the multiplex dependency structure of financial markets

Research ArticleThe Multiplex Dependency Structure of Financial Markets

Nicolograve Musmeci1 Vincenzo Nicosia2 Tomaso Aste34

Tiziana Di Matteo13 and Vito Latora25

1Department of Mathematics Kingrsquos College London The Strand London WC2R 2LS UK2School of Mathematical Sciences Queen Mary University of London Mile End Road London E1 4NS UK3Department of Computer Science University College London Gower Street London WC1E 6BT UK4Systemic Risk Centre London School of Economics and Political Sciences London WC2A 2AE UK5Dipartimento di Fisica ed Astronomia Universita di Catania and INFN 95123 Catania Italy

Correspondence should be addressed to Vito Latora vlatoraqmulacuk

Received 25 May 2017 Accepted 16 July 2017 Published 20 September 2017

Academic Editor Tommaso Gili

Copyright copy 2017 Nicolo Musmeci et alThis is an open access article distributed under theCreativeCommonsAttributionLicensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose here a multiplex network approach to investigate simultaneously different types of dependency in complex datasetsIn particular we consider multiplex networks made of four layers corresponding respectively to linear nonlinear tail and partialcorrelations among a set of financial time series We construct the sparse graph on each layer using a standard network filteringprocedure and we then analyse the structural properties of the obtained multiplex networks The study of the time evolution ofthe multiplex constructed from financial data uncovers important changes in intrinsically multiplex properties of the network andsuch changes are associated with periods of financial stressWe observe that some features are unique to themultiplex structure andwould not be visible otherwise by the separate analysis of the single-layer networks corresponding to each dependency measure

1 Introduction

In the last decade network theory has been extensivelyapplied to the analysis of financial markets Financial marketsand complex systems in general are comprised of many inter-acting elements and understanding their dependency struc-ture and its evolution with time is essential to capture the col-lective behaviour of these systems to identify the emergenceof critical states and tomitigate systemic risk arising from thesimultaneous movement of several factors Network filteringis a powerful instrument to associate a sparse network toa high-dimensional dependency measure and the analysisof the structure of such a network can uncover importantinsights on the collective properties of the underlying systemFollowing the line first traced by the preliminary work ofMantegna [1] a set of time series associated with financialasset values is mapped into a sparse complex network whosenodes are the assets and whose weighted links representthe dependencies between the corresponding time seriesFiltering correlation matrices has been proven to be veryuseful for the study and characterisation of the underlying

interdependency structure of complex datasets [1ndash5] Indeedsparsity allows filtering out noise and sparse networks canthen be analysed by using standard tools and indicatorsproposed in complex networks theory to investigate themultivariate properties of the dataset [6 7] Further thefiltered network can be used as a sparse inference structure toconstruct meaningful and computationally efficient predic-tive models [7 8]

Complex systems are often characterised by nonlinearforms of dependency between the variables which are hardto capture with a single measure and are hard to map into asingle filtered network Amultiplex network approach whichconsiders the multilayer structure of a system in a consis-tent way is thus a natural and powerful way to take intoaccount simultaneously several distinct kinds of dependencyDependencies among financial time series can be describedby means of different measures each one having its ownadvantages and drawbacks and this has led to the study ofdifferent types of networks namely correlation networkscausality networks and so on The most common approachuses Pearson correlation coefficient to define the weight of a

HindawiComplexityVolume 2017 Article ID 9586064 13 pageshttpsdoiorg10115520179586064

2 Complexity

link because this is a quantity that can be easily and quicklycomputed However the Pearson coefficient measures thelinear correlation between two time series [9] and thisis quite a severe limitation since nonlinearity has beenshown to be an important feature of financial markets [10]Other measures can provide equally informative pictureson assets relationships For instance the Kendall correlationcoefficient takes into accountmonotonic nonlinearity [11 12]while other measures such as the Tail dependence quantifydependence in extreme events It is therefore important todescribe quantitatively how these alternative descriptions arerelated but also differ from the Pearson correlation coefficientand also to monitor how these differences change in time ifat all

In this work we exploit the power of a multiplex approachto analyse simultaneously different kinds of dependenciesamong financial time seriesThe theory of multiplex networkis a recently introduced framework that allows describingreal-world complex systems consisting of units connected byrelationships of different kinds as networks with many layerswhere the links at each layer represent a different type ofinteraction between the same set of nodes [13 14] A mul-tiplex network approach combined with network filteringis the ideal framework to investigate the interplay betweenlinear nonlinear and Tail dependencies as it is specificallydesigned to take into account the peculiarity of the patternsof connections at each of the layers but also to describe theintricate relations between the different layers [15]

The idea of analysing multiple layers of interaction wasintroduced initially in the context of social networks withinthe theory of frame analysis [16] The importance of consid-ering multiple types of human interactions has been morerecently demonstrated in different social networks fromterrorist organizations [14] to online communities in all thesecases multilayer analyses unveil a rich topological structure[17] outperforming single-layer analyses in terms of networkmodeling and prediction as well [18 19] In particular mul-tilayer community detection in social networks has beenshown to be more effective than single-layer approaches [20]similar results have been reported for community detectionon the World Wide Web [21 22] and citation networks[23] For instance in the context of electrical power gridsmultilayer analyses have provided important insight into therole of synchronization in triggering cascading failures [2425] Similarly the analyses on transport networks have high-lighted the importance of a multilayer approach to optimizethe system against nodes failures such as flights cancellation[26] In the context of economic networks multiplex analyseshave been applied to study the World Trade Web [27]Moreover they have been extensively used in the context ofsystemic risk where graphs are used to model interbank andcredit networks [28 29]

Here we extend the multiplex approach to financialmarket time series with the purpose of analysing the roleof different measures of dependencies namely the PearsonKendall Tail and Partial correlation In particular we con-sider the so-called Planar Maximally Filtered Graph (PMFG)[2ndash4 7] as filtering procedure to each of the four layers Foreach of the four unfiltered dependence matrices the PMFG

filtering starts from the fully connected graph and uses agreedy procedure to obtain a planar graph that connects allthe nodes and has the largest sumofweights [3 4]ThePMFGis able to retain a higher number of links and therefore alarger amount of information than the Minimum SpanningTree (MST) and can be seen as a generalization of the latterwhich is always contained as a proper subgraph [2] Thetopological structures of MST and PMFG have been shownto provide meaningful economic and financial information[30ndash34] that can be exploited for risk monitoring [35ndash37]and asset allocation [38 39] The advantage of adopting afiltering procedure is not only in the reduction of noise anddimensionality but more importantly in the possibility ofgenerating sparse networks as sparsity is a requirement formost of the multiplex network measures that will be used inthis paper [14] Other kinds of filtering procedures includ-ing thresholding based methods [35 40] could have beenconsidered However PMFG has the advantage to producenetworks with fixed a priori (3119873 minus 6) number of linksthat make the comparison between layers and across timewindows easier It is worth mentioning that the filteringof the Partial correlation layer requires an adaptation ofthe PMFG algorithm to deal with asymmetric relations Wehave followed the approach suggested in [41] that rules outdouble links between nodes The obtained planar graphcorresponding to Partial correlations has been then convertedinto an undirected graph and included in the multiplex

2 Results

21 Multiplex Network of Financial Stocks We have con-structed a time-varying multiplex network with119872 = 4 layersand a varying number of nodes Nodes represent stocksselected from a dataset of 119873tot = 1004 US stocks which haveappeared at least once in SampP500 in the period between03011993 and 26022015 The period under study has beendivided into 200 rolling time windows each of 120579 = 1000trading days The network at time 119879 = 1 2 200 canbe described by the adjacency matrix 119886120572119894119895(119879) with 119894 119895 =1 119873(119879) and 120572 = 1 2 3 4 The network at time window119879 has 119873(119879) lt 119873tot nodes representing those stocks whichwere continuously traded in timewindow119879The links at eachof the four layers are constructed bymeans of the PMFG pro-cedure fromPearson Kendall Tail and Partial dependenciesThe reason for this choice is to provide a complete picture ofthe market dependency structure Pearson layer accounts forlinear dependency Kendall layer for monotonic nonlinearityand Tail dependency for correlation in the tails of returnsdistribution while Partial correlation detects direct asset-asset relationships which are not explained by themarket (seeMaterials and Methods for details)

Figure 1(a) shows how the average link weight of eachof the four dependency networks changes over time Wenotice that the average edge weight is a meaningful proxy forthe overall level of correlation in one of these dependencylayers since the distribution of edge weights within a layeris normally quite peaked around its mean The curvesshown in Figure 1(a) indicate an overall increase of thetypical weights in the examined period 1993ndash2015 and reveals

Complexity 3

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Figure 1 The multiplex nature of dependence among financial assets The plots report the network analysis of a multiplex whose four layersare PlanarMaximally FilteredGraphs (PMFGs) obtained from four classical dependencemeasures namely Pearson Kendall Tail and Partialcorrelation computed on rolling timewindows of 23 trading days between 1993 and 2015 Each of the four layers provides different informationon the dependency structure of a market Although market events and trends have a somehow similar effect on the average dependence ⟨119908119894119895⟩between nodes at the different layers (panel (a)) each layer has a distinct local structureThis is made evident by the plots of the average edgeoverlap ⟨119874⟩ (panel (b)) and of the fraction 119880[120572] of edges unique to each layer which confirm that an edge exists on average on less than twolayers and up to 70 of the edges of a layer are not present on any other layer Moreover the same node can have different degrees acrossthe four layers as indicated by the relatively low values of the pairwise interlayer degree correlation coefficient 120588[120572120573] reported in panel (d) forthree pairs of layers over the whole observation interval

a strongly correlated behaviour of the four curves (withlinear correlation coefficients between the curves range in[091 099]) In particular they all display a steep increase incorrespondence with the 2007-2008 financial crisis revealinghow the market became more synchronized regardless ofthe dependence measure used This strong correlation inthe temporal patterns of the four measures of dependencemay lead to the wrong conclusion that the four networkscarry very similar information about the structure of financialsystems Conversely we shall see that even basic multiplexmeasures suggest otherwise In Figure 1(b) we report theaverage edge overlap ⟨119874⟩ that is the average number of

layers of the financial multiplex network where a generic pairof nodes (119894 119895) is connected by an edge (see Materials andMethods for details) Since our multiplex network consistsof four layer ⟨119874⟩ takes values in [1 4] and in particularwe have ⟨119874⟩ = 1 when each edge is present only in onelayer while ⟨119874⟩ = 4 when the four networks are identicalThe relatively low values of ⟨119874⟩ observed in this case revealthe complementary role played by the different dependencyindicators It is interesting to note that the edge overlap ⟨119874⟩displays a quite dynamic pattern and its variations seem to berelated to the main financial crises highlighted by the verticallines in Figure 1(b) Overall what we observe is that periods

4 Complexity

of financial turbulence are linked to widening differencesamong the four layers Namely the effect of nonlinearity inthe cross-dependence increases as well as correlation on thetails of returns the dependence structure becomes richerand more complex during financial crisis This might berelated to the highly nonlinear interactions that characteriseinvestors activities in turbulent periods and that make fat-tailand power-law distributions distinctive features of financialreturns Indeed if returns were completely described by amultivariate normal distribution the Pearson layer would besufficient to quantify entirely the cross-dependence and itsrelation with the other layers would be trivial and would notchange with time Therefore any variation in the overlappingdegree is a signature of increasing complexity in the marketIn particular the first event that triggers a sensible decreasein the average edge overlap is the Russian crisis in 1998which corresponds to the overall global minimum of ⟨119874⟩ inthe considered interval Then ⟨119874⟩ starts increasing towardsthe end of year 2000 and reaches its global maximum atthe beginning of 2002 just before the market downturnof the same year We observe a marked decrease in 2005in correspondence with the second phase of the housingbubble which culminates in the dip associated with the creditcrunch at the end of 2007 A second even steeper dropoccurs during the Lehman Brothers default of 2008 Afterthat the signal appears more stable and weakly increasingespecially towards the end of 2014 Since each edge is presenton average in less than two layers each of the four layerseffectively provides a partial perspective on the dependencystructure of the marketThis fact is mademore evident by theresults reported in Figure 1(c) where we show for each layer120572 = 1 4 the fraction of edges119880[120572] that exist exclusively inthat layer (see Materials and Methods for details) We noticethat at any point in time from 30 to 70 of the edgesof each of the four layers are unique to that layer meaningthat a large fraction of the dependence relations capturedby a given measure are not captured by the other measuresFor instance despite the fact that Pearson and Kendall showsimilar behaviour in Figure 1(c) still between 30 and 40of the edges on each of those layers exist only on that layerThis indicates that the Pearson and Kendall layers differ forat least 60 to 80 of their edges In general each of thefour layers is contributing information that cannot be foundin the other three layers It was shown in a recent paper bysome of the authors [36] that information filtering networkscan be used to forecast volatility outburstsThe present resultssuggest that a multilayer approach could provide a furtherforecasting instrument for bearbull markets However thisrequires further explorations Interestingly we observe anincrease of 119880[120572] for all the layers since 2005 which indicatesa build-up of nonlinearity and tail correlation in the yearspreceding the financial crisis such dynamicsmight be relatedto early-risk warnings

Another remarkable finding is that also the relativeimportance of a stock in the network measured for instanceby its centrality in terms of degree [39 42] varies a lot acrosslayers This is confirmed by the degree correlation coefficient120588[120572120573] for pairs of layers 120572 and 120573 In general high values of

120588[120572120573] signal the presence of strong correlations between thedegrees of the same node in the two layers (see Materials andMethods for details) Figure 1(d) shows 120588[120572120573] as a functionof time for three pairs of dependence measures namelyPearson-Kendall Kendall-Tail and Tail-Partial Notice thatthe degrees of the layers corresponding to Pearson andKendall exhibit a relatively large correlation which remainsquite stable over the whole time interval Conversely thedegrees of nodes in the Kendall and Tail layers are on averageless correlated and the corresponding values of 120588[120572120573] exhibitlarger fluctuations For example in the tenth time windowwefind that General Electric stock (GE US) is a hub in Kendalllayer with 71 connections but it has only 16 connectionsin the Tail layer therefore the relevance of this stock inthe dependence structure depends sensitively on the layerA similar pattern in observed in the interlayer correlationbetween the degrees of nodes in Partial and Tail Thismight have important implications for portfolio allocationproblems since the asset centrality in the network is relatedto its risk in the portfolio

The presence of temporal fluctuations in ⟨119874⟩ in partic-ular the fact that ⟨119874⟩ reaches lower values during financialcrises together with the unique patterns of links at each layertestified by high values of 119880[120572] and by relatively weak inter-layer degree-degree correlations for some pairs of layers con-firms that an analysis of relations among stocks simply basedon one dependencemeasure can neglect relevant informationwhich can however be captured by othermeasures As we willshow below a multiplex network approach which takes intoaccount at the same time all the four dependence measuresbut without aggregating them into a single-layer network isable to provide a richer description of financial markets

22 Multiedges and Node Multidegrees As a first exampleof useful quantities that can be investigated in a multiplexnetwork we have computed the so-called multidegree 119896997888rarr119898119894for each node 119894 in the network corresponding to differentmultiedges (see Materials and Methods) [43] In particularwe have normalised the multidegree of node 119894 dividing it bythe corresponding node overlapping degree 119900119894 so that theresulting 119896997888rarr119898119894 119900119894 is the fraction of multiplex edges of node 119894that exist only on a given subset of layers In Figure 2 wereport the average normalised multidegree of each of the 10industry sectors of the Industry Classification Benchmark(ICB) classification We focus on the edges existing exclu-sively in one of the four layers and on the combinationof multiedges associated with edges existing in either ofthe Kendall Tail or Partial layer but not in the Pearsonlayer As shown in Figure 2 the multidegree exhibits strongvariations in time and high heterogeneity across differentindustries Industries such as Oil amp Gas Utilities and BasicMaterials show low values of normalised multidegree in allthe four panels (Figures 2(a)ndash2(d)) Conversely the edgesof nodes corresponding to Industrials Finance TechnologyTelecommunications and Consumer Services tend to con-centrate in one layer or in a small subset of layers onlyFor instance we observe a relatively high concentrationof edges at the Kendall layer for nodes corresponding to

Complexity 5

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(e) (Kendall cup Tail cup Partial) Pearson

Figure 2 Multidegrees reveal the different role of industrial sectors during crises The plots of the average multidegree of the nodes ofthe same industrial sector restricted to edges existing exclusively on the (a) Pearson (b) Kendall (c) Tail and (d) Partial layers clearlyshow that some dependence measures can reveal structures which are unnoticed by other measures In particular the plot of the averagemultidegree associated with edges existing on at least one layer among Kendall Tail and Partial but not on Pearson (panel (e)) reveals thatPearson correlation does not capture many important features such as the prominent role of Basic Materials Financial Consumer Goodsand Industrials during crises and the increasing importance of Technology and Consumer Services after the 2007-2008 crisis

6 Complexity

Finance Industrials and Consumer Goods stocks in theperiod preceding the dot-com bubble and the 2002 down-turn a feature not visible in the Pearson layer in Figure 2(a)This implies that for stocks belonging to those industriesnonlinearity was a feature of their cross-dependency moreimportant than for other stocks Analogously we notice asudden increase of edges unique to the Tail layer for nodesin Consumer Goods Consumer Services and Health Careafter the 2007-2008 crisisTherefore during this crisis periodthe synchronization in the tail region has become a morerelevant factor in their dependency structure than beforethe crisis this has important implications for portfolio riskas high Tail dependence can lead to substantial financiallosses in case of large price movementsThe presence of largeheterogeneity and temporal variations in the relative role ofdifferent industrial sectors confirms the importance of usinga multiplex network approach to analyse dependence amongassets Since industrial sectors have been often used for riskdiversification these findings point out that their use as adiversification benchmark might benefit remarkably from acontinuous monitoring based on multiplex an increase ofedges in one layer for an industry can indicate the need ofusing the corresponding dependency measure for assessingthe industryrsquos risk and diversification potential The fact thatdifferent industries display different degrees of nonlinearityand Tail dependence is not surprising after all given thateach industrial sector can be affected in a different way bynew information this industrial specific sensitivity mighttranslate in different cross-dependency properties

From this perspective it is particularly interesting todiscuss the plot of multidegree restricted to edges that arepresent on either Kendall Partial or Tail layer but arenot present in the Pearson layer as reported in Figure 2(e)Despite the fact that Pearson correlation coefficient is themost used measure to study dependencies the plot revealsthat until 2002 an analysis of the financial network basedexclusively on Pearson correlations would have missed from40 up to 60 of the edges of assets in sectors such as BasicMaterials Financial Consumer Goods and Industrials Thestudy of evolution with time in Figure 2(e) reveals that therelative role of such industrial sectors in Kendall Tail andPartial layers becomes relatively less important between thetwo crises in 2002 and in 2007 but then such sectors becomecentral again during the 2007-2008 crisis and beyond Thisprominent role is quite revealing but it would not had beenevident from the analysis of the Pearson layer alone Let usalso note that the period following the 2007-2008 crisis isalso characterised by a sensible and unprecedented increaseof the normalised multidegree on Kendall Partial and Taillayers of stocks belonging to Technology and Telecommuni-cations sectors whose importance in the market dependencestructure has been therefore somehow underestimated overthe last ten years by the studies based exclusively on Pearsoncorrelation

23 Multiplex Cartography of Financial Systems To betterquantify the relative importance of specific nodes and groupsof nodes we computed the overlapping degree and participa-tion coefficient respectively measuring the total number of

edges of a node and how such edges are distributed acrossthe layers (see Materials andMethods for details) We startedby computing the average degree 119896[120572]119868 at layer 120572 of nodesbelonging to each ICB industry sector 119868 defined as 119896[120572]119868 =(1119873119868) sum119894isin119868 119896[120572]119894 120575(119888119894 119868) where by 119888119894 we denote the industry ofnode 119894 and 119873119868 is the number of nodes belonging to industrysector 119868 Figures 3(a)ndash3(d) show 119896[120572]119868 as a function of time foreach of the four layers

Notice that nodes in the Financial sector exhibit a quitehigh average degree nomatter the dependencemeasure usedwith a noticeable peak before the dot-com bubble in 2002After that the average degree of Financials has droppedsensibly with the exception of the 2007-2008 crisis Apartfrom the existence of similarities in the overall trend ofFinancials across the four layers the analysis of the averagedegree suggests again the presence of high heterogeneityacross sectors and over time

In the Pearson layer Basic Materials is the second mostcentral industry throughout most of the observation intervalwhereas Industrials and Oil amp Gas acquired more connec-tions in the period following the 2007-2008 crisisThe degreein the Kendall layer is distributed more homogeneouslyamong the sectors than in the Pearson layer Interestinglythe plot of degree on the Tail layer looks similar to that ofthe Pearson layer Finally in the Partial layer we observe thehighest level of concentration of links in Finance (consistentlyto what was found in [41]) and after the 2007-08 crisis inBasic Materials

We have also calculated for each industry 119868 the averageoverlapping degree 119900119868 equiv ⟨119900119894⟩119894isin119868 where 119900119894 is the overlappingdegree of node 119894 which quantifies the overall importance ofeach industrial sector in the multiplex dependence networkThe average overlapping degree of each industry is shownas a function of time in Figure 3(e) As we can see 119900119868 isable to highlight the prominent role played in the multiplexnetwork by Financials BasicMaterials Oil ampGas and Indus-trials sectors revealing also the presence of four differentphases between 1997 and 2015 The first phase during whichFinancials is the only prominent industry covers the periodbetween 1997 and 2000 The second phase between 2000and the 2007-08 crisis is characterised by the emergence ofBasic Materials as the secondmost central sector In the thirdphase between 2009 and 2014 Financials loses its importancein favour of Industrials Oil amp Gas and Basic Materials(that becomes the most central one) Finally in 2014 a newequilibrium starts to emerge with Financials and Industrialsgaining again a central role in the system

The participation coefficient complements the informa-tion provided by the overlapping degree quantifying how theedges of a node are distributed over the layers of the multi-plex In particular the participation coefficient of node 119894 isequal to 0 if 119894 has edges in only one of the layers while it ismaximum and equal to 1when the edges of node 119894 are equallydistributed across the layers (see Materials and Methods fordetails) In Figure 3(f) we report as a function of time theaverage participation coefficient 119875119868 for each ICB industry 119868Interestingly the plot reveals that the increase of the over-lapping degrees of Financials Basic Materials Industrials

Complexity 7

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(f) Participation coefficient

Figure 3 Average node degree as a proxy of the importance of an industryThe plots of average degree of the nodes belonging to the differentindustrial sectors restricted to the (a) Pearson (b) Kendall (c) Tail and (d) Partial layers and of the average overlapping degree reported inpanel (e) confirm the relative importance of Financials However the average participation coefficient (panel (f)) suggests that the dependencestructure of some sectors such as BasicMaterials Industrials andOil ampGas has becomemore heterogeneous that is focusing only on a subsetof the four layers after the 2007-2008 crisis

8 Complexity

and Oil amp Gas sectors shown in Figure 3(e) is normallyaccompanied by a substantial decrease of the correspondingparticipation coefficients This indicates that those sectorsaccumulated degree on just one or two layers confirmingwhat we found in multidegree analysis A somehow moredetailed analysis of the temporal evolution of participationcoefficient for each sector is reported in Section 45

3 Discussion

By using filtered networks from different correlation mea-sures we have demonstrated that a multiplex networkapproach can reveal features that would have otherwise beeninvisible to the analysis of each dependency measure inisolation Although the layers produced respectively fromPearson Kendall Tail and Partial correlations show a certainoverall similarity they exhibit distinct features that areassociated with market changes For instance we observedthat average edge overlap between the first three layers dropssignificantly during periods of market stress revealing thatnonlinear effects are more relevant during crisis periods Theanalysis of the average multidegree associated with edges notpresent on the Pearson layer but existing on at least oneof the three remaining layers indicates that Pearson corre-lations alone can miss detecting some important featuresWe observed that the relative importance of nonlinearityand tails on market dependence structure as measuredby mean edge overlap between the last three layers hasdropped significantly in the first half of the 2000s and thenrisen steeply between 2005 and the 2007-08 crisis Overallfinancial crises trigger remarkable drops in the edge overlapwidening therefore the differences among the measures ofdependence just when evaluation of risk becomes of the high-est importance Different industry sectors exhibit differentstructural overlaps For instance Financials Industrials andConsumer Goods show an increasing number of connectionsonly on Kendall layer in the late 90searly 2000s at theedge of the dot-com bubble After the 2007-08 crisis theseindustries tend to have many edges on the Kendall Tailand Partial which are not present on the Pearson layer Thisobservation questions whether we can rely on the Pearsonestimator alone when analysing correlations between stocksA study of the overlapping degree and of the participationcoefficient shows that asset centrality an important feature forportfolio optimization [39 40] changes considerably acrosslayers with largest desynchronized changes occurring duringperiods of market distress Overall our analysis indicatesthat different dependency measures provide complementarypieces of information about the structure and evolution ofmarkets and that a multiplex network approach can be usefulin capturing systemic properties that would otherwise gounnoticed

4 Materials and Methods

41 Dataset The original dataset consists of the daily pricesof 119873tot = 1004 US stocks traded in the period between03011993 and 26022015 Each stock in the dataset has beenincluded in SampP500 at least once in the period considered

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15

Figure 4 Number of stocks in each ICB industry in time Numberof stocks that are continuously traded in each time window togetherwith their partition in terms of ICB industries

Hence the stocks considered provide a representative pictureof the US stock market over an extended time window of 22years and cover all the 10 industries listed in the IndustryClassification Benchmark (ICB) (Figure 4) It is importantto notice that most of the stocks in this set are not tradedover the entire period This is a major difference with respectto the majority of the works on dynamic correlation-basednetworks in which only stocks continuously traded over theperiod under study are considered leading to a significantldquosurvival biasrdquo For each asset 119894 we have calculated the seriesof log-returns defined as 119903119894(119905) = log(119875119894(119905)) minus log(119875119894(119905 minus 1))where 119875119894(119905) is stock price at day 119905 The construction of thetime-varying multiplex networks is based on log-returns andhas been performed in moving time windows of 120579 = 1000trading days (about 4 years) with a shift of 119889119879 = 23 tradingdays (about onemonth) adding up to 200 different multiplexnetworks one for each time window For each time window119879 four different119873(119879)times119873(119879) dependencematrices have beencomputed respectively based on the four different estimatorsillustrated in Section 42 Since the number of active stockschanges with time dependence matrices at different timescan have different number of stocks 119873(119879) as shown inFigure 4 In the figure the ICB industry composition of ourdataset in each time window is also shown confirming thatwe have a representative sample of all market throughout theperiod We have verified that the results we are discussing inthe following are robust against change of 120579 and 11988911987942 Dependence among Financial Time Series We have con-sidered four different measures of dependence between twotime series 119903119894(119906) and 119903119895(119906) 119894 119895 = 1 2 119873 119906 = 1 2 120579indicated in the following respectively as Pearson KendallTail and Partial

Complexity 9

421 Pearson Dependence It is a measure of linear depen-dence between two time series and is based on the evaluationof the Pearson correlation coefficient [44] We have usedthe exponentially smoothed version of this estimator [45] inorder to mitigate excessive sensitiveness to outliers in remoteobservations120588119908119894119895= sum120579119906=1 119908119906 (119903119894 (119906) minus 119903119894119908) (119903119895 (119906) minus 119903119895119908)

radicsum120579119906=1 119908119906 (119903119894 (119906) minus 119903119894119908)2radicsum120579119906=1 119908119906 (119903119895 (119906) minus 119903119895119908)2 (1)

with

119908119906 = 1199080 exp(119906 minus 120579119879lowast ) (2)

where 119879lowast is the weight characteristic time (119879lowast gt 0) thatcontrols the rate at which past observations lose importancein the correlation and 1199080 is a constant connected to thenormalisation constraint sum120579119906=1 119908119906 = 1 We have chosen 119879lowast =1205793 according to previously established criteria [45]

422 Kendall Dependence It is ameasure of dependence thattakes into account the nonlinearity of a time series It is basedon the evaluation of the so-called Kendallrsquos 120591 rank correlationcoefficient starting from the quantities 119889119896(119906 V) equiv sgn(119903119896(119906)minus119903119896(V)) The estimator counts the number of concordant pairsthat is pairs of observations such that 119889119894(119906 V) and 119889119895(119906 V)have equal signs minus the number of discordant pairs [11]As for the case of the Pearson dependence we have used theexponentially smoothed version of the estimator [45]

120591119908119894119895 =120579sum119906=1

120579sumV=119906+1

119908119906V119889119894 (119906 V) 119889119895 (119906 V) (3)

with

119908119906V = 1199080 exp(119906 minus 120579119879lowast ) exp(V minus 120579119879lowast ) (4)

where 119879lowast is again the weight characteristic time

423 Tail Dependence It is a nonparametric estimator of tailcopula that provides a measure of dependence focused onextreme events It is based on the evaluation of the followingestimator [46]

119862119894119895 (1199011 1199012) = sum120579119906=1 1119865119894(119903119894(119906))lt1199011and119865119895(119903119895(119906))lt1199012sum120579119906=1 1119865119894(119903119894(119906))lt1199011or119865119895(119903119895(119906))lt1199012 (5)

where 119865119894 and 119865119895 are the empirical cumulative probabilitiesof returns 119903119894(119906) and 119903119895(119906) respectively and 1199011 and 1199012 aretwo parameters representing the percentiles above which anobservation is considered (lower) tail We focus on lowertails since we are interested in risk management applicationswhere the attention is on losses It can be shown that this is aconsistent estimator of tail copula [46] In this work we havechosen 1199011 = 1199012 = 01 (ie we consider tail every observationbelow the 10th percentile) as a trade-off between the need ofstatistic and the interest in extreme events

424 Partial Dependence It is a measure of dependencethat quantifies to what extent each asset affects other assetscorrelation The Partial correlation 120588119894119896|119895 or correlation influ-ence between assets 119894 and 119896 based on 119895 is the Pearsoncorrelation between the residuals of 119903119894(119906) and 119903119896(119906) obtainedafter regression against 119903119895(119906) [47] It can be written in termsof a Pearson correlation coefficient as follows [41]

120588119894119896|119895 = 120588119894119896 minus 120588119894119895120588119896119895radic[1 minus 1205882119894119895] [1 minus 1205882

119896119895]

(6)

This measure represents the amount of correlation between119894 and 119896 that is left once the influence of 119895 is subtractedFollowing [41] we define the correlation influence of 119895 on thepair 119894 119896 as

119889 (119894 119896 | 119895) = 120588119894119896 minus 120588119894119896|119895 (7)

119889(119894 119896 | 119895) is large when a significant fraction of correlationbetween 119894 and 119896 is due to the influence of 119895 Finally in orderto translate this into a measure between 119894 and 119895 the so-calledPartial dependence we average it over the index 119896

119889 (119894 | 119895) = ⟨119889 (119894 119896 | 119895)⟩119896 =119894119895 (8)

119889(119894 | 119895) is the measure of influence of 119895 on 119894 based on Partialcorrelation It is worth noting that unlike the other measuresof dependence 119889(119894 | 119895) provides a directed relation betweenassets (as in general 119889(119894 | 119895) = 119889(119895 | 119894)) In the rest ofthe paper we refer to this indicator as ldquoPartial dependencerdquoeven though strictly speakingwe are analysing the correlationinfluence based on Partial correlation

43 Graph Filtering and the Construction of the MultiplexNetwork For each of the 200 time windows we have thenconstructed a multiplex network with119872 = 4 layers obtainedrespectively by means of the four dependence indicatorsIn order to reduce the noise and the redundance containedin each dependence matrix we have applied the PlanarMaximally Filtered Graph [2ndash4 7] It is worth mentioningthat the filtering of the correlation influence layer requires anadaptation of the PMFG algorithm to deal with asymmetricrelations We have followed the approach suggested in [41]that rules out double links between nodes The obtainedplanar graphs have been then converted into undirectedgraphs and included in the multiplex

44 Multiplex Measures Let us consider a weighted mul-tiplex network M on 119873 nodes defined by the 119872-dimensional array of weighted adjacency matrices W =119882[1]119882[2] 119882[119872] where119882[120572] = 119908[120572]119894119895 are the matricesof weights that determine the topology of the 120572th layerthough the PMFG filtering Here the weight 119908[120572]119894119895 representsthe strength of the correlation between node 119894 and node 119895 onlayer 120572 where the different layers are obtained through differ-ent correlation measures In the following we will indicate by119882[120572] the weighted adjacency matrix of the PMFG associatedwith layer 120572 and by 119860[120572] the corresponding unweighted

10 Complexity

adjacency matrix where 119886[120572]119894119895 = 1 if and only if 119908[120572]119894119895 = 0We denote by 119870[120572] = (12)sum119894119895 119886[120572]119894119895 the number of edges onlayer 120572 and by 119870 = (12)sum119894119895[1 minus prod120572(1 minus 119886[120572]119894119895 )] the numberof pairs of nodes which are connected by at least one edge onat least one of the 119872 layers Notice that since the network ateach layer is a PMFG then we have 119870[120572] = 3(119873 minus 2) forall120572 byconstruction

We consider some basic quantities commonly used tocharacterise multiplex networks [14 43] The first one is themean edge overlap defined as the average number of layerson which an edge between two randomly chosen nodes 119894 and119895 exists

⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)

Notice that ⟨119874⟩ = 1 only when all the119872 layers are identicalthat is 119860[120572] equiv 119860[120573] forall120572 120573 = 1 119872 while ⟨119874⟩ = 0if no edge is present in more than one layer so that theaverage edge overlap is in fact a measure of howmuch similarthe structures of the layers of a multiplex network are Asomehow dual quantity is the fraction of edges of layer 120572which do not exist on any other layer

119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)

which quantifies how peculiar the structure of a given layer 120572is since 119880[120572] is close to zero only when almost all the edgesof layer 120572 are also present on at least one of the other 119872 minus 1layers

More accurate information about the contribution of eachnode to a layer (or to a group of layers) can be obtained by theso-called multidegree of a node 119894 Let us consider the vector997888rarr119898 = (1198981 1198982 119898119872) with119872 equal to the number of layerswhere each 119898120572 can take only two values 1 0 We say that apair of nodes 119894 119895 has a multilink 997888rarr119898 if they are connected onlyon those layers120572 for which119898120572 = 1 in997888rarr119898 [43]The informationon the 119872 adjacency matrices 119886120572119894119895 (120572 = 1 119872) can then beaggregated in the multiadjacency matrix 119860997888rarr119898119894119895 where 119860997888rarr119898119894119895 = 1if and only if the pair 119894 119895 is connected by a multilink 997888rarr119898Formally [13 43]

119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)

From the multiadjacency matrix we can define the multide-gree 997888rarr119898 of a node 119894 as the number of multilinks 997888rarr119898 connecting119894

119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)

This measure allows us to calculate for example how manyedges node 119894 has on layer 1 only (119896997888rarr119898119894 choosing 1198981 = 1 119898120572 =0 forall120572 = 1) integrating the global information provided by119880[120572]

The most basic measure to quantify the importance ofsingle nodes on each layer is by means of the node degree119896[120572]119894 = sum119895 119886[120572]119894119895 However since the same node 119894 is normallypresent at all layers we can introduce two quantities tocharacterise the role of node 119894 in the multiplex [14] namelythe overlapping degree

119900119894 = sum120572

119896[120572]119894 (13)

and the multiplex participation coefficient

119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)

The overlapping degree is just the total number of edgesincident on node 119894 at any layer so that node are classified ashubs if they have a relatively large value of 119900119894 The multiplexparticipation coefficient quantifies the dispersion of the edgesincident on node 119894 across the layers In fact 119875119894 = 0 if theedges of 119894 are concentrated on exactly one of the119872 layers (inthis case 119894 is a focused node) while 119875119894 = 1 if the edges of 119894are uniformly distributed across the 119872 layers that is when119896[120572]119894 = 119900119894119872 forall120572 (in which case 119894 is a truly multiplex node)The scatter plot of 119900119894 and119875119894 is calledmultiplex cartography andhas been used as a synthetic graphical representation of theoverall heterogeneity of node roles observed in a multiplex

In a multiplex network it is important also to look atthe presence and sign of interlayer degree correlations Thiscan be done by computing the interlayer degree correlationcoefficient [15]

120588[120572120573] = sum119894 (119877[120572]119894 minus 119877[120572]) (119877[120573]119894 minus 119877[120573])radicsum119894 (119877[120572]119894 minus 119877[120572])2sum119895 (119877[120573]119895 minus 119877[120573])2

(15)

where 119877[120572]119894 is the rank of node 119894 according to its degree onlayer 120572 and 119877[120572] is the average rank by degree on layer 120572 Ingeneral 120588[120572120573] takes values in [minus1 1] where values close to +1and minus1 respectively indicate the strong positive and negativecorrelations while 120588[120572120573] ≃ 0 if the degrees at the two layersare uncorrelated

45 TimeEvolution of theAverage ParticipationCoefficient InFigure 5 we plot the time evolution of the average participa-tion coefficient 119875119868 (119909-axis) of stocks in the industrial sector 119868against the average overlapping degree 119900119868 (119910-axis) Each circlecorresponds to one of the 200 time windows while the sizeand colour of each circle represent different time windowsEach panel corresponds to one industrial sector 119868 Thediagrams reveal that in the last 20 years the role of differentsectors has changed radically and in different directionsFor instance stocks in the Financials sector evolved from arelatively large overlapping degree and a small participationcoefficient in the late 1990s to a smaller number of edgesdistributed more homogeneously across the layers towardsthe end of the observation period Conversely Industrials

Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

23

24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21

22 Consumer Services

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

20

21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099 1

(e)

18

20

22

24

26

28

30

32

34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity

13141516171819202122 Telecommunications

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)

Figure 5 Industries evolution in the overlapping degreeparticipation coefficient plane Fixing an industry 119868 we have plotted for each timewindow a circle whose 119910 coordinate is the average overlapping degree 119900119868 and whose 119909 coordinate is the average participation coefficient119875119868 Points at different times are characterised by different sizes (small to large) and colours (legend on the right) In (a)ndash(j) we show theresults respectively for Basic Materials Consumer Goods Consumer Services Financials Health Care Industrials Oil amp Gas TechnologyTelecommunications and Utilities

stocks have acquired degree on some of the layers resultingin a considerable decrease of participation coefficient Thisis another indication of the importance of monitoring allthe layers together as an increase in the structural roleof an industry (as measured by the overlapping degree) istypically due to only a subset of layers (as indicated by thecorresponding decrease of the participation coefficient)

Conflicts of Interest

The authors declare no competing financial interests

Authorsrsquo Contributions

Nicolo Musmeci and Vincenzo Nicosia contributed equallyto this work Nicolo Musmeci Vincenzo Nicosia TomasoAste Tiziana Di Matteo and Vito Latora devised the studyperformed the experiments and simulations analysed theresults wrote the paper and approved the final draft

Acknowledgments

The authors wish to thank Alessandro Fiasconaro foruseful discussions at the beginning of this project VitoLatora acknowledges support from the EPSRC ProjectsEPK0206331 and EPN0134921 The authors wish to thankBloomberg for providing the data Tiziana Di Matteo wishesto thank the COST Action TD1210 for partially supportingthis work

References

[1] R Mantegna ldquoHierarchical structure in financial marketsrdquoTheEuropean Physical Journal B vol 11 no 1 pp 193ndash197 1999

[2] M Tumminello T Aste T DiMatteo and RMantegna ldquoA toolfor filtering information in complex systemsrdquo Proc Natl AcadSci vol 102 pp 10421ndash10426 2005

[3] T Aste T Di Matteo and S T Hyde ldquoComplex networks onhyperbolic surfacesrdquo Physica A 346 2005

[4] T Aste and T Di Matteo ldquoDynamical networks from correla-tionsrdquo Physica A Statistical Mechanics and its Applications vol370 no 1 pp 156ndash161 2006

[5] T Aste R Grammatica and T Di Matteo ldquoExploring complexnetworks via topological embedding on surfacesrdquo Phys Rev Evol 86 Article ID 036109 2012

[6] S Boccaletti V Latora Y Moreno M Chavez and D WHwang ldquoComplex networks Structure and dynamicsrdquo PhysicsReports vol 424 no 4-5 pp 175ndash308 2006

[7] G P Massara T Di Matteo and T Aste ldquoNetwork filtering forbig data Triangulated maximally filtered graphrdquo J ComplexNetworks vol 5 no 2 2017

[8] W Barfuss G P Massara T Di Matteo and T Aste ldquoParsimo-nious modeling with information filtering networksrdquo CoveringStatistical Nonlinear Biological and SoftMatter Physics vol 94no 6 2016

[9] P Embrechts A McNeil and D Straumann ldquoCorrelation anddependency in risk management properties and pitfallsrdquo inRisk Management Value at Risk and Beyond M Dempster etal Ed Cambridge University Press Cambridge UK 2001

[10] D Sornette and J Andersen ldquoA nonlinear super-exponentialrational model of speculative financial bubblesrdquo Int J ModPhys C vol 13 pp 171ndash188 2002

[11] M G Kendall ldquoA New Measure of Rank Correlationrdquo Biomet-rika vol 30 no 12 p 81 1938

[12] G Meissner Correlation Risk Modeling and Management Har-vard University Press 2014

[13] S Boccaletti et al ldquoThe structure and dynamics of multilayernetworksrdquo Phys Rep vol 544 pp 1ndash122 2014

[14] F Battiston V Nicosia and V Latora ldquoMetrics for the analysisof multiplex networksrdquo Phys Rev E vol 89 Article ID 0328042013

[15] V Nicosia and V Latora ldquoMeasuring and modelling correla-tions in multiplex networksrdquo Phys Rev E vol 92 Article ID032805 2015

Complexity 13

[16] E Goffman Frame Analysis An Essay on the Organization ofExperience Harvard University Press 1974

[17] M Szell R Lambiotte and S Thurner ldquoMultirelational orga-nization of large-scale social networks in an online worldrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 107 no 31 pp 13636ndash13641 2010

[18] P Klimek and SThurner ldquoTriadic closure dynamics drives scal-ing laws in social multiplex networksrdquo New Journal of Physicsvol 15 Article ID 063008 2013

[19] B Corominas-Murtra B Fuchs and S Thurner ldquoDetection ofthe elite structure in a virtual multiplex social system by meansof a generalised K-corerdquo PLoS ONE vol 9 no 12 Article IDe112606 2014

[20] L Tang X Wang and H Liu ldquoCommunity detection via het-erogeneous interaction analysisrdquo Data Min Knowl Discov vol25 pp 1ndash33 2012

[21] T G Kolda B W Bader and J P Kenny ldquoHigher-order weblink analysis usingmultilinear algebrardquo in Proceedings of the 5thIEEE International Conference on DataMining ICDM 2005 pp242ndash249 November 2005

[22] G A Barnett HW Park K Jiang C Tang and I F Aguillo ldquoAmulti-level network analysis of web- citations among the worldsuniversitiesrdquo Scientometrics vol 99 p 26 2014

[23] Z Wu W Yin J Cao G Xu and A Cuzzocrea ldquoCommunitydetection in multirelational social networksrdquo in Web Informa-tion Systems Engineering WISE 2013 S B Heidelberg Ed vol8181 of Lecture Notes in Computer Science pp 43ndash56 2014

[24] S V Buldyrev R Parshani G Paul H E Stanley and S HavlinldquoCatastrophic cascade of failures in interdependent networksrdquoNature vol 464 no 7291 pp 1025ndash1028 2010

[25] C D Brummitt R M DSouza and E A Leicht ldquoSuppressingcascades of load in interdependent networksrdquo Proc Natl AcadSci USA vol 109 pp E680ndashE689 2012

[26] A Cardillo M Zanin J Gomez-Gardenes M Romance AGarcıa del Amo and S Boccaletti ldquoModeling the multi-layernature of the european air transport network Resilience andpassengers re-scheduling under random failuresrdquo Eur Phys JSpec Top vol 215 pp 23ndash33 2013

[27] M Barigozzi G Fagiolo and G Mangioni ldquoIdentifying thecommunity structure of the international-trade multi-net-workrdquo Physica A Statistical Mechanics and its Applications vol390 no 11 pp 2051ndash2066 2011

[28] M Montagna and C Kok Multi-layered interbank model forassessing systemic risk Kiel 2013

[29] R BurkholzM V Leduc A Garas and F Schweitzer ldquoSystemicrisk in multiplex networks with asymmetric coupling andthreshold feedbackrdquo Physica D Nonlinear Phenomena vol 323-324 pp 64ndash72 2016

[30] D J FennM A Porter P JMucha et al ldquoDynamical clusteringof exchange ratesrdquoQuantitative Finance vol 12 no 10 pp 1493ndash1520 2012

[31] T Di Matteo F Pozzi and T Aste ldquoThe use of dynamical net-works to detect the hierarchical organization of financialmarketsectorsrdquo Eur Phys J B vol 73 pp 3ndash11 2010

[32] T Aste W Shaw and T Di Matteo ldquoCorrelation structure anddynamics in volatile marketsrdquo New J Phys vol 12 Article ID085009 2010

[33] C Borghesi M Marsili S MiccichΦ and S Micciche ldquoEmer-gence of time-horizon invariant correlation structure in finan-cial returns by subtraction of the market moderdquo Phys Rev Evol 76 Article ID 026104 2007

[34] N Musmeci T Aste and T Di Matteo ldquoRelation betweenfinancial market structure and the real economyrdquo The PLOSONE 2015

[35] J-P Onnela A Chakraborti K Kaski and J Kertesz ldquoDynamicasset trees and Black Mondayrdquo Physica A Statistical Mechanicsand its Applications vol 324 no 1-2 pp 247ndash252 2003

[36] N Musmeci T Aste and T Di Matteo ldquoRisk diversificationa study of persistence with a filtered correlation-networkapproachrdquo Journal of Network Theory in Finance vol 1 pp 1ndash22 2015

[37] NMusmeci T Aste and T DiMatteo ldquoWhat does past correla-tion structure tell us about the future an answer from networktheoryrdquo Portfolio Management 2016

[38] V Tola F LilloMGallegati and RNMantegna ldquoCluster anal-ysis for portfolio optimizationrdquo Journal of Economic Dynamicsand Control vol 32 no 1 pp 235ndash258 2008

[39] F Pozzi T Di Matteo and T Aste ldquoSpread of risk acrossfinancial markets better to invest in the peripheriesrdquo ScientificReports vol 3 2013

[40] H Kaya ldquoEccentricity in asset managementrdquo The Journal ofNetwork Theory in Finance vol 1 no 1 pp 45ndash76 2015

[41] D Y Kenett M Tumminello A Madi G Gur-GershgorenR N Mantegna and E Ben-Jacob ldquoDominating clasp of thefinancial sector revealed by partial correlation analysis of thestock marketrdquo PLoS ONE vol 5 no 12 Article ID e15032 2010

[42] D B West Introduction to graph theory ( Prentice-Hall Pren-tice-Hall Englewood Cliffs NJ USA 1996

[43] G Bianconi ldquoStatistical mechanics of multiplex networksentropy and overlaprdquo Physical Review EmdashStatistical Nonlinearand Soft Matter Physics vol 87 no 6 Article ID 062806 2013

[44] K Pearson ldquoNote on regression and inheritance in the case oftwo parentsrdquo Proceedings of the Royal Society of London vol 58pp 240ndash242 1895

[45] F Pozzi T Di Matteo and T Aste ldquoExponential smoothingweighted correlationsrdquo Eur Phys J B vol 85 2012

[46] R Schmidt and U Stadmuller ldquoNonparametric estimation oftail dependencerdquo Scandinavian Journal of Statistics vol 33 pp307ndash335 2006

[47] K Baba R Shibata andM Sibuya ldquoPartial correlation and con-ditional correlation as measures of conditional independencerdquoAustralian and New Zealand Journal of Statistics vol 46 no 4pp 657ndash664 2004

Submit your manuscripts athttpswwwhindawicom

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MathematicsJournal of


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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Complexity

link because this is a quantity that can be easily and quicklycomputed However the Pearson coefficient measures thelinear correlation between two time series [9] and thisis quite a severe limitation since nonlinearity has beenshown to be an important feature of financial markets [10]Other measures can provide equally informative pictureson assets relationships For instance the Kendall correlationcoefficient takes into accountmonotonic nonlinearity [11 12]while other measures such as the Tail dependence quantifydependence in extreme events It is therefore important todescribe quantitatively how these alternative descriptions arerelated but also differ from the Pearson correlation coefficientand also to monitor how these differences change in time ifat all

In this work we exploit the power of a multiplex approachto analyse simultaneously different kinds of dependenciesamong financial time seriesThe theory of multiplex networkis a recently introduced framework that allows describingreal-world complex systems consisting of units connected byrelationships of different kinds as networks with many layerswhere the links at each layer represent a different type ofinteraction between the same set of nodes [13 14] A mul-tiplex network approach combined with network filteringis the ideal framework to investigate the interplay betweenlinear nonlinear and Tail dependencies as it is specificallydesigned to take into account the peculiarity of the patternsof connections at each of the layers but also to describe theintricate relations between the different layers [15]

The idea of analysing multiple layers of interaction wasintroduced initially in the context of social networks withinthe theory of frame analysis [16] The importance of consid-ering multiple types of human interactions has been morerecently demonstrated in different social networks fromterrorist organizations [14] to online communities in all thesecases multilayer analyses unveil a rich topological structure[17] outperforming single-layer analyses in terms of networkmodeling and prediction as well [18 19] In particular mul-tilayer community detection in social networks has beenshown to be more effective than single-layer approaches [20]similar results have been reported for community detectionon the World Wide Web [21 22] and citation networks[23] For instance in the context of electrical power gridsmultilayer analyses have provided important insight into therole of synchronization in triggering cascading failures [2425] Similarly the analyses on transport networks have high-lighted the importance of a multilayer approach to optimizethe system against nodes failures such as flights cancellation[26] In the context of economic networks multiplex analyseshave been applied to study the World Trade Web [27]Moreover they have been extensively used in the context ofsystemic risk where graphs are used to model interbank andcredit networks [28 29]

Here we extend the multiplex approach to financialmarket time series with the purpose of analysing the roleof different measures of dependencies namely the PearsonKendall Tail and Partial correlation In particular we con-sider the so-called Planar Maximally Filtered Graph (PMFG)[2ndash4 7] as filtering procedure to each of the four layers Foreach of the four unfiltered dependence matrices the PMFG

filtering starts from the fully connected graph and uses agreedy procedure to obtain a planar graph that connects allthe nodes and has the largest sumofweights [3 4]ThePMFGis able to retain a higher number of links and therefore alarger amount of information than the Minimum SpanningTree (MST) and can be seen as a generalization of the latterwhich is always contained as a proper subgraph [2] Thetopological structures of MST and PMFG have been shownto provide meaningful economic and financial information[30ndash34] that can be exploited for risk monitoring [35ndash37]and asset allocation [38 39] The advantage of adopting afiltering procedure is not only in the reduction of noise anddimensionality but more importantly in the possibility ofgenerating sparse networks as sparsity is a requirement formost of the multiplex network measures that will be used inthis paper [14] Other kinds of filtering procedures includ-ing thresholding based methods [35 40] could have beenconsidered However PMFG has the advantage to producenetworks with fixed a priori (3119873 minus 6) number of linksthat make the comparison between layers and across timewindows easier It is worth mentioning that the filteringof the Partial correlation layer requires an adaptation ofthe PMFG algorithm to deal with asymmetric relations Wehave followed the approach suggested in [41] that rules outdouble links between nodes The obtained planar graphcorresponding to Partial correlations has been then convertedinto an undirected graph and included in the multiplex

2 Results

21 Multiplex Network of Financial Stocks We have con-structed a time-varying multiplex network with119872 = 4 layersand a varying number of nodes Nodes represent stocksselected from a dataset of 119873tot = 1004 US stocks which haveappeared at least once in SampP500 in the period between03011993 and 26022015 The period under study has beendivided into 200 rolling time windows each of 120579 = 1000trading days The network at time 119879 = 1 2 200 canbe described by the adjacency matrix 119886120572119894119895(119879) with 119894 119895 =1 119873(119879) and 120572 = 1 2 3 4 The network at time window119879 has 119873(119879) lt 119873tot nodes representing those stocks whichwere continuously traded in timewindow119879The links at eachof the four layers are constructed bymeans of the PMFG pro-cedure fromPearson Kendall Tail and Partial dependenciesThe reason for this choice is to provide a complete picture ofthe market dependency structure Pearson layer accounts forlinear dependency Kendall layer for monotonic nonlinearityand Tail dependency for correlation in the tails of returnsdistribution while Partial correlation detects direct asset-asset relationships which are not explained by themarket (seeMaterials and Methods for details)

Figure 1(a) shows how the average link weight of eachof the four dependency networks changes over time Wenotice that the average edge weight is a meaningful proxy forthe overall level of correlation in one of these dependencylayers since the distribution of edge weights within a layeris normally quite peaked around its mean The curvesshown in Figure 1(a) indicate an overall increase of thetypical weights in the examined period 1993ndash2015 and reveals

Complexity 3

0

02

04

06

08

PearsonKendall

PartialTail

1998

2000

2002

2004

2006

2008

2010

2012

2014

⟨wij⟩

(a)

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150

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Creditcrunch

Euro debtcrisis

Russiancrisis

2002market

downturn

LehmanBr defaults

1998

2000

2002

2004

2006

2008

2010

2012

2014

⟨O⟩

(b)

02

03

04

05

06

07

08

PearsonKendall

PartialTail

1998

2000

2002

2004

2006

2008

2010

2012

2014

U[

]

(c)

03

04

05

06

07

08

09

1


1998

2000

2002

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2006

2008

2010

2012

2014

[

]

(d)




4 Complexity






Complexity 5



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Complexity 7

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8 Complexity


3 Discussion




050

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T

N(T

)

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Complexity 9



with

119908119906 = 1199080 exp(119906 minus 120579119879lowast ) (2)



120591119908119894119895 =120579sum119906=1

120579sumV=119906+1

119908119906V119889119894 (119906 V) 119889119895 (119906 V) (3)

with








119896119895]

(6)


119889 (119894 119896 | 119895) = 120588119894119896 minus 120588119894119896|119895 (7)


119889 (119894 | 119895) = ⟨119889 (119894 119896 | 119895)⟩119896 =119894119895 (8)




10 Complexity



⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)


119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)



119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)


119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)



119900119894 = sum120572

119896[120572]119894 (13)


119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)




(15)



Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

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(b)

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21


1996199719981999200020012002200320042005200620072008200920102011201220132014

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(c)

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1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

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(d)

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21 Health Care

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(e)

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34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

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(f)

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30

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34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

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092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

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24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

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27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 3

0

02

04

06

08

PearsonKendall

PartialTail

1998

2000

2002

2004

2006

2008

2010

2012

2014

⟨wij⟩

(a)

140

150

160

Asiancrisis

Creditcrunch

Euro debtcrisis

Russiancrisis

2002market

downturn

LehmanBr defaults

1998

2000

2002

2004

2006

2008

2010

2012

2014

⟨O⟩

(b)

02

03

04

05

06

07

08

PearsonKendall

PartialTail

1998

2000

2002

2004

2006

2008

2010

2012

2014

U[

]

(c)

03

04

05

06

07

08

09

1


1998

2000

2002

2004

2006

2008

2010

2012

2014

[

]

(d)




4 Complexity






Complexity 5



TechnologyTelecomm

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6 Complexity









Complexity 7

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8 Complexity


3 Discussion




050

100150200250300350400450500



T

N(T

)

171

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Complexity 9



with

119908119906 = 1199080 exp(119906 minus 120579119879lowast ) (2)



120591119908119894119895 =120579sum119906=1

120579sumV=119906+1

119908119906V119889119894 (119906 V) 119889119895 (119906 V) (3)

with








119896119895]

(6)


119889 (119894 119896 | 119895) = 120588119894119896 minus 120588119894119896|119895 (7)


119889 (119894 | 119895) = ⟨119889 (119894 119896 | 119895)⟩119896 =119894119895 (8)




10 Complexity



⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)


119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)



119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)


119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)



119900119894 = sum120572

119896[120572]119894 (13)


119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)




(15)



Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

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24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

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21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

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094 095 096 097 098 099 1

(e)

18

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34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

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094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




4 Complexity






Complexity 5



TechnologyTelecomm

Utilities

171

219

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050

120

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0240220201801601401201008006

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024026

0220201801601401201008006



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0220201801601401201008006004002



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01801601401201008006004002



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6 Complexity









Complexity 7

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TechnologyTelecomm

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8 Complexity


3 Discussion




050

100150200250300350400450500



T

N(T

)

171

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050

120

00

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Complexity 9



with

119908119906 = 1199080 exp(119906 minus 120579119879lowast ) (2)



120591119908119894119895 =120579sum119906=1

120579sumV=119906+1

119908119906V119889119894 (119906 V) 119889119895 (119906 V) (3)

with








119896119895]

(6)


119889 (119894 119896 | 119895) = 120588119894119896 minus 120588119894119896|119895 (7)


119889 (119894 | 119895) = ⟨119889 (119894 119896 | 119895)⟩119896 =119894119895 (8)




10 Complexity



⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)


119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)



119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)


119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)



119900119894 = sum120572

119896[120572]119894 (13)


119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)




(15)



Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

23

24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

20

21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099 1

(e)

18

20

22

24

26

28

30

32

34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 5



TechnologyTelecomm

Utilities

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

0240220201801601401201008006

BasicMaterials

ConsumerGoods

ConsumerServices


171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

024026

0220201801601401201008006



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices


171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

0220201801601401201008006004002



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices


171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

01801601401201008006004002



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices


171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

06

055

05

045

04

035

03

025



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices



6 Complexity









Complexity 7

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

12

11

10

9

8

7

6

5

4



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(a) Degree Pearson

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

1213

1110987654



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(b) Degree Kendall

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

1213

1110987654



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(c) Degree Tail

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

16

14

12

10

8

6

4



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(d) Degree Partial

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

50

45

40

35

30

25

20

15



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices


171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

099

098

097

096

095

094

093



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices



8 Complexity


3 Discussion




050

100150200250300350400450500



T

N(T

)

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15



Complexity 9



with

119908119906 = 1199080 exp(119906 minus 120579119879lowast ) (2)



120591119908119894119895 =120579sum119906=1

120579sumV=119906+1

119908119906V119889119894 (119906 V) 119889119895 (119906 V) (3)

with








119896119895]

(6)


119889 (119894 119896 | 119895) = 120588119894119896 minus 120588119894119896|119895 (7)


119889 (119894 | 119895) = ⟨119889 (119894 119896 | 119895)⟩119896 =119894119895 (8)




10 Complexity



⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)


119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)



119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)


119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)



119900119894 = sum120572

119896[120572]119894 (13)


119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)




(15)



Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

23

24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

20

21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099 1

(e)

18

20

22

24

26

28

30

32

34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




6 Complexity









Complexity 7

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

12

11

10

9

8

7

6

5

4



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(a) Degree Pearson

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

1213

1110987654



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(b) Degree Kendall

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

1213

1110987654



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(c) Degree Tail

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

16

14

12

10

8

6

4



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(d) Degree Partial

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

50

45

40

35

30

25

20

15



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices


171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

099

098

097

096

095

094

093



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices



8 Complexity


3 Discussion




050

100150200250300350400450500



T

N(T

)

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15



Complexity 9



with

119908119906 = 1199080 exp(119906 minus 120579119879lowast ) (2)



120591119908119894119895 =120579sum119906=1

120579sumV=119906+1

119908119906V119889119894 (119906 V) 119889119895 (119906 V) (3)

with








119896119895]

(6)


119889 (119894 119896 | 119895) = 120588119894119896 minus 120588119894119896|119895 (7)


119889 (119894 | 119895) = ⟨119889 (119894 119896 | 119895)⟩119896 =119894119895 (8)




10 Complexity



⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)


119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)



119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)


119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)



119900119894 = sum120572

119896[120572]119894 (13)


119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)




(15)



Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

23

24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

20

21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099 1

(e)

18

20

22

24

26

28

30

32

34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 7

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

12

11

10

9

8

7

6

5

4



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(a) Degree Pearson

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

1213

1110987654



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(b) Degree Kendall

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

1213

1110987654



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(c) Degree Tail

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

16

14

12

10

8

6

4



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices

(d) Degree Partial

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

50

45

40

35

30

25

20

15



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices


171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15

099

098

097

096

095

094

093



TechnologyTelecomm

Utilities

BasicMaterials

ConsumerGoods

ConsumerServices



8 Complexity


3 Discussion




050

100150200250300350400450500



T

N(T

)

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15



Complexity 9



with

119908119906 = 1199080 exp(119906 minus 120579119879lowast ) (2)



120591119908119894119895 =120579sum119906=1

120579sumV=119906+1

119908119906V119889119894 (119906 V) 119889119895 (119906 V) (3)

with








119896119895]

(6)


119889 (119894 119896 | 119895) = 120588119894119896 minus 120588119894119896|119895 (7)


119889 (119894 | 119895) = ⟨119889 (119894 119896 | 119895)⟩119896 =119894119895 (8)




10 Complexity



⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)


119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)



119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)


119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)



119900119894 = sum120572

119896[120572]119894 (13)


119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)




(15)



Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

23

24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

20

21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099 1

(e)

18

20

22

24

26

28

30

32

34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




8 Complexity


3 Discussion




050

100150200250300350400450500



T

N(T

)

171

219

96

050

120

00

300

120

03

170

220

06

110

320

09

280

320

12

210

420

15



Complexity 9



with

119908119906 = 1199080 exp(119906 minus 120579119879lowast ) (2)



120591119908119894119895 =120579sum119906=1

120579sumV=119906+1

119908119906V119889119894 (119906 V) 119889119895 (119906 V) (3)

with








119896119895]

(6)


119889 (119894 119896 | 119895) = 120588119894119896 minus 120588119894119896|119895 (7)


119889 (119894 | 119895) = ⟨119889 (119894 119896 | 119895)⟩119896 =119894119895 (8)




10 Complexity



⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)


119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)



119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)


119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)



119900119894 = sum120572

119896[120572]119894 (13)


119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)




(15)



Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

23

24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

20

21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099 1

(e)

18

20

22

24

26

28

30

32

34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 9



with

119908119906 = 1199080 exp(119906 minus 120579119879lowast ) (2)



120591119908119894119895 =120579sum119906=1

120579sumV=119906+1

119908119906V119889119894 (119906 V) 119889119895 (119906 V) (3)

with








119896119895]

(6)


119889 (119894 119896 | 119895) = 120588119894119896 minus 120588119894119896|119895 (7)


119889 (119894 | 119895) = ⟨119889 (119894 119896 | 119895)⟩119896 =119894119895 (8)




10 Complexity



⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)


119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)



119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)


119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)



119900119894 = sum120572

119896[120572]119894 (13)


119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)




(15)



Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

23

24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

20

21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099 1

(e)

18

20

22

24

26

28

30

32

34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




10 Complexity



⟨119874⟩ = 12119870sum119894119895

sum120572

119886[120572]119894119895 (9)


119880[120572] = 12119870[120572]sum119894119895 119886[120572]119894119895 prod120573 =120572

(1 minus 119886[120573]119894119895 ) (10)



119860997888rarr119898119894119895 equiv119872prod120572=1

[119886120572119894119895119898120572 + (1 minus 119886120572119894119895) (1 minus 119898120572)] (11)


119896997888rarr119898119894 = sum119895

119860997888rarr119898119894119895 (12)



119900119894 = sum120572

119896[120572]119894 (13)


119875119894 = 119872119872 minus 1 [1 minus sum120572

(119896[120572]119894119900119894 )] (14)




(15)



Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

23

24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

20

21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099 1

(e)

18

20

22

24

26

28

30

32

34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 11

15

20

25

30

35

40 Basic Materials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(a)

16

17

18

19

20

21

22

23

24 Consumer Goods

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(b)

16

17

18

19

20

21


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

095 096 097 098 099 1

(c)

2628303234363840424446 Financials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099

(d)

15

16

17

18

19

20

21 Health Care

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099 1

(e)

18

20

22

24

26

28

30

32

34 Industrials

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

094 095 096 097 098 099

(f)

22

24

26

28

30

32

34 Oil amp Gas

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(g)

Technology

1996199719981999200020012002200320042005200620072008200920102011201220132014

16

18

20

22

24

26

28

oI

PI

0955 096 0965 097 0975 098 0985 099

(h)

Figure 5 Continued

12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




12 Complexity


1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

092 094 096 098 1

(i)

21

22

23

24

25

26

27

28 Utilities

1996199719981999200020012002200320042005200620072008200920102011201220132014

oI

PI

093 094 095 096 097 098 099 1

(j)







Acknowledgments


References
















Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 13








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




the multiplex dependency structure of financial markets

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