
Degree distribution?
 Again, it’s scale-free!
Economic agents are connected via
“inhomogeneous” scale-free network!
Comparison with random return model
Model 0:
 Too simple!
(One factor model)
Model 1:
: return of company i
: random noise term
: market factor (e.g. S&P500 index)
Real data
Random data
Can be used for Portfolio analysis
Strength between stocks
cross-correlation of log-return
Si t   ln Pi (t  t )  ln Pi (t )
Log-return
Si S j  Si S j
Ci , j 
( S
2
i
 Si )( S
2
2
j
 Sj
2
( 1  Ci , j  1)
)
Better to use shifted log-return
1
Gi (t )  Si (t ) 
N
 S (t )
i
i
Gi G j  Gi G j
i , j 
2
i
(G
 Gi )( G
2
2
j
 Gj
2
( 1  i , j  1)
)
SUMMARY
Complex system
Network
Sex Web
WWW
Food Web
Scale-free network
Econo-network
Internet
Cell
UNCOVERING ORDER HIDDEN WITHIN COMPLEX SYSTEMS!
References
• R. Albert, H. Jeong, A.L. Barabasi, Nature 401 130 (1999).
• R. Albert, A.L. Barabasi, Science 286 509 (1999).
• A.L. Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
• R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000).
• H.Jeong, B.Tombor, R.Albert, Z.N.Oltvai, A.L.Barabasi, Nature 407 651 (2000).
• H. Jeong, S.P. Mason, A.L. Barabasi, Z.N. Oltvai, Nature 411 41 (2001).
• J. Podani, Z.N. Oltvai, H. Jeong, B. Tombor, A.L. Barabasi, Nature Genetics
(2001).
• S.H. Yook, H. Jeong, A.-L. Barabasi, PNAS (2002)
URL: http://stat.kaist.ac.kr
Email: Hawoong Jeong
[email protected]
ORIGIN OF SCALE-FREE NETWORKS
(1) The number of nodes (N) is NOT fixed.
Networks continuously expand by
the addition of new nodes
Examples:
WWW : addition of new documents
Citation : publication of new papers
(2) The attachment is NOT uniform.
A node is linked with higher probability to a node
that already has a large number of links.
Examples :
WWW : new documents link to well known sites
(CNN, YAHOO, NewYork Times, etc)
Citation : well cited papers are more likely to be cited again
(1) GROWTH :
Scale-Free Model
At every timestep we add a new node with m edges
(connected to the nodes already present in the system).
(2) PREFERENTIAL ATTACHMENT :
The probability Π that a new node will be connected to
node i depends on the connectivity ki of that node
ki
 ( ki ) 
 jk j
P(k) ~k-3
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
Scale-free Network in Financial Correlations
http://archive.org/abs/cond-mat/0107449
•S&P500 지수는 WSF Behavior를 가진다.
  1.8
Universal characteristic
in financial market
Price: P(t)
Price change: P(t)=P(t+)-P(t)
Return: R(t)=[P(t+)-P(t)]/P(t)
Log-return: S(t)=log[P(t+)/P(t)]
• Probability distribution of returns
shows power law tail
Stylized Facts about
Financial Time Series
• 가격 변동에 대한 correlation은 수십분
C ( ) 
1
 ( P(t )  P )( P(t   )  P ) 
2
 P(t   ) P(t )    P  2

2
• 가격 변동의 분포는 정규분포를 따르지
않음
Kurtosis : the fourth-order cumulant
• 간헐성 : volatility clustering
Volatility : vol (t ) 
1



i 1
| S (t  i ) |
(or r.m.s of S(t)
Autocorrelation of
Financial Time Series
Probability Density Function for
High-Frequency Price Changes
Kurtosis the fourth-order cumulant
Kurtosis is based on the size of a distribution's tails.
leptokurtic ; distributions with relatively large tails (so called “fat tail”)
platykurtic ; distributions with small tails
Mesokurtic ; distribution with the same kurtosis as the normal distribution.
Kurtosis of yt is defined as
 ( yt   ) 4 
Ky  E

4



For the normal distributi on : K y  3
http://www.ruf.rice.edu/~lane/
hyperstat/A53638.html
Volatility Clustering
Agent-based Computational
Economics (ACE)
• What is ACE?
– Computational study of economies
modeled as evolving systems of
autonomous interacting agents
– Specialization to economics of the basic
complex adaptive systems paradigm
• Refer to
http://www.econ.iastate.edu/tesfatsi/ace.htm
http://www.bradeis.edu/~blebaron
Adaptive Agents, Intelligence, and
Emergent Human Organization
(PNAS 99, 7187)
• ABM : social structure 와 social fact
가 개개의 agent의 상호작용에 의해
bottom up으로 형성
• Agent의 local interaction 어떻게
global social structure와 behavior의
pattern을 만들어내느냐에 관심
Agent-Based Computational
Economics
(ISU Economics Working Paper No. 1)
• Agent-based computational
economics (ACE) : 자치적이며 서로
상호작용하는 agent의 evolving system
으로 경제 모델을 만들어 computer를
이용하여 연구 → 복잡 적응계라는 새로
운 패러다임을 경제학에 잘 적용
Agent-Based Computational
Economics
(ISU Economics Working Paper No. 1)
• ACE의 기본적인 두가지 관심사
- 어떻게 autonomous interacting agent
의 local interaction으로 global
regularity를 만들어내는가?
- 자신에게 유리한 mechanism를 추구하
는 agent의 반복된 행위가 어떠한 사회
적 결과를 나타내는가?
Agent-Based Computational
Economics (ACE)
• Agent가 잘 정의된 동적이면서 단순한
전략에 따라 시장에 참여하는 경우
- 정확한 결과
- 시장 상황의 변화에 따른 전략 변화 없
음
Agent-Based Computational
Economics (ACE)
• Agent가 시장 가격이 확률보행과정이라
고 간주하고 무작위 투자를 하는 경우
- 양호한 결과
- 실제 투자 성향을 반영하지 못함
Agent-Based Computational
Economics (ACE)
• 인공신경망, 유전알고리즘 등 인공지능
을 활용하여 agent가 학습하고 변화 적
응하는 경우
- 현실과 가장 근접
- 계산의 복잡도 증가
- 인공지능 전략의 현실성
Herd behavior and aggregate fluctuations
in financial markets
(Macroeconomic Dynamics 4, 170)
 N agents connected random network with probability p  c / N
 Each clusters have random variable (+1: buy, -1: sell, 0: stay)
   1,0,1 P(1)  P(1)  a
P(0)  1  2a
 Price change
1
W (t )

 Cluster size distribution
 (c  1)W 
A

P(W )  5 / 2 exp  
W0 
W

R
 Kurtosis

2c  1
2aN (1  c / 2) A(c)(1  c) 3
For small a, power-law with exp. Cutoff
for large a, Gaussian
Time-Reversal Asymmetry in Cont-Bouchaud Stock
Market Model (Physica A 299, 547)
 After downward movement of price agents have higher prob. to
sell (PANIC)
 After upward movement of price agents have higher prob. to buy
 Panic effect is taken as ten times stronger than influence of a
price increase
 ==> Downward Crash
 Add fundamentalist’s activity to stabilize crash:
 High prices cause more people to sell than to buy
 Low prices increase buying prob.
Asymmetries, Correlations and Fat Tails in
Percolation Market Model (cont-mat/0108345)
 N agents connected random network with probability p  c / N
 Each clusters have random variable
P(1)  2apb
   1,0,1
P(0)  1  2a

 Price change
P(1)  2aps
R   ns s   ns s
buy
sell
 r<0
pb  0.5  5 10 7 x  5  10 4 r
r>0
pb  0.5  5 10 7 x  5 10 5 r
Transmission of information and herd behavior:
An application to financial markets (PRL 85, 5659)
 There are N agents.
 The state of agent j is represented by  j   1,0,1
 An agent j is selected at random
 With probability a , agent j choose state  1 or  1 randomly & transmit  j
 With probability 1  a , agent j choose state 0
 Price change
R~
1

 (t )
i
i
 Define herding parameter
h
1
1
a
h<h* : power-law
h>h* : crash!
& link to other agent
Volatility Clustering and Scaling for Financial Time
Series due to Attractor Bubbling (PRL 89, 158701)
1.0
1.0
1.0
0.8
0.6
0.5
0.5
0.4
0.0
x(t)
x(t)
x(t)
0.2
0.0
0.0
-0.2
-0.5
-0.4
-0.5
-0.6
-0.8
-1.0
0
200
400
600
t
N=100
800
1000
-1.0
0
200
400
600
t
N=1,000
800
1000
-1.0
0
200
400
600
t
N=5,000
800
1000
More realistic model?
• 고정된 혹은 변화하는 경제agent 의 숫자
• 정보 및 경제 네트워크 위에서 서로 상호
작용 (via communication on several
networks: Random, WS, Scale-Free)
• 각각의 행위자는 신뢰도 등 개개인의 경
제성향을 가짐
==> Work in progress
Volatility Clustering and Scaling for Financial Time
Series due to Attractor Bubbling (PRL 89, 158701)
• Agent (spin) = N개
• Opinion :  i (t )  1
1
Ii 
N
• Local field :
A
ij
(t ) j (t )  hi (t )
j
Aij (t )  A (t )  a ij (t )
hi (t )  h i (t )
Volatility Clustering and Scaling for Financial Time
Series due to Attractor Bubbling (PRL 89, 158701)
• Update orientation of spin
p  1 /{1  exp[ 2I i (t )]}
with probability p
1
 i (t  1)  
with probability 1-p

1

• Price change
1
x(t ) 
N
  (t )
i