Mapping Coupled Time-series Onto Complex Network
Jamshid Ardalankia, Jafar Askari, Somaye Sheykhali, Emmanuel Haven, G. Reza Jafari
MMapping Coupled Time-series Onto Complex Network
Jamshid Ardalankia a,b , Jafar Askari c , Somaye Sheykhali b,d,e , Emmanuel Haven f , G. Reza Jafari b,c,g a Department of Financial Management, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran b Center for Complex Networks and Social Datascience, Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran c Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran d Department of Physics, University of Zanjan (ZNU), Zanjan, 45371-38791, Iran e Instituto de Fsica Interdisciplinary Sistemas Complejos IFISC (CSIC-UIB), Palma de Mallorca, E07122, Spain f Faculty of Business Administration, Memorial University, St. John’s, Canada and IQSCS, UK g Department of Network and Data Science, Central European University, 1051 Budapest, Hungary
Abstract
For the sake of extracting hidden mutual and coupled information from possibly uncoupled time-series, we explored the profoundmeasures of network science on time-series. Alongside common methods in time-series analysis of coupling between financial andeconomic markets, mapping coupled time-series onto networks is an outstanding measure to provide insight into hidden aspectsembedded in couplings intrinsically. In this manner, we discretize the amplitude of coupled time-series and investigate relativesimultaneous locations of the corresponding amplitudes (nodes). The transmissions between simultaneous amplitudes are clarifiedby edges in the network. In this sense, by segmenting magnitudes, the scaling features, volatilities’ size and also the direction of thecoupled amplitudes can be described. The frequency of occurrences of the coupled amplitudes is illustrated by the weighted edges,that is to say, some coupled amplitudes in the time-series can be identified as communities in the network. The results show thatdespite apparently uncoupled joint probabilities, the couplings possess some aspects which diverge from random Gaussian noise.Thereby, with the aid of the network’s topological and statistical measurements, we distinguished basic structures of coupling ofcross-market networks. Meanwhile, it was discovered that even two possibly known uncoupled markets may possess coupledpatterns with each other. Thereby, those markets should be examined as coupled and weakly coupled markets!
Keywords:
Coupled Time-series, Complex Networks, Financial Markets
1. Introduction
It is intriguing to study coupled markets by mapping theircoupling onto a network. The reasoning behind this includesthe numerous measures introduced by network science.In addition to previously applied measures by time-seriesanalysis, network analysis measures help scholars to exploredeeper characteristics of economic and financial networks.In the present study, we show that coupling of time-series(which may be apparent so far, or highly coupled with eachother) contains information which may have been remaineduncovered. This discovery is demonstrated by measures ofdescribing the features of a network obtained from the couplingof two time-series.In order to extract more hidden information from time series,network science has been utilized for analyzing the extractionof information about the time series of a wide variety of fieldsthrough the analysis of the derived network [1–5]. Consideringthat network science [6, 7] has become highly applicable in timeseries analysis and multi-agent based models [8], such as finan-cial markets [9–15] to clarify the intrinsic structure of the cou- ∗ [email protected] (G.Reza Jafari) Email address: [email protected] (Emmanuel Haven) plings between the price volatility structure [16] and macroe-conomic measures [17, 18], the economic recovery plans [19]relating to financial and economic crises [20] and systemicrisk [21–25], risk measurement in financial directed causalitynetworks [26], multivariate financial time-series [27], analysisof nonlinear time-series [28], analysis of time-series by map-ping to weighted and directed networks [29], analysis of linearregression patterns in a non-stationary time-series [30], laggede ff ects of a dynamical system by network science [31]. More-over, these methodologies have successfully been applied forchaotic time-series [32], higher-order network analysis [33, 34],and also tourism management [35]. Along with giving usthe ability to reason about network topology and communitystructures [36], the internal interactions and information trans-missions in magnitude-wise aspects are crucial. Hence, as amulti-scale approach in mapping time-series onto a complexnetwork, a novel algorithm in transmission of regression pat-terns between two time-series was developed in a complex net-work viewpoint [30]. Other proposed methods include mappingmulti-variate time-series onto multiplex [37, 38] networks todevelop multi-dimensional signal processing [27] to assess fi-nancial instability [1, 39, 40] by visibility graphs [41], machinelearning algorithms by visibility graphs to multiplex learningnetworks [42] and financial minimum spanning tree [36, 43].At the heart of the above applications are measures for reveal- Preprint submitted to Elsevier April 29, 2020 a r X i v : . [ q -f i n . C P ] A p r ng the structural topology of networks of coupled time-serieswhether the time-series are random, periodic (ordered) or frac-tal which will cause the mapping process to result in a ran-dom network, regular network and scale-free network, respec-tively [1].Cross-correlation in financial time-series intrinsically con-tains scaling behaviors [44, 45]. Those scaling behavioursnot only emerge in temporal aspects, but also, they emerge inhigher statistical moments of price return distribution— corre-lation coe ffi cient. In this context, the present study casts lightsinto the behavior of couplings between financial time-series byapplying novel measures of network science. In respect to thecross-correlation networks [46], the network properties such asclustering coe ffi cient, e ffi ciency, the cross-correlation degreeof cross-correlation interval and also modularity of dynamicstates have been investigated [32].In particular, we are supposed to capture / estimate tempo-ral / dynamic behaviors of the financial time-series by mappingonto a network perspective as follows, by: I. introducing the mapping algorithm from coupled time-seriesonto a network; II. constructing of the networks obtained by coupling of twofinancial time-series, and;
III. the networks obtained by fractional Gaussian noise (fGns)coupled by their corresponding 1-step lag with a diverse rangeof Hurst exponents;
IV. comparing the obtained networks and extracting the hiddenfeatures of couplings.Hurst exponent is a criterion to perceive to what extent twosignals are coupled in various time-scales. We observed thataccording to some features both coupled and uncoupled time-series are di ff erent. On the other hand, some features of thosecouplings are significantly close to the networks obtained byfGns. However, there exist features where none of the net-works converge to a definite value. Based on the segregationof those networks, the information transitions and common fea-tures among couplings are revealed.
2. Mapping Single Time-series Onto Network
Considering scaling features of amplitude-wise financial cor-relations [45, 47], alongside with the fact that correlation coef-ficients just reveal the linear co-behaviors of the time-series,there exists a vital need to consider the e ff ects of the directionand the size of amplitudes which may consist of nonlinear be-haviours. In this way, without the need for necessarily linearrelation, the couplings are defined. This procedure can be ex-plored by temporal-interval [32] and amplitude-interval pointsof view [2]. To explore amplitude intervals (amplitude bins),we generate a method to couple the amplitude of a time-serieswith its corresponding 1-step lag. Hence, we make a segmenta-tion on the amplitudes and convert the amplitudes onto severalbins [2]—from now on, we consider these amplitude bins asnodes in a certain network. For illustrative purposes, Fig. 1, topsubfigure, shows the way we design the process. Figure 1: The algorithm of mapping is demonstrated. This figure depicts theway links in the network are generated. As shown, when the relative ampli-tudes corresponding to two time-series are located in the same amplitude bins(nodes), a weighted loop is considered. On the contrary, when the amplitudesare not located in the same bin, two nodes connecting with an weighted edge are generated. The term weight , here, implies the frequency of this directedsituation– and can be applied with the timing resolution considerations andthe weighting threshold sampling [48] and the persistence of the edges [49].Thereby, the outcome will be a temporal coupling network.
The Hurst exponent of a system implies how two time-series–also one single time-series and its lags– are coupled (uncou-pled). In this regard, financial time-series contain some struc-tural and intrinsic information whether they are a developed oran emerging market [44, 50, 51]. Initially, we generate somefractional Gaussian noise (fGns), and we will further comparethe characteristics of real-world time-series. To cast light intothe e ff ects of Hurst exponents in the behavior of a certain frac-tional Gaussian noise, in Fig. 2, the auto-correlation matricespertaining to the Hurst exponents are depicted. It is notablethat for higher Hurst exponents (as an identification of strongercoupling), we observe higher correlations around the diameterof the correlation matrix.From Fig. 2, one observes that a high Hurst exponent leadsto a high elongation around its main diameter of joint probabil-ities, refer to Eq. 1.For the sake of quantifying the elongation of couplings, weintroduce a deformation parameter, R, which widely clarifiesthe couplings behavior. To quantify this deformation, we intro-duce a deformation parameter based on the standard deviationsalong diameters of joint probabilities matrix, R , as Eq. 1 illus-trates; R = σ i − σ j max { σ i , σ j } ; (1)where, σ denotes the standard deviations along diameters ofthe joint probability matrix. Further, the relationship betweenthe individual parameter R relative to the corresponding Hurst2 igure 2: 1st row) Auto-correlation matrices relating to Hurst = = = R versus their corresponding Hurst exponents are shown. exponents will be clarified in Fig. 2, bottom subfigure.For now, according to Fig. 2 and applying Eq. 1, one is able toestimate the couplings’ Hurst exponent corresponding to eachcross-market based on its R parameter–which has been alreadyobtained by the cross-market joint probabilities.
3. Mapping Coupled Time-series Onto Network
In the following, we map the coupling of cross-markettime-series onto a network. The algorithm which is appliedhere is the same as the previous one, but two signals withthe same chronological time-stamp (no lag) is considered.Next, the outcome will be compared with the fGns which aremapped onto the network before. This process is showed inFig. 3. By focusing on this phenomenon for cross-marketsjoint probabilities, the extent to which the strength of cou-plings causes elongation is carried out in [53]. Besides our e ff orts on considering the Hurst exponent of the couplings ofcross-market time-series [47, 54], in the present study, we willinvestigate the cross-market couplings in a financial networkapproach.From Fig. 1, it should be highly emphasized that in addi-tion to considering the positive and negative directions of am-plitudes, we account for the di ff erences between the size andthe locations in the amplitude bins by generating edges. In thisregard, the placement of amplitudes in the same amplitude binsleads to a loop. Meanwhile the placement of amplitudes in dif-ferent amplitude bins leads to a an edge. In this regard, thedirection of edges stands for emphasizing on the di ff erence be-tween whether the first signal is in bin A and the other one in B,as opposed to whether the first signal in bin B and the secondone in A.Here, the role of the Hurst exponent should be highlighted –asan intrinsic structure of time series– during the process of map-ping the coupling onto the joint probability matrix.In this research we generate discrete intervals to evaluate thecoupled amplitude of the markets by mapping the time seriesonto coupled networks. Smaller amplitude bins lead to higheramount of bins and consequently more noise detection in thecouplings. Accordingly, wider amplitude bins imply more sim-ilar events between the markets, and, smaller amplitude binsyield to more noise detection among amplitudes. Results and Discussion - Firstly, it is notable that fGn with Hurst = uncoupled situation, which is shown in Fig. 3. - Deformation Ratio, R . According to Fig. 3, the processof mapping coupling onto a network is clarified. As shownfrom the comparison between joint probability matrices ofDJIA-SSEC and DJIA-S&P500, the strength of couplings arevisually showing that the coupling of DJIA-S&P500 is strongerthan that of DJIA-SSEC. This feature is quantified based onEq. 1 with the R parameter which is considered in the radarplot in Fig. 3. - Degree Measurements . The measurements correspondingto degrees, such as mean squared out-degrees < k out > ,mean squared in-degrees < k in > , mean squared total-degrees < k total > , mean out-degrees < k out > , mean in-degrees < k in > and mean total-degrees < k total > , significantly contain thepower of proving the segregation among cross-markets andfGn with Hurst = igure 3: Mapping the cross-correlation of time-series onto a network is shown: from left, the 1st column shows the time-series of the markets such as SSEC,DJIA and S&P500 in daily resolution during 2000 days until Jul. 31 st ff erent from each other where prove the types of couplings aretypically distinguishable. Also, the standard deviation of total-degree < k totalstd > turns upto identify cross-market couplings. - Clustering Measurements [55]. When it comes to cluster-ing features, the standard deviation of global clustering coef-ficient Cl . Coe f . stdglobal is capable of exploring the di ff erence be-tween cross-market couplings. Moreover, the undirected localclustering coe ffi cient Cl . Coe f . undirectedlocal can distinguish amongthe networks of coupled and uncoupled cross-markets. It isshown that this feature converges to fGn with Hurst = ffi cient Cl . Coe f . directedlocal is di ff erent foruncoupled and coupled outcomes. Whereas, the global cluster-ing coe ffi cient Cl . Coe f . global for fGns, uncoupled and coupledcross-markets are approximately similar. - Length (Shortest Path Between Pair-wise Vertices) Measure-ments . It is striking that the directed mean length < L directed > and the undirected mean length < L undirected > significantly ex-plore the di ff erences between coupled cross-markets from un-coupled cross-markets and the fGns. - Assortativity Measurements [56]. The variance of scalarassortativity coe ffi cient S c . Ass . Coe f . var . for fGns, uncoupled and coupled cross-markets are approximately similar. Con-versely, the assortativity coe ffi cient variance Ass . Coe f . var . , as-sortativity coe ffi cient Ass . Coe f . , scalar assortativity coe ffi -cient S c . Ass . Coe f . markedly distinguish among coupled cross-markets from fGns and uncoupled cross-markets. - Modularity Measurements [57, 58]. By extracting commu-nity structures, the modularity measurements enable scholarsto distinguish the dynamic states of the network. Thereby,following the approach of mapped networks [32], we find ithighly useful to investigate the coupling networks structuresby comparing their modularity, as shown in Fig. 3. Asdepicted, notwithstanding that out-degree modularity enablesone to identify the cross-markets from fGn, the total-degreemodularity
Modularity total − degree is highly capable of showingthe divergence between uncoupled and coupled cross-markets.This implies, that based on the out-degree modularity mea-surement, there exists mutual information among markets andit is not uncoupled in this manner. In this regard, markets arecoupled or weakly coupled (not necessarily uncoupled)!To further explore the payo ff s and to assess the di ff erences,DJIA-S&P500’s coupling is adequately far from Gaussian ran-dom noise (fGn with Hurst = = ff erent. In such a situ-ation we can conclude that although the joint probabilities ofDJIA and SSEC time-series show that they are uncoupled (re-fer to deformation ratio, R, in Eq. 1 and Fig. 3), by mappingcoupled time-series onto a complex network perspective, morehidden properties are revealed and the obtained results illustratethat according to some other network criteria, those marketsstill possess coupling information! Hence, being a market con-tributes to being coupled with others! Thus, it is better to usethe term, weakly coupled markets rather than the term, uncou-pled markets . As depicted in Fig. 3, the joint probability matri-ces can be considered as the adjacency matrices correspondingto the network of couplings. Along with giving us the abil-ity to reason about the couplings formation, Fig. 3 will extendour knowledge toward realistic simulations in intrinsic coupledstructures in the network.
4. Conclusion
Taken together, mapping cross-correlation of time-seriesonto a network contributes to defining a coupling network .Topological and statistical parameters along with the de-formation ratio of adjacency matrix –regarding the cross-correlations– are able to reveal the coupling information whichhave previously been beyond the reach of researchers. Not onlycomparing fGns with cross-market coupled networks provespair-wise interconnectedness, but also it clarifies the diversestructure of the coupling. The reasoning behind this claim isthat couplings with di ff erent Hurst exponents show a diverserange of behaviors (anti-persistent, random, persistent) whichcan be reflected in a so-obtained mapped network. In thepresent study, cross-correlation of some time-series are mappedonto a network. Accordingly, a mapped coupling network is ex-plored. Meanwhile, a deformation parameter which is extractedfrom a relation for the standard deviations alongside both diam-eters of the directed weighted adjacency matrix (joint probabil-ity matrix) is introduced. The above mentioned approach is ap-plied on cross-markets coupling networks such as DJIA-SSECand DJIA-S&P500.With the aid of the network topological and statistical mea-surements, we distinguished a basic structure of coupling ofcross-market networks. Meanwhile, it was discovered that eventwo previously known uncoupled markets may possess coupledaspects with each other. Thereby, those markets should be ex-amined as coupled and weakly coupled markets! References [1] L. Lacasa, B. Luque, F. Ballesteros, J. Luque, J. C. Nuno, From timeseries to complex networks: The visibility graph, Proceedings of the Na-tional Academy of Sciences 105 (13) (2008) 4972–4975.[2] A. H. Shirazi, G. R. Jafari, J. Davoudi, J. Peinke, M. R. R. Tabar,M. 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