Degree distribution of the visibility graphs mapped from fractional Brownian motions and multifractal random walks
aa r X i v : . [ phy s i c s . s o c - ph ] M a y Degree distributions of the visibility graphs mapped from fractional Brownianmotions and multifractal random walks
Xiao-Hui Ni a,b,c , Zhi-Qiang Jiang a,b,c,d , Wei-Xing Zhou ∗ ,a,b,c,e,f a School of Business, East China University of Science and Technology, Shanghai 200237, China b School of Science, East China University of Science and Technology, Shanghai 200237, China c Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China d Chair of Entrepreneurial Risks, D-MTEC, ETH Zurich, Kreuplatz 5, CH-8032 Zurich, Switzerland e Engineering Research Center of Process Systems Engineering (Ministry of Education), East China University of Science and Technology,Shanghai 200237, China f Research Center on Fictitious Economics & Data Science, Chinese Academy of Sciences, Beijing 100080, China
Abstract
The dynamics of a complex system is usually recorded in the form of time series, which can be studied through itsvisibility graph from a complex network perspective. We investigate the visibility graphs extracted from fractionalBrownian motions and multifractal random walks, and find that the degree distributions exhibit power-law behaviors,in which the power-law exponent α is a linear function of the Hurst index H of the time series. We also find that thedegree distribution of the visibility graph is mainly determined by the temporal correlation of the original time serieswith minor influence from the possible multifractal nature. As an example, we study the visibility graphs constructedfrom two Chinese stock market indexes and unveil that the degree distributions have power-law tails, where the tailexponents of the visibility graphs and the Hurst indexes of the indexes are close to the α ∼ H linear relationship. Key words:
Visibility graph; complex networks; power-law distribution; fractional Brownian motion; multifractalrandom walk
PACS:
1. Introduction
Complex systems are ubiquitous in natural and social sciences, where the constituents interact with one anotherand form a complex network. In recent years, complex network theory has stimulated explosive interests in thestudy of social, informational, technological, and biological systems, resulting in a deeper understanding of complexsystems [1, 2, 3, 4]. However, for many complex systems, it is hard to obtain detailed information of interactingconstituents and their ties, which makes the underlying network invisible. Instead, we are able to observe and record atime series generated by the system. For such cases, time series analysis becomes a crucial way to unveil the dynamicsof complex systems. There are also some e ff orts to map time series into graphs to study time series from the networkperspective, which amounts to investigating the dynamics from the associated network topology.For a pseudoperiodic time series, one can partition it into disjoint cycles according to the local minima or maxima,and each cycle is considered a basic node of a network, in which two nodes are deemed connected if the phase spacedistance or the correlation coe ffi cient between the corresponding cycles is less than a predetermined threshold [5]. Wenote that a weighted network can also be constructed if the phase space distance or the correlation coe ffi cient is treatedas the weight of a link. This method for pseudoperiodic time series can also be generalized to other time series, wherea node is defined by a sub-series of a fixed length as the counterpart of a cycle, which has been applied to stock prices[6]. ∗ Corresponding author. Address: 130 Meilong Road, P.O. Box 114, School of Business, East China University of Science and Technology,Shanghai 200237, China, Phone: +
86 21 64253634, Fax: +
86 21 64253152.
Email address: [email protected] (Wei-Xing Zhou )
Preprint submitted to Physics Letters A October 28, 2018 nother method for network construction from time series is based on the fluctuation patterns [7, 8]. In thisapproach, each data point is encoded as a symbol R , r , D , or d , corresponding to big rise, small rise, big dropand small drop, respectively. The time series is then transformed into a symbol sequence. Defining a fluctuationpattern as an n -tuple consist of a string of n symbols, the symbol sequence can be further mapped into a sequenceof non-overlapping n -tuples. The n -tuples are treated as nodes of the constructed network. Therefore, the numberof nodes does not exceed n . Two nodes are connected if the associated n -tuples appear one after the other in thepattern sequence. Furthermore, the edge weight between two nodes can be defined as the occurrence number of twosuccessive patterns in the sequence. This approach has been applied to study the price trajectory of Hang Seng index[7, 8].A third method is to convert time series into visibility graphs based on the visibility of nodes [9]. For simplicity,consider an evenly sampled time series { y t : t = , , · · · , N } . Each data point of the time series is encoded into a nodeof the visibility graph. Two arbitrary data points y i and y j have visibility if any other data point y k located betweenthem fulfills y j − y k j − k > y j − y i j − i . (1)Two visible nodes become connected in the associated graph. An example of a time series containing 16 data pointsand the associated visibility graph derived from the visibility algorithm is illustrated in Fig. 1. By definition, anyvisibility graph extracted from a time series is always connected since each node sees at least its nearest neighbor(s)and the degree of any node y t with 1 < t < N is not less than 2. In addition, a periodic time series converts intoa regular graphs, whose degree distribution is formed by a finite number of peaks related to the series period, whilerandom time series lead to irregular random graphs [9]. It is also found that visibility graph is invariant under a ffi netransformations of the series data since the visibility criterion is invariant under rescaling of both horizontal andvertical axes, and under horizontal and vertical translations [9]. Figure 1: Example of a time series containing 16 data points (upper panel) and the associated visibility graph derived from the visibility algorithm(lower panel).
Degree distribution p ( k ) is one of the most important characteristic properties of complex networks [10]. Thedegree distribution of the visibility graphs of several specific examples of time series have been investigated [9]. Fora random time series extracted from an uniform distribution in [0 , p ( k ) ∼ e − k / k . Alternatively, the visibility graphs of Brownian motions and Conway series arescale-free, characterized by a power-law tail in the degree distribution: p ( k ) ∼ k − α , (2)where α = . ± .
01 for Brownian motions and α = . ± . H ) might have influence on the degreedistribution of its visibility graph [9]. In this work, we test this projection based on extensive numerical simulations.Specifically, fractional Brownian motions (FBMs) [11] and multifractal random walks (MRWs) [12] are synthesized2o investigate the influence of autocorrelation and multifractality on the degree distribution.
2. Numerical analysis
There are many di ff erent algorithms for the generation of fractional Brownian motions [13] and we adopt awavelet-based algorithm to simulate FBMs [14]. On the other hand, a multifractal random walk can be generatedby the cumulative summation of the increments ∆ y t = ǫ t e ω t , (3)where ǫ t is a fractional Gaussian noise with Hurst index H in , ω t is a correlated Gaussian noise, and they are independent[12]. We use the detrended fluctuation analysis [15, 16] to verify if the resultant Hurst index of the generated signals isidentical to the input value of H in in the algorithms. For each H in , we generate 10 realizations and calculate the meanHurst index H . The results are presented in Fig. 2. The top panel of Fig. 2 shows that the estimated Hurst indexes ofthe synthesized FBMs are very close to the input value H in with minor deviation for small H in . For MRWs, we findthat H = H in when H in > . H > H in when H in .
5. In addition, we have confirmedthat the generated MRW signals possess multifractal nature. H in H (a) H in H (b) Figure 2: (color online.) Dependence of the Hurst index H of the simulated (a) FBMs and (b) MRWs, determined by detrended fluctuation analysison the input Hurst index H in in the two synthesis algorithms. We have investigated FBMs with the input Hurst index H in ranging from 0.1 to 0.9 in the spacing of 0.1. For each H in , we repeat the simulation 100 times and each simulation gives a FBM signal with the size N = , ff erent Hurst indexes are depicted in Fig. 3(a). Nice power-law behaviorsare observed in the distributions, followed by faster relaxation. We note that the visibility graphs of other FBMs alsoexhibit power-law tails in the degree distribution.The situation is very similar for the MRW case. We have simulated MRW signals of size N = ,
000 withdi ff erent input Hurst index H in ranging from 0.05 to 0.95 with an increment of 0.05. For each H in , 100 MRW signalsare simulated and then converted to 100 visibility graphs. Three typical empirical degree distributions of the visibilitygraphs are depicted in Fig. 3(b) for di ff erent Hurst indexes H (not H in ). All the distributions exhibits nice power lawswith faster decay for large degrees. 3 −6 −5 −4 −3 −2 −1 k P ( k ) H = 0.2 H = 0.5 H = 0.8 −6 −5 −4 −3 −2 −1 k P ( k ) H = 0.33 H = 0.5 H = 0.8 Figure 3: (color online.) (a) Empirical degree distributions of the visibility graphs converted from MRW series with di ff erent Hurst indexes H = .
33, 0 . .
8. (b) Empirical degree distributions of the visibility graphs converted from FBM series with di ff erent Hurst indexes H = . . . The power-law exponents α of the distributions are calculated in the scaling ranges. Figure 4 shows the dependenceof the power-law exponents α on the Hurst indexes H . Both curves show a nice linear relationship: α ( H ) = a − bH . (4)A least-squares regression gives a = .
35 and b = .
87 for FBMs and a = .
19 and b = .
55 for MRWs. We find thatthe multifractal nature of the MRWs has minor influence on the degree distributions of the visibility graphs. H α H α Figure 4: (color online.) Dependence of the power-law exponent α on the Hurst index H for (a) FBM visibility graphs and (b) MRW visibilitygraphs. The straight lines are the least-squares fits of Eq. (4). We note that, the linear relationship between the tail exponent and the Hurst index was also found for fractionalBrownian motions independently [17].
3. Applications to financial data
In this section, we apply the visibility graph method to financial data. Specifically, two indices of the Chinesestock market are considered. The organized stock market in mainland China is composed of two stock exchanges, theShanghai Stock Exchange (SHZE) and the Shenzhen Stock Exchange (SZSE). Shanghai Stock Exchange Compositeindex (SHCI) and Shenzhen Stock Exchange Component index (SZCI) are the representative indices for the two stock4xchange, respectively. Our analysis is based on the 1-min data of the two indices. The time series spans from 2January 2001 to 28 December 2007 for the SHCI and from 4 January 2002 to 28 December 2007 for the SZCI. Thetemporal evolution of the price trajectories of the two indices are illustrated in Fig. 5. According to Fig. 5, the Chinesestock market was in a bearish phase from 2001 to 2005, which is known as an antibubble [18], and then the marketreversed and produced a very marked bubble which burst at the end of 2007. In order to test if the market phase hasimpact on the results, we partition each time series into two sub-series delimited on 31 December 2005, correspondingto the bear period and the bull period, respectively. year i nd e x SZCISHCI
Figure 5: Temporal evolution of the price trajectory of the Shanghai Stock Exchange Composite index and the Shenzhen Stock Exchange Compo-nent index.
For each index, we obtain three visibility graphs corresponding to the bear period, the bull period, and the wholetime period. The empirical degree distributions of the three visibility graphs are determined. Fig. 6 shows the resultsof SHCI. We note that the results for the SZCI are very similar. −11 −9 −7 −5 −3 −1 k P ( k ) SHCI
Whole periodBear periodBull period
Figure 6: (color online.) Empirical degree distributions of the three visibility graphs of the Shanghai Stock Exchange Composite index.
We observe that all the distributions have heavy tails and the degree distribution in the bear period can be wellmodeled by a power law (2) in the right part. The rapid decay does not mean that the power law is truncated but ratherreflects the fluctuations at finite size [19]. The tail exponent is estimated to be α = .
95. For the case of the bullperiod, there is a hump in the tail. This phenomenon is more evident for the whole period, which is caused by the factthat there are more points in the bull period that can see more previous points (large degrees). There are two powerlaws in the degree distributions for the whole time period case, illustrated by two parallel dashed lines in Fig. 6. Theestimates of the tail exponents α for SHCI and SZCI are listed in Table 1.In order to compare the empirical results with the numerical results in Section 2, we determine the Hurst indexes5 able 1: Comparison of the estimated tail exponents α and the “predicted” tail exponents α ′ . The last row is e = ( α − α ′ ) /α ′ . SHCI SZCIBear Bull Whole Bear Bull Whole H α α ′ e H = .
5. The multifractal detrended fluctuation analysis [20]confirms that all the time series possess multifractal nature. Therefore, the tail exponents can be “predicted” accordingto the linear relationship α ′ = . − . H . The predicted tail exponents are also presented in Table 1. It is found thatthe discrepancy between α and α ′ is not large, which is quantified by the relative di ff erence e = ( α − α ′ ) /α ′ shown inTable 1. Fig. 7 further illustrate this point by the scatter plot of the six data points ( H , α ). These data points are veryclose to the dot-dashed line α ′ = . − . H , showing that the empirical results are consistent with the numericalanalysis in Section 2. H α SHCISZCI
Figure 7: (color online.) Scatter plot of ( H , α ) of the six time series comparing the empirical results with the numerical results predicted by α ′ = . − . H (dot-dashed line).
4. Conclusion
In summary, we have studied the degree distributions of visibility graphs extracted from fractional Brownianmotions and multifractal random walks. We found that the degree distributions exhibit power-law behaviors, in whichthe power-law exponent is a linear function of the Hurst index inherited in the time series. In addition, the degreedistribution of the visibility graph is mainly determined by the temporal correlation of the corresponding time series,and contains minor information about the multifractal nature of the time series. The linear relation (4) provide apossible tools for the determination of H of a time series from its visibility graph [9]. However, cations should betaken since the increments distribution of the time series might also have impact on the degree distribution. Acknowledgments:
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