Introduction to MATLAB

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Introduction to MATLAB
Financial Networks with
Static and dynamic
thresholds
Tian Qiu
Nanchang Hangkong University
Outline





Motivation
Financial networks with static and
dynamic thresholds
Topology dynamics
Economic sectors
Conclusions
2
Motivation
We introduce a dynamic financial network with both static and
dynamic thresholds based on the daily data of the American
and Chinese stock markets, and investigate the topology
dynamics, such as the average clustering coefficient, the
average degree and the cross correlation of degrees. Special
attention is focused on dynamic effect of the thresholds on the
network structure and network stability.
3
Financial networks with static and
dynamic thresholds
4
Financial networks with static and
dynamic thresholds
We define the price return
Ri (t , t )  ln Pi (t  t )  ln Pi (t )
We normalize the price return to
ri (t ) 
where
i 
Ri  Ri
i
R  Ri
2
i
2
5
Financial networks with static and
dynamic thresholds
We define an instantaneous equal-time cross-correlation
between two stocks by
Gij (t )  ri (t )rj (t )

take individual stocks as nodes and set a threshold
to create
links. At each time step, if the cross correlation G (t )   ,
ij
then add a link between stocks i and j ; otherwise, cut the link.
6
Financial networks with static and
dynamic thresholds
static threshold
2
Qs 
N ( N  1)T
N 1
N
T
  G (t )
i 1 j i 1 t 1
ij
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Financial networks with static and
dynamic thresholds
dynamic threshold
N 1 N
2
Qd (t ) 
Gij (t )


N ( N  1) i 1 j i 1
8

Topology dynamics

detrended fluctuation analysis(DFA)

Average clustering coefficient

Average degree

cross correlation of degrees
9
Topology dynamics-detrended
fluctuation analysis(DFA)
For a time series A(t’’), we eliminate the average trend
from the time series by introducing
B(t ' )  t ''1[ A(t ' ' Aave ]
t'
Uniformly dividing [1, T ] into windows of size t and fitting B(t’) to
a linear function Bt (t ' ) in each window, we define the DFA
function as
1 T
F (t )  ( [ B(t ' )  Bt (t ' )]2
T t '1
10
Topology dynamics-detrended
fluctuation analysis(DFA)
In general, F(t) will obey a power-law scaling behavior
F (t ) ~ t 
0    0.5 indicate anti-correlated time series
  0.5
indicate the Gaussian white noise
0.5    1.0 indicate long-range correlating time series
indicate 1 / f noise
  1.0
  1.0
indicate unstable time series
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Topology dynamics-Average
clustering coefficient
The average clustering coefficient is defined by
1
C (t ) 
N
N
 c (t )
i 1
i
where ci (t ) is the clustering coefficient of node i
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Topology dynamics-Average
clustering coefficient
1 T
C   C (t )
T t 1
0.78
0.88
0.68
0.85
13
Topology dynamics-Average
clustering coefficient
14
Topology dynamics-Average degree
The average degree is defined by
1
K (t ) 
N
N
 k (t )
i 1
i
where ki (t ) is the degree of node i
15
Topology dynamics-Average
degree
N  259
16
Topology dynamics-Average degree
1 T
K   K (t )
T t 1
17
Why is the dynamic threshold crucial?
One important reason is that the volatilities fluctuate strongly in the
dynamic evolution, especially on the crash days. It induces large
temporal fluctuations of the cross correlations of price returns.
Thus the static threshold leads to dramatic changes in the
topological structure of the network. However, the dynamic
threshold proportional to Qd (t ) suppresses such kinds of
fluctuations and results in a stable topological structure of the
network.
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Why is the dynamic threshold crucial?
Extreme market
(30 days)
Stable market
(30 days)
Static
threshold
K  74
K  74
K e  166
C  0.78
C e  0.95
Dynamic
threshold
K  108
C  0.88
K  108
K e  102
C e  0.83
K s  109 C s  0.89
C  0.78
K s  56 C s  0.73
C  0.88
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Degree distribution
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Topology dynamics-cross correlation
of degrees
The so-called assortative or disassortative
mixing on the degrees refers to the cross
correlation of degrees. ‘Assortative mixing’
means that high-degree nodes tend to directly
connect with high-degree nodes, while
‘disassortative mixing’ indicates that highdegree nodes prefer to directly connect with
low-degree nodes.
21
Topology dynamics-cross correlation
of degrees
The cross correlation of degrees is defined as
1
M  j k  [ M  ( j  k )]2
2
r (t ) 
1 2
1
2
1
1
M  ( j  k )  [ M  ( j  k )]2
2
2
1
1
where j and k are the degrees of the nodes at both
ends of the  th link, with   1,..., M
r  0, r  0, r  0 represent assortative mixing, no
assortative mixing and disassortative mixing, respectively.
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Topology dynamics-cross
correlation of degrees
0.00
1 T
r   r (t )
T t 1
0.36
-0.20
0.22
23
Topology dynamics-cross correlation
of degrees
24
Economic sectors
25
Topology dynamics- Economic
sectors
we first introduce the normalized individual degrees
~
ki  ( ki  ki ) /  k i
We then construct the cross correlation matrix F of
~
individual degrees ki (t ) whose elements are
1 T ~ ~
Fij   ki (t )k j (t )
T t 1
and compute its eigenvalues and eigenvectors.
26
Topology dynamics- Economic
sectors
  Qs
  Qd (t )
A:basic materials; B: conglomerates; C: consumer goods;
D: finance; E: healthcare;F: industrial goods; G: services; H:
technology; I: utilities.
27
Topology dynamics- Economic
sectors
  1.1Qs
  6Qd (t )
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Conclusions


the dynamic threshold properly suppresses
the large fluctuation induced by the cross
correlations of individual stock prices and
creates a rather robust and stable network
structure during the dynamic evolution, in
comparison to the static threshold.
Long-range time correlations are revealed
for the average clustering coefficient, the
average degree and the cross correlation of
degrees.
29
Thank You!
30

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