Universal and nonuniversal allometric scaling behaviors in the visibility graphs of world stock market indices

Abstract

The investigations of financial markets from a complex network perspective have unveiled many phenomenological properties, in which the majority of these studies map the financial markets into one complex network. In this work, we investigate 30 world stock market indices through their visibility graphs by adopting the visibility algorithm to convert each single stock index into one visibility graph. A universal allometric scaling law is uncovered in the minimal spanning trees, whose scaling exponent is independent of the stock market and the length of the stock index. In contrast, the maximal spanning trees and the random spanning trees do not exhibit universal allometric scaling behaviors. There are marked discrepancies in the allometric scaling behaviors between the stock indices and the Brownian motions. Using surrogate time series, we find that these discrepancies are caused by the fat-tailedness of the return distribution, the nonlinear long-term correlation, and a coupling effect between these two influence factors.

Full text

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arXiv:0910.2524v1 [q-fin.ST] 14 Oct 2009
Universal and nonuniversal allometric scaling behaviors in the visibility graphs of
world stock market indices
Meng-Cen Qian,1Zhi-Qiang Jiang,2,3,4and Wei-Xing Zhou2,3,4,5,6, ∗
1School of Management, Fudan University, Shanghai 200433, China
2School of Business, East China University of Science and Technology, Shanghai 200237, China
3School of Science, East China University of Science and Technology, Shanghai 200237, China
4Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China
5Engineering Research Center of Process Systems Engineering (Ministry of Education),
East China University of Science and Technology, Shanghai 200237, China
6Research Center on Fictitious Economics & Data Science,
Chinese Academy of Sciences, Beijing 100080, China
(Dated: October 14, 2009)
The investigations of financial markets from a complex network perspective have unveiled many
phenomenological properties, in which the majority of these studies map the financial markets into
one complex network. In this work, we investigate 30 world stock market indices through their
visibility graphs by adopting the visibility algorithm to convert each single stock index into one
visibility graph. A universal allometric scaling law is uncovered in the minimal spanning trees,
whose scaling exponent is independent of the stock market and the length of the stock index.
In contrast, the maximal spanning trees and the random spanning trees do not exhibit universal
allometric scaling behaviors. There are marked discrepancies in the allometric scaling behaviors
between the stock indices and the Brownian motions. Using surrogate time series, we find that
these discrepancies are caused by the fat-tailedness of the return distribution, the nonlinear long-
term correlation, and a coupling effect between these two influence factors.
PACS numbers: 05.45.Tp, 05.40.-a, 05.45.Df, 89.75.Da, 89.65.Gh
I.INTRODUCTION
Econophysics is an interdisciplinary field which adopts
ideas, tools and theories from statistical mechanics, non-
linear science, complexity science and applied math-
ematics to understand the emerging complexity and
self-organized macroscopic behaviors of economic sys-
tems, with special interest paid to financial markets
[1, 2, 3, 4, 5]. Econophysists are particularly interested in
unveiling different universal behaviors in financial mar-
kets [6, 7, 8, 9, 10, 11], following the phenomenological
framework [12, 13].
In recent years, complex network theory has witnessed
a flourishing progress [14, 15, 16, 17, 18, 19]. It is nat-
ural that a wealth of studies have been carried out from
a complex network perspective. The current economic
crisis calls for a deeper understanding of the dynamics of
economic activities on the global economic network [20].
The studies in this field can be classified into two types
based on how the network is constructed. The studies of
the first type deal with many time series to form a com-
plex network with each node standing for a time series
and the weight of a link between two nodes character-
ized by the correlation coefficient of the two time series
[21, 22, 23] or by the distance between the two time series
[24, 25, 26].
Concerning the studies of the second type, different
∗Electronic address: wxzhou@ecust.edu.cn
mapping methods have been proposed to convert time
series into different kinds of networks, including cycle
networks based on the local extrema and their distance
in the phase space [27, 28], segment correlation networks
[29, 30], nearest neighbor networks [31], n-tuple networks
based on the fluctuation patterns [32, 33], the visibility of
nodes [34], space state networks based on conformational
fluctuations [35], bin transition networks [36], and recur-
rence networks [37, 38]. The visibility algorithm has been
diversely used to investigate stock market indices [39],
human strive intervals [40], occurrence of hurricanes in
the United States [41], foreign exchange rates [42], and
energy dissipation rates in three-dimensional fully devel-
oped turbulence [43].
In this work, we study the allometric scaling behavior
of spanning trees extracted from the visibility graphs of
30 stock market indices all over the world. Both uni-
versal and nonuniversal scaling behaviors are reported,
which is reminiscent of the universal scaling behavior of
the weighted world trade networks [44]. The paper is
organized as follows. In Sec. II, we describe briefly the
methodology adopted. Section III presents the empirical
findings. We study the impact of the length of the stock
indices on the allometric behaviors in Sec. IV. Section
V explores the driving factors that cause the different
behaviors between the stock indices and the Brownian
motions. And Section VI concludes.

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II.METHODOLOGY
Here we briefly explain the methodology used in this
study. For each stock market index, a unique visibility
graph is constructed. A maximal spanning tree (MaxST),
a minimal spanning tree (MinST), and 100 random span-
ning trees (RanSTs) are extracted from the constructed
visibility graph. For each spanning tree, an allometric
scaling analysis is carried out and a scaling exponent is
determined.
A.Construction of visibility graph
Consider the price series p(t) of a stock market index
with a length of N. We can transform the time series
into complex networks by applying the visibility algo-
rithm [34]. Each data point in the time series is regarded
as a node in the complex network, and an edge is drew
connecting two nodes according to the rule that the two
corresponding data points can see each other in the di-
agram of the time series. Mathematically, two arbitrary
data points p(ti) and p(tj) have visibility if any other
data point p(tk) located between them fulfills
p(tj) − p(tk)
tj− tk
>p(tj) − p(ti)
tj− ti
. (1)
We assign the average acceleration of price movement,
defined as [lnp(tj) − lnp(ti)]/(tj− ti), as the weight of
the edge [ti,tj], where ti< tj.
B.Construction of spanning trees
We extract three different kinds of spanning trees
(MaxST, MinST, RanST) from each visibility graph. The
algorithms for building MaxST, MinST, and RanST are
given in the following.
MaxST: First, select an edge with a maximum weight
as the first edge of the MaxST. Second, select a
new edge with a maximum weight among the edges
which connect the tree and make sure that no loop
is introduced. Third, repeat the second step till all
nodes are added into the tree.
MinST: First, select an edge with a minimum weight
as the first edge of the MinST. Second, select a
new edge with a minimum weight among the edges
which connect the tree and make sure that no loop
is introduced. Third, repeat the second step till all
nodes are added into the tree.
RanST: First, arbitrarily select an edge as the first edge
of the RanST. Second, randomly select a new edge
among the edges which connect the tree but make
sure that no loop is introduced. Third, repeat the
second step till all nodes are added into the tree.
C.Allometric scaling
Allometric scaling laws are ubiquitous in complex
systems evolving on complex networks, such as the
metabolism of organisms and ecosystems river networks
[45, 46, 47, 48, 49, 50], the food webs [51], the world
trade webs [44], the world investment networks [52], and
so forth. The original model of the allometric scaling on
a spanning tree was developed by Banavar, Maritan, and
Rinaldo [49]. The node with the maximum degree is con-
sidered as the root of a spanning tree. Each node of a
spanning tree is assigned a number 1, and two values Ai
and Ciare defined for each node i in an iterative manner
as follows:
Ai=
?
j
Aj+ 1 and Ci=
?
j
Cj+ Ai,(2)
in which j stands for all the nodes linked from i [49].
The allometric scaling relation is then highlighted by the
power law relation between Ciand Ai:
C ∼ Aη,(3)
where the leaf nodes with A = C = 1 should be excluded
from the estimation of the scaling exponent η [51].
Any spanning tree can range in principle between two
extremes, that is, the chain-like trees and the star-like
trees. For chain-like trees we have η = 2−, while for
start-like trees we have η = 1+. Therefore, 1 < η < 2 for
all spanning trees. It should be note that not all trees
exhibit such an allometric scaling behavior, for instance
the classic Cayley trees [53].
III.UNIVERSAL SCALING IN THE
VISIBILITY GRAPHS CONSTRUCTED FROM
STOCK MARKET INDICES
The data sets we analyzed contain 30 stock market
indices all over the world, which are retrieved from Ya-
hoo!Finance at http://finance.yahoo.com.
the index names is given below together with the ab-
breviations, countries or areas and starting dates of the
time series used for analysis in the ensuing parenthe-
ses: Amsterdam Exchange Index (AEX, Netherlands, 3
January 2000), ATX Vienna (ATX, Austria, 11 Novem-
ber 1992), Euronext BEL-20 (BFX, Belgium, 11 Febru-
ary 2005), BSE Sensex (BSESN, India, 1 July 1997),
Ibovespa (BVSP, Brazil, 27 April 1993), CMA GENL In-
dex (CMA, Egypt, 26 May 2003), Dow Jones Industrial
Average (DJIA, USA, 1 October 1928), CAC 40 Index
(FCHI, France, 1 March 1990), FTSE 100 Index (FTSE,
UK, 2 April 1984), DAX Index (GDAXI, Germany, 26
November 1990), S&P/TSX Composite Index (GSPTSE,
Canada, 3 January 2000), Hang Seng Index (HSI, Hong
Kong, 31 December 1996), Jakarta Composite Index
(JKSE, Indonesia, 1 July 1997), FTSE Bursa Malasia
KLCI Index (KLSE, Malaysia, 3 December 1993), Ko-
rea Composite Stock Price Index (KOSPI, Korea, 1 July
A list of

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1997), Merval Buenos Aires (MERV, Argentina, 8 Octo-
ber 1996), MIBTEL Index (MIBTEL, Italy, 3 January
2000), MXX IPC (MXX, Mexico, 1 November 1991),
NIKKEI 225 (N225, Japan, 4 January 1984), NASDAQ
Composite (NASDAQ, USA, 2 February 1971), NZX
50 Index (Gross) (NZ50, New Zealand, 30 April 2004),
OMXS All Share Index (OMXSPI, Sweden, 8 January
2001), Oslo Exchange All Share Index (OSEAX, Nor-
way, 7 February 2001), IGBM (SMSI, Spain, 2 January
2002), Standard and Poor’s 500 Index (S&P500, USA,
3 January 1950), Shanghai Stock Exchange Composite
Index (SSEC, China, 4 January 2000), Swiss Market In-
dex (SMI, Switzerland, 9 November 1990), Straits Times
Index (STI, Singapore, 28 December 1987), Tel Aviv TA-
100 Index (TA100, Israel, 1 July 1997), and Taiwan Stock
Exchange Corporation Weighted Index (TWII, Taiwan,
2 July 1997). The ending dates of all the indices are 25
August 2009.
For each stock index, a visibility graph is constructed
and its maximum spanning tree (MaxST) and minimum
spanning tree (MinST) are determined uniquely. In ad-
dition, 100 random spanning trees (RanSTs) are also de-
rived from the visibility graph. For each tree, an allomet-
ric analysis is carried out and the two sequences of A and
C are calculated. Figure 1 shows the allometric scaling
behaviors of the MaxST, the MinST and a randomly se-
lected RanST associated with the FSTE 100 Index. Nice
power-law relationships are observed between A and C.
A linear regressionto the data finds that η = 1.271±0.002
for the MaxST, η = 1.264 ± 0.002 for the MinST, and
η = 1.308± 0.002 for the RanST, respectively.
100
101
102
A
103
104
10−2
100
102
104
106
C


MaxST
MinST
RanST
FIG. 1: Allometric scaling behavior of spanning trees ex-
tracted from the visibility graph of the FTSE 100 Index. The
data points for MinST and RanST are transformed vertically
by a factor of 10 and 0.1 for better visibility. The solid lines
are the best power-law fits.
We find that all the spanning trees exhibit excellent
allometric scaling behaviors for all the indices. The cor-
responding power-law exponents for different indices are
reported in table I. We find that the allometric scaling
exponents of the MaxSTs for different indices are very
close to each other, centered around 1.281±0.009 (mean
± std). The situation for the MinSTs is the same as
the MaxSTs, in which the power-law exponents fluctuate
slightly around 1.274±0.011. In contrast, the power-law
exponents of the RanSTs are basically larger than 1.3
except for DJIA, NASDAQ and S&P 500, which are sig-
nificantly larger than the exponents of both the MaxSTs
and the MinSTs. We infer that the appearance of smaller
values of ηRanSTfor DJIA, NASDAQ, and S&P 500 is due
to the fact that these three indices have much more data
points compared with other indices.
TABLE I: Allometric scaling exponents η of spanning trees
(MaxST, MinST, and RanST) for different indices.
stock index
AEX
ATX
BFX
BSESN
BVSP
CMA
DJIA
FCHI
FTSE
GDAXI
GSPTSE
HSI
JKSE
KLSE
KOSPI
MERV
MIBTEL
MXX
N225
NASDAQ
NZ50
OMXSPI
OSEAX
SMSI
SP500
SSEC
SMI
STI
TA100
TWII
ηMaxST
1.276(3) 1.278(3) 1.329(3)
1.282(3) 1.270(3) 1.312(2)
1.270(5) 1.269(4) 1.349(4)
1.288(3) 1.281(3) 1.310(3)
1.299(3) 1.290(3) 1.315(2)
1.277(12) 1.250(8) 1.315(7)
1.284(1) 1.267(1) 1.274(1)
1.280(2) 1.269(2) 1.313(2)
1.271(2) 1.264(2) 1.308(2)
1.284(2) 1.270(2) 1.312(2)
1.273(3) 1.260(4) 1.333(3)
1.276(2) 1.276(2) 1.304(2)
1.284(3) 1.280(3) 1.309(3)
1.276(3) 1.284(3) 1.318(2)
1.279(3) 1.292(3) 1.307(3)
1.278(3) 1.282(3) 1.318(2)
1.274(3) 1.274(3) 1.327(3)
1.281(3) 1.280(3) 1.312(2)
1.276(2) 1.271(2) 1.308(2)
1.308(2) 1.267(2) 1.271(2)
1.260(4) 1.276(4) 1.348(4)
1.278(3) 1.283(4) 1.313(3)
1.278(3) 1.266(4) 1.309(3)
1.297(5) 1.294(5) 1.345(5)
1.280(1) 1.256(1) 1.282(1)
1.285(4) 1.264(4) 1.313(3)
1.279(2) 1.267(2) 1.313(2)
1.273(2) 1.276(2) 1.317(2)
1.290(3) 1.286(3) 1.324(3)
1.291(3) 1.285(3) 1.325(3)
ηMinST
ηRanST
IV.FINITE-SIZE EFFECT
Table I shows that the allometric scaling exponents
for a given type of spanning tree are very close to each
other. However, there are still fluctuations around the
corresponding average values. It is possible that the ex-
ponent is dependent on the length of the index time se-
ries. It is thus necessary to further investigate if there
is a finite-size effect, which is crucial to the validation of
universality.

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A.The case of MaxST and MinST
Figure 2 illustrates the dependence of ηMaxST and
ηMinST as a function of the stock index length L. No
evident trend is identified by eye-balling. We fit the data
for MaxSTs and MinSTs using a linear model
ηMaxST,MinST= a + bL.(4)
For the MaxSTs, a = 1.279 and b = 3.949 × 10−7and
the corresponding p-values from the Student’s t-test are
0 and 0.355, respectively. For the MinSTs, a = 1.278
and b = −8.315 × 10−7and the corresponding p-values
are 0 and 0.082, respectively. If we regress the exponent
against lnL, the p-value of b is 0.322 for the MaxSTs and
0.314 for the MinSTs. It is clear that the coefficient b is
identical to 0 and the exponents ηMaxSTand ηMinSTare
independent of L.
102
103
104
105
1.24
1.26
1.28
1.3
1.32
L
η


MaxST
MinST
FIG. 2: (Color online) Dependence of the allometric scaling
exponents ηMaxST and ηMinSTon the length L of stock indices.
No evident finite-size effect is observed.
B.The case of RanST
For the RanSTs extracted from the visibility graphs
of stock indices, the three allometric scaling exponents
ηRanSTof the US market indices (DJIA, S&P 500, NAS-
DAQ) are significantly less than the others, which is a
signal of the possible presence of finite-size effect. Figure
3 plots ηRanST as a function of L for all the indices. It
is evident that ηRanSTdecreases logarithmically with the
increase of index length:
ηRanST= a + blnL,(5)
where a = 1.471 and b = −0.019, both of which are
significantly different from 0 with the p-values less than
10−6.
102
103
104
105
1.26
1.28
1.3
1.32
1.34
1.36
1.38
L
ηRanST
FIG. 3: Dependence of the allometric scaling exponent ηRanST
on the length L of stock indices. An evident finite-size effect
is observed.
C. Numerical tests
In order to further investigate the impact of the time
series length on the allometric scaling exponents of the
three types of spanning trees, we design and conduct two
numerical tests. We take subseries with different lengths
from the longest index DJIA to perform the allometric
scaling analysis. For each length L, 100 subseries are
randomly extracted and the corresponding scaling expo-
nents are calculated from their MaxSTs, MinSTs, and
RanSTs. Figure 4 illustrates the dependence of the av-
eraged exponent η with respect to the length L for the
DJIA subseries. A linear regression of ηMinSTagainst L
using Eq. (4) gives that a = 1.261 and b = 4.093 × 10−8
with the p-values being 0 and 0.16, respectively. It means
that ηMinSTis independent of L. For the MaxSTs, we find
that ηMaxSTincreases with L linearly
ηMaxST= 1.276 + 4.555 × 10−7L,(6)
where the linear coefficients are statistically significantly
different from zero with both the p-values being zero. In
contrast, ηRanSTexhibits an evident decreasing trend. A
linear regression gives a = 1.327 and b = −2.581× 10−6,
whose p-values are both nulls. These observations are
consistent with the results in Table I. In addition, we find
that ηMinST< ηMaxSTfor all L’s, while ηRanSTbecomes
less than ηMaxSTwhen L > 16500 and less than ηMinST
for much greater L.
For comparison, we synthesize Brownian motions with
different lengths. For each length, 100 time series are
generated and the allometric scaling exponents are de-
termined. The results are also illustrated in Fig. 4. The
scaling exponents ηMaxSTand ηMinSTare found to be in-
dependent of L since linear regressions show that the p-
values of the corresponding b’s are 0.17 and 0.06, respec-
tively. On the other hand, the scaling exponent ηMinST
exhibits a decreasing trend for small L and then reaches
a constant. For all L’s, ηMinST≈ ηMaxST < ηRanST for
the Brownian motions.

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00.51
L
1.52
x 104
1.22
1.24
1.26
1.28
1.3
1.32
1.34
1.36
η


DJIA: MaxST
DJIA: MinST
DJIA: RanST
Bm: MaxST
Bm: MinST
Bm: RanST
FIG. 4: (Color online) Dependence of the averaged exponent
η with respect to the length L for the DJIA and the Brownian
motions.
We stress that there are marked discrepancies between
the scaling exponents of the MinSTs (and the MaxSTs
as well) of the DJIA and Brownian motions.
for an investigation of influence factors on the allometric
behaviors of the MinSTs and MaxSTs of the stock market
indices. Note that a comparison between the behaviors
of the stock indices and the Brownian motions is not
out of blue. Indeed, stock prices are assumed to follow
a geometric Brownian motion in the celebrated Black-
Scholes model of option pricing [54].
It calls
V.
ALLOMETRIC SCALING BEHAVIOR
INFLUENCE FACTORS ON THE
To understand the marked discrepancies between the
allometric behaviors of the stock market indices and the
Brownian motions, further numerical experiments are
needed.Comparing with Brownian motions, we find
there are three potential factors that may have influ-
ence on the behavior of any time series, that is, the fat-
tailedness of the probability distribution, the linear long-
term correlation, and the nonlinear long-term correlation
[55]. It is well established that there is no linear long-
term correlation in the returns of stock indices. Hence,
two factors remain for further investigations. For com-
pleteness, we first study the impact of linear long-term
correlations based on fractional Brownian motions.
A.Fractional Brownian motions
We generate fractional Brownian motions through
wavelet transform with the Hurst indexes varying from
0.05 to 0.95. We obtain 100 realizations for each Hurst
index, and each realization has 5000 data points.
In the case of Brwonian motions where the Hurst index
H = 0.5, remarkable power-law behaviors are observed
between A and C for the three spanning trees. The least
square linear fit to the data for each spanning tree yields
an estimate of the power-law exponent, which results in
η = 1.233 ± 0.004 for the MaxST, η = 1.233 ± 0.004 for
the MinST, and η = 1.314 ± 0.004 for the RanST. Note
that the node (1,1) is excluded in the implementation of
fitting. The plots of A with respect to C for the other
Hurst indexes share the similar pattern as the plots for
H = 0.5, but the power-law exponents are different.
Figure 5 shows the allometric scaling exponent η as a
function of the Hurst index H for three different spanning
trees. For the MaxST and the MinST, the two curves
almost overlap onto a same curve. Furthermore, both
curves show a linear decreasing trend
η = a + bH,(7)
where a = 1.238 and b = −0.008 for the MaxSTs and
a = 1.238 and b = −0.007 for the MinSTs with all the p-
values less than 0.01. The variation of these two scaling
exponents is actually very slight and all the exponents
are embedded in the interval [1.23,1.24]. In contrast, the
power-law scaling exponent η for the RanSTs increases
with the Hurst index H, which can also be modeled by
Eq.(7).A linear regression finds that a = 1.301 and
b = 0.027 with the associated p-values being zero.
0 0.20.40.60.81
1.22
1.24
1.26
1.28
1.3
1.32
1.34
1.36
H
η


RanST
MinST
MaxST
FIG. 5: Plots of the dependence of allometric scaling exponent
η on the Hurst index H for three spanning trees.
B.
exponents between financial series and Brownian
motions
Origin of the difference of allometric scaling
Compared with the Brownian motions, the financial
series exhibit fat-tailed PDF and nonlinearity as well [55].
We design several tests to verify the contribution of the
two factors on the allometric scaling behavior. In the
tests, different surrogate data are used. There is no linear
long-term correlation in the surrogate data. We use the
FTSE index as an example, and all the surrogate time
series have the same length as the FTSE data.
The surrogates of the first type (termed “Surr 1”) have
the same probability distribution of returns as the FTSE