traders’ networks of interactions and structural...

Research ArticleTradersrsquo Networks of Interactions and Structural Properties ofFinancial Markets An Agent-Based Approach

Linda Ponta 12 and Silvano Cincotti 1

1Department of Mechanics Energetics Management and Transportation University of Genova Genova Italy2Department of Management Engineering LIUC Cattaneo University Castellanza Italy

Correspondence should be addressed to Linda Ponta lindapontaunigeit

Received 27 October 2017 Accepted 21 December 2017 Published 29 January 2018

Academic Editor Ilaria Giannoccaro

Copyright copy 2018 Linda Ponta and Silvano Cincotti This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

An information-based multiasset artificial stock market characterized by different types of stocks and populated by heterogeneousagents is presented and studied so as to determine the influences of agentsrsquo networks on themarketrsquos structure Agents are organizedin networks that are responsible for the formation of the sentiments of the agents In themarket agents trade risky assets in exchangefor cash and share their sentiments by means of interactions that are determined by sparsely connected graphs A central marketmaker (clearing house mechanism) determines the price process for each stock at the intersection of the demand and the supplycurves A set of marketrsquos structure indicators based on the main single-assets and multiassets stylized facts have been defined inorder to study the effects of the agentsrsquo networks Results point out an intrinsic structural resilience of the stock market In fact thenetwork is necessary in order to archive the ability to reproduce themain stylized facts but also themarket has some characteristicsthat are independent from the network and depend on the finiteness of tradersrsquo wealth

1 Introduction

The large availability of financial data has allowed thestudy of financial markets by means of the cooperation ofdifferent fields such as engineering physics mathematicsand economics [1ndash5] This new multidisciplinary approachovercomes the limits of the classical approach and improvesthe knowledge about the price processes discovering the socalled stylized facts that is the main statistical properties offinancial markets In particular focusing on the distributionof intertrade time between different financial transactionsprevious works have demonstrated the presence of Weibulldistribution [6]Moreover empirical study has demonstratedthat the dynamics of price and volume of transactionsincluding the volatility over different time horizons areinfluenced by the correlations and temporal patterns of theintertrade times Furthermore the rules that regulate theinteractions among agents strongly depend on the regula-tory mechanisms of each individual market [7] In orderto evaluate the correlations and to identify and quantifyintegrations among dynamical entities such as agents on

the stock market special methods have been developed [89] Generally speaking according to the classical approachsimple analytically tractable models with a representativeperfectly rational agent have been themain corner stones andmathematics has been the main tool of analysis Converselythe complexity science approach considers financial marketsas complex systems where a large number of heterogeneousagents interact In particular the markets are populatedby boundedly rational heterogeneous agents using rule ofthumb strategies This approach fits much better with agent-based simulation models and computational and numericalmethods have become an important tool of analysis [10]Thus a number of computer-simulated artificial financialmarkets have been born with the aim of becoming a frame-work to perform computational experiments Following thepioneering work done at the Santa Fe Institute [11ndash13] a largenumber of researchers have proposed model for artificialmarkets populated by heterogeneous agents endowed withlearning and optimization capabilities [14 15] Moreoverthe artificial financial markets are a useful framework tostudy the role of fraudulent agents and the corruption in

HindawiComplexityVolume 2018 Article ID 9072948 9 pageshttpsdoiorg10115520189072948

2 Complexity

financial markets that is how the fraudulent agents impacton the markets [16] In particular empirical analysis showsthat corruption influences the economic growth rate andforeign investment [17] For a detailed review onmicroscopic(ldquoagent-basedrdquo) models of financial markets see [18 19]

In this paper using the Genoa Artificial Stock Market(GASM) developed in Genoa the impact of the structuralproperties of tradersrsquo networks of interaction on the emergentoutcome in financial markets has been studied In particularstarting from the information-based single-assets artificialmarket a multiassets artificial stock market version of theGASM has been used [20ndash23] In order to investigate thisrelationship themarket is populated by heterogeneous agentsthat are seen as nodes of sparsely connected graphs Themarket is characterized by different types of stocks and agentstrade risky assets in exchange for cash Agents share theirinformation by means of interactions that are determined bythe graphs Besides the amount of cash and assets ownedeach agent is characterized by sentiments that summarizethe agentrsquos information about the market and agent worldThe sentiments include in one element the influence of themarket trend the influence of the neighbours agents and thepropensity for the market Agents are subject to a portfoliochoice on number and type of risky securities The allocationstrategy is based on sentiments and wealth A central marketmaker (clearing house mechanism) determines the priceprocess for each stock at the intersection of the demand andthe supply curves

The paper presents a study on how the tradersrsquo networksand thus the sentimentsrsquo components influence the marketstructure In particular this paper investigates the effects ofchanges in tradersrsquo networks of interaction in the financialmarket In order to perform this investigation five differentldquomarketrsquos structure indicatorsrdquo have been defined The indi-cators have been defined considering themain univariate andmultivariate stylized facts Concerning univariate processesthe threemain stylized facts taken as reference are the unitaryroot of price processes the fat tails distribution of returnsand the volatility clustering Concerning the multiassetsenvironment the set of stylized facts consists in the statisticalproperties of the cross-correlation matrices of returns [24ndash26] and of the variance-covariancematrices of prices [27] thatmake reference to static and dynamic factors respectively

Thus the indicators defined are the number of I(1)processes the number of heteroscedastic processes thenumber of processes with fat tails the number of sectorpresented in the market and the number of common trendsThe computational experiments show an intrinsic structuralresilience of the stock market In fact some characteristics ofthe market are ldquostructuralrdquo and depend on the agentsrsquo budgetconstraint whereas others are an emerging properties of thetradersrsquo network of interactions

The paper is organized as follows Section 2 presentsthe model and Section 3 the ldquomarketrsquos structure indicatorsrdquoand Section 4 shows the computational experiments andSection 5 the discussion of results Finally Section 6 providesthe conclusion of the study

2 The GASM Model

21 Overview of the GASM Model The model presented inthis paper is an enrichment of the Genoa Artificial StockMarket (GASM) developed at the University of Genoa [2028] The GASM is an agent-based artificial stock marketwhose baseline originally includes heterogeneous agents thattrade risky assets in exchange for cash [29]They aremodeledas liquidity traders that is decision making process isconstrained by the finite amount of financial resources (cash+ stocks) they own At the beginning of the simulation cashand stocks are distributed randomly among agents

22 Agentsrsquo Networks In order to investigate the effectsof agentsrsquo networks in financial markets for each stockpresented in the market the heterogeneous agents have beenorganized in graphs and in particular according to a directedrandom graph where the agents are the nodes and thebranches represent the interactions among agentsThe graphsare responsible for the changes in agentrsquos sentiments Thegraphs are directed that is the interactions are assumedunidirectional (ie if agent 119895-th influences agent 119894-th notnecessarily agent 119894-th influences agent 119895-th) and characterizedby a strength 119892119896119895119894 assuming a positive real number Generallyspeaking due to the presence of a directed graph both anoutput node degree related to the output branches of a givennode and an input node degree related to the input branchesshould be defined

The agents are organized according to a Zipf law Foreach stock an agent is randomly connected to a set of otheragents whose number and strength (of the connection)119892119896119894119895 areinversely proportional to hisher rank that is richer agentsinfluence a larger number of agents with a higher strengthConsequently the output degree distributions over the nodesare set to power laws and the input degree distributions resultin power laws too Each agent has a different belief about the119870 assets depending on hisher rank Agent 119894 is characterizedby a sentiment 119878119896119894 (ie real number in the interval [minus1 1]) thatrepresents a propensity to invest in asset 119896 A positive averagesentiment denotes a propensity to buy whereas a negativeaverage sentiment corresponds to a propensity to sell Thegraphs are responsible for the changes in agentrsquos sentimentsAt each time step ℎ information is propagated through themarket and sentiments 119878119896119894 of agent 119894 are updated

Let I119896119894 be the set of agents that influences the behaviorof trader i-th for the asset 119896 and 119901119896 the market price of therisky asset 119896Thenew sentiments 119878119896119894 of agent i-th for each asset119896 are functions of the previous sentiments of the log return(market feedback) of the influence of interacting agents andof average sentiment of the agent about the market behaviorThe expression is

119878119896119894 (ℎ) = 119865 (120572119878119894119878119896119894 (ℎ minus 1) + 120572119872119894119903119896 (ℎ minus 1)+ 120572119873119894119878119896119894 (ℎ minus 1) + 120572119875119894119878119894 (ℎ minus 1)) (1)

where

119865 (119909) = tanh (119909) (2)

Complexity 3

is a smooth function that constrains agent sentiments in therange [minus1 1]

Furthermore

119903119896 (ℎ minus 1) = log [119901119896 (ℎ minus 1)] minus log [119901119896 (ℎ minus 2)] (3)

represents the market feedback

119878119896119894 (ℎ minus 1) = sum119895isinI119896119894

119892119896119895119894119878119896119895 (ℎ minus 1)sum119895isinI119896

119894

119892119896119895119894 (4)

represents the influence of interacting agents and

119878119894 (ℎ minus 1) = sum119896 119878119896119894119870 (5)

models the global vision of agent 119894-th for the market trendThe 120572119878119894 coefficients in (1) are inversely proportional to agentrsquosrank that is richer agents have stronger beliefs Moreover aconstraint on graph intersection is considered

10038161003816100381610038161205721198731198941003816100381610038161003816 = (120578 minus 10038161003816100381610038161205721198781198941003816100381610038161003816) (6)

that is self-interaction is a counterpart of graph interactionswith random (ie uniform distribution) changes in sign ateach time step Eq (6) models a specific behavior of agentsthat is the fact that sometimes an agent changes idea aboutthe sentiments of neighbour and so he changes his reactionIn fact (6) points out that agent that are strongly influencedby their previous sentiment (big traders bank mutual fundsetc) and are poorly influenced by the neighbouring agentsrsquosentiment (eg small single investors) and 120578 represents theself-neighbouring sentiment balance coefficient [20]

Moreover the amplitude of market feedback depends onrank so that the coefficients120572119872119894 are inversely proportional toagent ranks that is agents with higher ranks are less sensitiveto the single asset trends Finally the 119878119894(ℎ minus 1) term is astabilizing element for the sentiment so that the coefficient120572119875119894 in (1) is always negative

Agentrsquos trading decision is based on cash and stocksowned and on sentiment In particular the stock priceprocess depends on the propagation of information amongthe interacting agents on budget constraints and on marketfeedback In this respect also the 120572119878119894 coefficients in (1)are proportional to agentrsquos rank that is richer agents havestronger beliefs

23 Allocation Strategy At each time step ℎ a subset of agentsis randomly chosen from a uniform distribution to operate astraders on the market Let 119878119896119894 (ℎ) be the sentiment 119862119894(ℎ) theamount of cash and 119902119896119894 (ℎ) the amount of asset 119896 owned by the119894-th trader at time ℎ

If 119901119896(ℎ minus 1) is the market price of the risky asset 119896 at timestep ℎminus1 the risky wealth119882119903119894 (ℎminus1) owned by trader 119894 at timestep ℎ is

119882119903119894 (ℎ minus 1) = sum119896

119902119896119894 (ℎ minus 1) 119901119896 (ℎ minus 1) (7)

whereas 119882119894(ℎ minus 1) = 119888119894(ℎ minus 1) + 119882119903119894 (ℎ minus 1) represents the totalwealth of agent 119894-th

At each simulation step trader 119894-th tries to allocate inrisky assets a fraction 120574119903 of his total wealth related to his visionof the market trend that is

119903119894 (ℎ) = 120574119903 (ℎ minus 1) 119882119894 (ℎ minus 1) (8)

where 120574119903 = (1 + 119878119894(ℎ minus 1))2119878119894 is the average sentiments of all assets described by (5)The symbol sdot denotes that 119903119894 (ℎ) is the amount that agent i-th is willing to allocate in the risky investment whereas thereal amount 119882119894(ℎ) effectively allocated in stocks will dependon the trading process with the other agents For each asset 119896a positive sentiment denotes a propensity to allocate while anegative sentiment denotes a propensity to sell all the assets 119896in the portfolio In thismodel only long positions are allowedThus if 119878119896119894 (ℎ) gt 0 the quantity desired by agent 119894 of risky asset119896 is given by

119902119896119894 (ℎ) = lfloor 120574119896119886119903119894 (ℎ)119901119896 (ℎ minus 1) rfloor (9)

where 120574119886 is given by

120574119896119886 = 119878119896119894sum119896isin119860119894 119878119896119894 (10)

119860 119894 is the set of assets with positive sentiment The symbol lfloorsdotrfloorin (9) denotes the integer part Conversely if 119878119896119894 (ℎ) lt 0 asset119896 is characterized by a desired quantity 119902119896119894 (ℎ) = 0

The amount Δ119896119894(ℎ) of the order issued by trader 119894-th attime step ℎ relative to stock 119896 is

Δ119896119894 (ℎ) = 119902119896119894 (ℎ) minus 119902119896119894 (ℎ minus 1) (11)

Δ119896119894 is the difference between the desired amount of stock 119896 attime step ℎ and the real amount held in the portfolio by agent119894-th If Δ119896119894 gt 0 the order is a buy order Conversely if Δ119896119894 lt 0the agent issues a sell order Every order is associated with alimit price Each limit price119889119896119894 is determined according to (12)

119889119896119894 (ℎ) = 119901119896 (ℎ minus 1) sdot 119873119894 (120583119896119894 120590119896119894 ) (12)

where 119873119894(120583119896119894 120590119896119894 ) is a random draw from a Gaussian distribu-tion with average

120583119896119894 = (1 + sgn (Δ119896119894) 10038161003816100381610038161003816119878119896119894 10038161003816100381610038161003816) (13)

According to previous models [20 30] we assume that buy(sell) orders cannot be executed at prices above (below) theirlimit price 119889119896119894 It is worth noting that for a buy order (ieΔ119896119894 gt 0) in average 119889119896119894 (ℎ) gt 119901119896(ℎ minus 1) Conversely for a sellorder (ieΔ119896119894 lt 0) in average 119889119896119894 (ℎ) lt 119901119896(ℎminus1) Furthermorethe standard deviation 120590119896119894 is proportional to the historicalvolatility 120590119896(119879119894) of the price 119901119896(ℎ minus 1) of stock 119896 through the

4 Complexity

20

21

22

23

24Pr

ices

(eur

o)

0 100 200 300 400 500 600 700(Days)

(a)

0 100 200 300 400 500 600 700(Days)

minus001

minus0005

0

0005

001

Retu

rns

(b)

Figure 1 Price process (a) and return process (b) for a reference asset of the GASM

equation 120590119896119894 = 120585120590119896(119879119894) Linking limit orders to volatility takesinto account a realistic aspect of trading psychology whenvolatility is high uncertainty on the ldquotruerdquo price of a stockgrows and traders place orders with a broader distributionof limit prices In our model 120585 is a constant for all agentswhereas 120590119896(119879119894) is the standard deviation of log-price returnsof asset 119896 computed in a timewindow119879119894 proper for agent 119894-thrandomly associated with the agent [20 28]

All buy and sell orders issued at time step ℎ are collectedand the demand and supply curves are consequently com-putedThe intersection of the two curves determines the newprice (clearing price) 119901119896(ℎ) of stock 119896 (see [20 28] for moredetails on market clearing)

Buy and sell orders with limit prices compatible with119901119896(ℎ) are executed After any transactions tradersrsquo cashportfolio and sentiments are updated Orders that do notmatch the clearing price are discarded

3 Marketrsquos Structure Indicators

As discussed in the previous Sections we aim to investigatehow the structural properties of tradersrsquo networks of inter-actions affect the emergent outcome of financial markets Inorder to measure the influence of the tradersrsquo networks onfinancial markets five different indicators have been definedthat is

(a) The number of prices processes that are integratedI(1) processes

(b) Thenumber of returns processes that exhibit volatilityclustering (heteroscedastic processes)

(c) The number of returns processes whose distributionshows fat tails (power law distributions)

(d) The number of static factors(e) The number of dynamic factors

These indicatorsmake reference to themain stylized factsempirically derived by the international literature on stockmarkets In fact the large availability of financial data hasallowed both qualitative and quantitative investigations offinancial markets by means of stylized facts In particularthe indicators (a) (b) and (c) referred to the single assetstatistical properties [31ndash35] whereas indicators (d) and (e)referred to the multiassets statistical properties [24ndash27]

The first indicator chosen is the number of prices pro-cesses that are integrated I(1) processes and is indicated with119868119886 [31] In order to verify if a time series is integrated I(1)process the Augmented Dickey-Fuller and the KwiatkowskiPhillips Schmidt and Shin (KPSS) tests at the significancelevel of 5 are employed It is worth remembering that thenull hypothesis of the ADF test is that a univariate time seriespresents a unit root whereas the null hypothesis of the KPSStest is that the time series is stationary [36 37] Figure 1(a)shows an I(1) price process for a typical asset generated bythe artificial stock market (GASM) presented in this paper

The second indicator chosen is the number of returnsprocesses that present volatility clustering and is indicatedwith 119868119887 [33] In order to test the presence of heteroscedas-tic effect Englersquos autoregressive conditional heteroscedastic(ARCH) test and the Ljung-Box 119876-test (LBQ test) at thesignificance level of 5 are employed It is worth notingthat the null hypothesis of the ARCH test is no conditionalheteroscedasticity whereas the null hypothesis of the LBQtest is that the residuals of the absolute value of returnsare autocorrelated [38ndash40] Figure 1(b) shows the volatilityclustering of the return process for the price process shownin Figure 1(a)

The third indicator chosen is the number of returns pro-cesses whose distribution presents ldquofat tailsrdquo and is indicatedwith 119868119888 [35] It is worth remembering that the expressionldquofat tailsrdquo means that the distribution of the analyzed datadecays with a power law that is if compared with a Gaussiandistribution it decays slowly compared to the Gaussiandistribution The Gaussian distribution has kurtosis equal to3 that is if the datarsquos kurtosis is larger than 3 the data exhibitsldquofat tailsrdquo and the distribution is leptokurtic Figure 2 showsthe ldquofat tailsrdquo of returns for asset shown in Figure 1 comparedwith a Gaussian distribution Moreover the presence of ldquofattailsrdquo is checked by means of the Jarque-Bera (JB) test It isworth remembering that the null hypothesis of JB test is thatthe data comes from a normal distribution with an unknownmean and variance [41]

The fourth and fifth indicators are focused on thestatistical properties of the multivariate process of pricesand returns and deal with the definition and analysis offactor models In the context of factor models two mainclasses can be identified that is static and dynamic factorsConcerning the former class attention is paid to returnsas the return processes result (in the first approximation)

Complexity 5

Returns

Surv

ival

func

tion

GASMGaussian data

100

10minus1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus2

Figure 2 Probability density function (PDF) for returnsrsquo process fora reference asset of GASM data shown in Figure 1 (blue) comparedwith a Gaussian distribution (red)

in quasi-stationary In particular the risk of a security canbe described as superposition of different source of risks(also described by stationary processes) and this generalformulation is basic for classical portfolio theory and riskmanagement CAPM multifactors CAPM APT and so on[42ndash45]

Conversely in the case of dynamic factors attention ispaid to asset prices and the main employed concept iscointegration In particular statistical analysis on empiricaldata points out that in financial markets it is not possible toreject the hypothesis of integrated univariate price processesbut at the aggregate level the price processes are not inde-pendent Indeed only few independent integrated processescan be identified whereas all the others price processes arecointegrated with them that is it is possible to identify linearcombinations of I(1) price processes that result in stationaryI(0) processes (so called cointegration equations) [42ndash45]

The number of static factor 119868119889 is evaluated using thecross-correlation matrix of returns In particular followingthe approach introduced in the econophysics literature [24ndash26] the cross-correlations of returns are analyzed in theframework of the random matrix theory (RMT) Thus theindicator 119868119889 is equal to the number of the eigenvalues ofthe cross-correlation matrix that are larger than the largesteigenvalue of the random matrix It is worth noting thatthe largest eigenvalue represents the market whereas theeigenvalues larger than the largest eigenvalue of the randommatrix represent the sectors Figure 3 shows the probabilitydensity function (PDF) of eigenvalues of the cross-correlationmatrix for the GASM data Furthermore for the sake of com-parison the theoretical PDF of a randommatrix (representedby the continuous line) and the PDF of eigenvalues for the100 stocks randomly chosen among the SampP 500 index arealso shown The presence of outliers well above the bounds

minus5 0 10

10

15

15

20

20

25

25

30

30

35Eigenvalues

0

02

04

06

08

1

12

PDF

0

005

01

GASMSampP500 sampleRandom matrix

5

5

Figure 3 Probability density function (PDF) for eigenvalues cross-correlation matrix of returns in the case of GASM data and SampP500data

determined according to RMT (ie eigenvalues larger thanthe largest eigenvalue determined by theRMT) is highlighted

Finally the fifth indicator the number of dynamic factors119868119890 makes reference to assets prices and their cointegrationThis indicator has been defined by means of the variance-covariance matrix of prices According to empirical analysisonly a reduced number of assets prices series in a largemarket are independent integrated processes [27] In factthe analysis of prices processes shows that financial assetsare random walk that is I(1) processes but aggregate offinancial assets exhibits cointegration The analysis of thisproperty is performed following the procedure describedby Stock and Watson [27] In particular the PCA analysison the variance-covariance matrix of prices allows one toidentify portfolios with minimum variance Conversely toprice processes these portfolios that is linear combinationof prices generally accept the hypothesis of stationarity [27]that is verified by the ADF and KPSS test at significance levelof 5 Thus the indicator 119868119890 is evaluated as the number ofportfolios that reject the hypothesis of stationarity Figure 4shows the results of the ADF test for the GASM data and for100 stocks randomly chosen among the SampP 500 index Asclearly stated in Figure 4 only a reduced number of portfoliosreject the hypothesis of stationarity These series are the onlyindependent I(1) processes that is the common trends of theaggregate

4 Computational Experiments

All the computational experiments performed make refer-ence to an artificial stock market with 100 different stockseach of them related to a specific firm The number ofagents is set to 2278 that are initially endowed by a randomdistribution of cash and number of stocks Furthermore

6 Complexity

Eigenvalue number

minus25

minus20

minus15

minus10

minus5

0

AD

F

GASM dataSampP500 data

Critical value at 5Critical value at 1

100 1009080706050403020

Figure 4 ADF test statistics of the cointegration portfolios in thecase of GASM and SampP500 data

time window 119879119894 for the calculation of the historical standarddeviation is randomly chosen from a uniform distribution inthe range (10 100)

Furthermore the influence of the tradersrsquo networks onthe financial market has been investigated by varying the120572 coefficients in (1) and analyzing the behavior of the fiveindicators defined in Section 3 In particular the performedanalysis addresses the influence of the market return (120572119872) ofthe neighbouring agents (120572119873) and of the agent global vision ofthemarket sentiment (120572119875) respectivelyThe influence of each120572 varies among three values none (N) average (A) or strong(S) It is worth noting that in the case of N the correspondinginteraction is absent in the case of A the interaction is ata reference value and in the case of S the influence is anorder of magnitude larger than the reference value Table 1summarizes the all scenarios considered in the analysis

5 Results and Discussion

The 27 scenarios have been considered and Table 2 shows themarketrsquo structure indicators for all cases

The indicators 119868119886 119868119887 and 119868119888 refer to the behavior of thesingle asset whereas 119868119889 and 119868119890 refer to the behavior of themultiasset market

Concerning the 119868119886 indicator that is the number of assetsthat are I(1) processes Table 2 shows the results of theADF and KPSS tests It is worth noting that from scenario(i) to scenario (ix) and from scenario (xix) to scenario(xxvii) that is when 120572119873 is equal to zero or it is very largeaccording to the ADF test almost all the price processesreject the null hypothesis of unitary root Conversely fromscenario (x) to scenario (xviii) the price processes do notreject the hypothesis of unitary root These results are alsoconfirmed by the KPSS test for which the null hypothesis ofstationarity is rejected for every cases Based on these resultswe can conclude that the influence of the neighbour is crucial

Table 1 Economic scenarios In the case of N the correspondinginteraction is absent in the case of A the interaction is at a referencevalue and in the case of S the influence is an order of magnitudelarger than the reference value

Scenario 120572119873rsquos influence 120572119872119894rsquos influence 120572119870119894rsquos influence(i) N N N(ii) N N A(iii) N N S(iv) N A N(v) N A A(vi) N A S(vii) N S N(viii) N S A(ix) N S S(x) A N N(xi) A N A(xii) A N S(xiii) A A N(xiv) A A A(xv) A A S(xvi) A S N(xvii) A S A(xviii) A S S(xix) S N N(xx) S N A(xxi) S N S(xxii) S A N(xxiii) S A A(xxiv) S A S(xxv) S S N(xxvi) S S A(xxvii) S S S

in order to reproduce the single asset stylized facts Theindicator 119868119887 that is the number of returns processes that areheteroscedastic shows that the volatility clustering is alwayspresent for every scenario In fact Table 2 reports the resultsof theARCH test that is the number of returns processes thatreject the null hypothesis of no conditional heteroscedasticityThis result has been also verified performing the Ljung-Boxtest on the absolute value of returns processes It is worthnoting that indicator 119868119887 suggests that the heteroscedasticityis a strong feature of financial markets not depending on thetradersrsquo networks of interaction

The indicator 119868119888 that is the number of returns processwith kurtosis larger than three shows that if the 120572119873 is nullor average that is from scenario (i) to scenario (xviii) theprice processes exhibit fat tails whereas if 120572119873 is strong thatis from scenario (xix) to scenario (xxvii) the 119868119888 indicator issmall and that the returns processes do not exhibit fat tailsThese results are further confirmed by the Jarque-Bera testTable 2 reports the number of assets that do not reject tonull hypothesis of normal distribution It is worth notingthat from scenario (i) to scenario (xviii) almost all the priceprocesses reject the null hypothesis of normal distribution

Complexity 7

Table 2 Marketsrsquo structure indicators results for scenarios described in Table 1 and for 100 assets randomly chosen from the SampP500 indexThe ADF and KPSS and JB statistical tests report the number of assets that do not reject the null hypothesis whereas the ARCH and LBQtests report the number of assets that reject the null hypothesis

Indicators 119868119886 119868119887 119868119888 119868119889 119868119890Scenario ADF KPSS ARCH LBQ LeptoK JB Sectors ADF KPSS(i) 1 0 99 100 100 0 9 1 56(ii) 3 0 99 99 100 0 8 2 58(iii) 5 0 99 100 100 0 8 2 74(iv) 0 0 99 99 99 0 4 2 73(v) 0 0 98 99 98 0 4 3 73(vi) 0 0 98 99 95 0 4 3 70(vii) 0 0 99 99 98 0 4 2 71(viii) 1 0 97 99 100 0 4 2 72(ix) 1 0 97 99 98 0 5 2 74(x) 90 0 94 100 98 0 8 6 74(xi) 90 0 93 100 100 0 7 8 73(xii) 93 0 92 100 100 0 9 9 75(xiii) 88 0 92 100 100 0 7 10 74(xiv) 92 0 93 100 100 0 7 7 74(xv) 93 0 96 100 100 0 7 8 76(xvi) 90 0 94 100 100 1 8 13 75(xvii) 90 0 95 100 100 0 6 14 75(xviii) 91 0 93 100 100 1 7 15 74(xix) 4 0 7 10 4 43 - 17 71(xx) 6 0 5 3 3 33 - 14 73(xxi) 7 0 5 5 3 43 - 17 70(xxii) 5 0 10 2 3 46 - 14 70(xxiii) 6 0 5 9 2 46 - 16 70(xxiv) 6 0 4 4 1 38 - 16 67(xxv) 5 0 6 7 4 40 - 16 68(xxvi) 4 0 9 6 3 41 - 15 70(xxvii) 7 0 2 7 2 44 - 18 69SampP500 85 0 77 100 100 0 6 7 76

whereas from scenario (xix) to scenario (xxvii) the JB testcannot reject the null hypothesis of Gaussian distribution

Summarizing the results of indicators 119868119886 119868119887 and 119868119888 wecan conclude that the influence of the neighbour is crucial inorder to reproduce the single asset stylized facts Moreover if120572119873 influence is strong the 119868119887 and 119868119888 indicator are significantlydifferent from the reference values of SampP500 thus allowingus to conclude that the presence of agentsrsquo neighbour isnecessary but it should be not too large Moreover theseresults are in good agreementwith the observations ofmarketdynamics in historical and archeological data [46]

Once the impacts of the traderrsquo network of interactionson single asset are addressed the attention has been focusedon the indicators 119868119889 and 119868119890 that deal with the aggregatebehavior of the market that is the statistical properties of themultivariate process of prices and returns

In particular the indicator 119868119889 shows the number of staticfactor that is the number of sectors presented in the marketFirst of all it is worth noting that even for the case ofabsence of interactions (ie 120572119873 = 120572119872 = 120572119875 = 0) at

least one sector is presented It is worth remarking that thisis not a trivial outcome as it points out that the CAPM isoriginated by the finiteness of traderrsquos wealth rather thanby traderrsquos interactions and strategies of portfolio allocationFurthermore if the influence of 120572119875 (average sentiment) islarger than the other 120572 the market shows an increasingnumber of sectorsMoreover if 120572119873 influence is strong that isfrom scenario (xix) to scenario (xxvii) some price processesare not traded This result can be explained focusing onthe traders allocationrsquos choice As 120572119873rsquos influence is strongeach trader is extremely influenced by the neighbours agentsand thus heshe makes the same portfolio allocation ofhisher neighbours Thus the financial market is becominghomogeneous and the heterogeneous agents start to act as inthe case of a representative agent

Finally the presence of the common trends of the aggre-gate is a strong feature that is present for every scenario andbecomes stronger increasing the influence of tradersrsquo networkof interactions It is worth remarking that the number ofcommon trends is influenced by tradersrsquo networks and by the

8 Complexity

market feedback as confirmed by the results in Table 1 Fromscenario (i) to scenario (xxvii) the number of common trendsincreases thus showing that the presence of cointegratedprocesses becomes larger as the 120572119873 and 120572119872 influence isstronger

In isworth noting that the tradersrsquo network of interactionsinfluences positively the market but irrespective to suchpositive outcome the main statistical properties of financialmarket are emergent features even in the case of absenceof any interactions among the traders This allows us toconclude that the multiassets stylized facts of the financialmarket are a direct consequence of realistic mechanism forportfolio allocation with budget constraint rather than oftradersrsquo interactions network All these emergent propertiespoint out an intrinsic structural resilience of the stockmarket

6 Conclusions

An analysis of the influences of agentsrsquo networks on themarket structure has been presented using an information-based multiasset artificial stock market characterized bydifferent types of stocks and populated by heterogeneousagents In this complex system agents are characterized bycash stocks and sentiments Sentiments denote propensitiesto buy or to sell Agents are organized in networks modeledas nodes of sparsely connected graph so that each agent isinfluenced by a subset of other agent the only ones that areldquonearrdquo to him Five network influence indicators consideringthe main single-assets and multiassets stylized facts havebeen defined in order to investigate the effects of the agentsnetworks Results have pointed out an intrinsic structuralresilience of the stockmarket In fact the network is necessaryin order to archive the ability to reproduce the main stylizedfacts but also the market has some characteristics that areindependent from the network and depend on the finitenessof tradersrsquo wealth

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work has been supported by the University of Genoaunder Grant FRA2016

References

[1] BMandelbrot ldquoThe variation of certain speculative pricesrdquoTheJournal of Business vol 36 no 4 pp 394ndash413 1963

[2] M Tumminello F Lillo and R N Mantegna ldquoCorrelationhierarchies and networks in financial marketsrdquo Journal ofEconomic Behavior amp Organization vol 75 no 1 pp 40ndash582010

[3] D-M Song M Tumminello W-X Zhou and R N MantegnaldquoEvolution of worldwide stock markets correlation structureand correlation-based graphsrdquo Physical Review E Statistical

Nonlinear and Soft Matter Physics vol 84 no 2 Article ID026108 2011

[4] M Riccaboni F Pammolli S V Buldyrev L Ponta and H EStanley ldquoThe size variance relationship of business firm growthratesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 105 no 50 pp 19595ndash19600 2008

[5] L PontaM Raberto A Teglio and S Cincotti ldquoAn agent-basedstock-flow consistent model of the sustainable transition in theenergy sectorrdquo Ecological Economics vol 145 pp 274ndash300 2018

[6] L Ponta E Scalas M Raberto and S Cincotti ldquoStatisticalanalysis and agent-based microstructure modeling of high-frequency financial tradingrdquo IEEE Journal of Selected Topics inSignal Processing vol 6 no 4 pp 381ndash387 2012

[7] P C Ivanov A Yuen and P Perakakis ldquoImpact of stock marketstructure on intertrade time and price dynamicsrdquo PLoS ONEvol 9 no 4 Article ID e92885 2014

[8] B Podobnik I GrosseDHorvatic S Ilic P C Ivanov andH EStanley ldquoQuantifying cross-correlations using local and globaldetrending approachesrdquo The European Physical Journal B vol71 no 2 pp 243ndash250 2009

[9] B Podobnik D F Fu H E Stanley and P C Ivanov ldquoPower-law autocorrelated stochastic processes with long-range cross-correlationsrdquoThe European Physical Journal B vol 56 no 1 pp47ndash52 2007

[10] L Fraccascia I Giannoccaro andV Albino ldquoEfficacy of landfilltax and subsidy policies for the emergence of industrial symbio-sis networks an agent-based simulation studyrdquo Sustainabilityvol 9 no 4 article 521 2017

[11] R G Palmer W B Arthur J H Holland B LeBaron and PTayler ldquoArtificial economic life a simple model of a stockmar-ketrdquo Physica D Nonlinear Phenomena vol 75 no 1ndash3 pp 264ndash274 1994

[12] W B Arthur J H Holland B D LeBaron R Palmer and PTayler ldquoAsset pricing under endogeneous expectations in anartificial stock marketrdquo inThe Economy as an Evolving ComplexSystem II W Arthur S Durlauf and D Lane Eds SFI Studiesin the Sciences of Complexity Addison Wesley Longman 1997

[13] B LeBaronWBArthur andR Palmer ldquoTime series propertiesof an artificial stockmarketrdquo Journal of Economic Dynamics andControl vol 23 no 9-10 pp 1487ndash1516 1999

[14] F Westerhoff ldquoAn agent-based macroeconomic model withinteracting firms socio-economic opinion formation and opti-misticpessimistic sales expectationsrdquo New Journal of Physics vol 12 Article ID 075035 2010

[15] M Goykhman ldquoWealth dynamics in a sentiment-driven mar-ketrdquo Physica A Statistical Mechanics and Its Applications vol488 pp 132ndash148 2017

[16] B Podobnik J Shao D Njavro P C Ivanov and H EStanley ldquoInfluence of corruption on economic growth rateand foreign investmentrdquo The European Physical Journal BCondensed Matter and Complex Systems vol 63 no 4 pp 547ndash550 2008

[17] J Shao P C Ivanov B Podobnik and H E Stanley ldquoQuanti-tative relations between corruption and economic factorsrdquoTheEuropean Physical Journal B Condensed Matter and ComplexSystems vol 56 no 2 pp 157ndash166 2007

[18] C H Hommes ldquoHeterogenous agent models in economicsand financerdquo in Handbook of Computational Economics LTesfatsion and K L Judd Eds 2006

[19] E Samanidou E Zschischang D Stauffer and T Lux ldquoAgent-based models of financial marketsrdquo Reports on Progress inPhysics vol 70 no 3 pp 409ndash450 2007

Complexity 9

[20] S Pastore L Ponta and S Cincotti ldquoHeterogeneous information-based artificial stock marketrdquo New Journal of Physics vol 12Article ID 053035 2010

[21] L Ponta S Pastore and S Cincotti ldquoInformation-based multi-assets artificial stock market with heterogeneous agentsrdquo Non-linear Analysis Real World Applications vol 12 no 2 pp 1235ndash1242 2011

[22] L Ponta S Pastore and S Cincotti ldquoStatic and dynamic factorsin an information-based multi-asset artificial stock marketrdquoPhysica A Statistical Mechanics and Its Applications vol 492pp 814ndash823 2018

[23] L Ponta Modeling and statistical analysis of financial marketsand firm growth [PhD dissertation] University of Genoa 2008

[24] L Laloux P Cizeau J-P Bouchaud and M Potters ldquoNoisedressing of financial correlation matricesrdquo Physical ReviewLetters vol 83 no 7 pp 1467ndash1470 1999

[25] V Plerou P Gopikrishnan B Rosenow L A N Amaral andH E Stanley ldquoUniversal and nonuniversal properties of crosscorrelations in financial time seriesrdquo Physical Review Lettersvol 83 no 7 pp 1471ndash1474 1999

[26] V Plerou P Gopikrishnan B Rosenow L A N Amaral TGuhr and H E Stanley ldquoRandom matrix approach to crosscorrelations in financial datardquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 65 no 6 pp 066126-1ndash066126-18 2002

[27] J H Stock and M W Watson ldquoTesting for common trendsrdquoJournal of the American Statistical Association vol 83 no 404pp 1097ndash1107 1988

[28] M Raberto S Cincotti SM Focardi andMMarchesi ldquoAgent-based simulation of a financial marketrdquo Physica A StatisticalMechanics and Its Applications vol 299 no 1-2 pp 319ndash3272001

[29] L Ponta M Raberto and S Cincotti ldquoA multi-assets artificialstock market with zero-intelligence tradersrdquo Europhysics Let-ters vol 93 no 2 Article ID 28002 2011

[30] S Cincotti S M Focardi M Marchesi and M Raberto ldquoWhowins Study of long-run trader survival in an artificial stockmarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 324 no 1-2 pp 227ndash233 2003

[31] B B Mandelbrot Fractals and Scaling in Finance SpringerBerlin Germany 1997

[32] U A Muller M M Dacorogna R B Olsen O V PictetM Schwarz and C Morgenegg ldquoStatistical study of foreignexchange rates empirical evidence of a price change scaling lawand intraday analysisrdquo Journal of Banking amp Finance vol 14 no6 pp 1189ndash1208 1990

[33] R N Mantegna and H E Stanley ldquoScaling behaviour in thedynamics of an economic indexrdquo Nature vol 376 no 6535 pp46ndash49 1995

[34] P C Ivanov A Yuen B Podobnik and Y Lee ldquoCommonscaling patterns in intertrade times of US stocksrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 69no 5 Article ID 056107 2004

[35] P Gopikrishnan V Plerou X Gabaix and H E StanleyldquoStatistical properties of share volume traded in financialmarketsrdquo Physical Review E Statistical Physics Plasmas Fluidsand Related Interdisciplinary Topics vol 62 no 4 pp R4493ndashR4496 2000

[36] D A Dickey and W A Fuller ldquoDistribution of the estimatorsfor autoregressive time series with a unit rootrdquo Journal of theAmerican Statistical Association vol 74 no 366 pp 427ndash4311979

[37] D Kwiatkowski P C B Phillips P Schmidt and Y Shin ldquoTest-ing the null hypothesis of stationarity against the alternative of aunit rootrdquo Journal of Econometrics vol 54 no 1ndash3 pp 159ndash1781992

[38] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[39] GM Ljung andG E P Box ldquoOn ameasure of lack of fit in timeseries modelsrdquo Biometrika vol 65 no 2 pp 297ndash303 1978

[40] G E Box and D A Pierce ldquoDistribution of residual autocorre-lations in autoregressive-integrated moving average time seriesmodelsrdquo Journal of the American Statistical Association vol 65no 332 pp 1509ndash1526 1970

[41] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[42] W F Sharpe ldquoCapital asset prices a theory of market equilib-rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[43] R C Merton ldquoAn intertemporal capital asset pricing modelrdquoEconometrica vol 41 pp 867ndash887 1973

[44] S A Ross ldquoThe arbitrage theory of capital asset pricingrdquo Journalof Economic Theory vol 13 no 3 pp 341ndash360 1976

[45] M Forni M Hallin M Lippi and L Reichlin ldquoThe generalizeddynamic factor model consistency and ratesrdquo Journal of Econo-metrics vol 119 no 2 pp 231ndash255 2004

[46] N E Romero Q D Ma L S Liebovitch C T Brown and P CIvanov ldquoCorrelated walks down the Babylonian marketsrdquo EPL(Europhysics Letters) vol 90 no 1 Article ID 18004 2010

Hindawiwwwhindawicom Volume 2018

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Applied MathematicsJournal of


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Complex AnalysisJournal of


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International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Decision SciencesAdvances in


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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

2 Complexity

financial markets that is how the fraudulent agents impacton the markets [16] In particular empirical analysis showsthat corruption influences the economic growth rate andforeign investment [17] For a detailed review onmicroscopic(ldquoagent-basedrdquo) models of financial markets see [18 19]

In this paper using the Genoa Artificial Stock Market(GASM) developed in Genoa the impact of the structuralproperties of tradersrsquo networks of interaction on the emergentoutcome in financial markets has been studied In particularstarting from the information-based single-assets artificialmarket a multiassets artificial stock market version of theGASM has been used [20ndash23] In order to investigate thisrelationship themarket is populated by heterogeneous agentsthat are seen as nodes of sparsely connected graphs Themarket is characterized by different types of stocks and agentstrade risky assets in exchange for cash Agents share theirinformation by means of interactions that are determined bythe graphs Besides the amount of cash and assets ownedeach agent is characterized by sentiments that summarizethe agentrsquos information about the market and agent worldThe sentiments include in one element the influence of themarket trend the influence of the neighbours agents and thepropensity for the market Agents are subject to a portfoliochoice on number and type of risky securities The allocationstrategy is based on sentiments and wealth A central marketmaker (clearing house mechanism) determines the priceprocess for each stock at the intersection of the demand andthe supply curves

The paper presents a study on how the tradersrsquo networksand thus the sentimentsrsquo components influence the marketstructure In particular this paper investigates the effects ofchanges in tradersrsquo networks of interaction in the financialmarket In order to perform this investigation five differentldquomarketrsquos structure indicatorsrdquo have been defined The indi-cators have been defined considering themain univariate andmultivariate stylized facts Concerning univariate processesthe threemain stylized facts taken as reference are the unitaryroot of price processes the fat tails distribution of returnsand the volatility clustering Concerning the multiassetsenvironment the set of stylized facts consists in the statisticalproperties of the cross-correlation matrices of returns [24ndash26] and of the variance-covariancematrices of prices [27] thatmake reference to static and dynamic factors respectively

Thus the indicators defined are the number of I(1)processes the number of heteroscedastic processes thenumber of processes with fat tails the number of sectorpresented in the market and the number of common trendsThe computational experiments show an intrinsic structuralresilience of the stock market In fact some characteristics ofthe market are ldquostructuralrdquo and depend on the agentsrsquo budgetconstraint whereas others are an emerging properties of thetradersrsquo network of interactions

The paper is organized as follows Section 2 presentsthe model and Section 3 the ldquomarketrsquos structure indicatorsrdquoand Section 4 shows the computational experiments andSection 5 the discussion of results Finally Section 6 providesthe conclusion of the study

2 The GASM Model

21 Overview of the GASM Model The model presented inthis paper is an enrichment of the Genoa Artificial StockMarket (GASM) developed at the University of Genoa [2028] The GASM is an agent-based artificial stock marketwhose baseline originally includes heterogeneous agents thattrade risky assets in exchange for cash [29]They aremodeledas liquidity traders that is decision making process isconstrained by the finite amount of financial resources (cash+ stocks) they own At the beginning of the simulation cashand stocks are distributed randomly among agents

22 Agentsrsquo Networks In order to investigate the effectsof agentsrsquo networks in financial markets for each stockpresented in the market the heterogeneous agents have beenorganized in graphs and in particular according to a directedrandom graph where the agents are the nodes and thebranches represent the interactions among agentsThe graphsare responsible for the changes in agentrsquos sentiments Thegraphs are directed that is the interactions are assumedunidirectional (ie if agent 119895-th influences agent 119894-th notnecessarily agent 119894-th influences agent 119895-th) and characterizedby a strength 119892119896119895119894 assuming a positive real number Generallyspeaking due to the presence of a directed graph both anoutput node degree related to the output branches of a givennode and an input node degree related to the input branchesshould be defined

The agents are organized according to a Zipf law Foreach stock an agent is randomly connected to a set of otheragents whose number and strength (of the connection)119892119896119894119895 areinversely proportional to hisher rank that is richer agentsinfluence a larger number of agents with a higher strengthConsequently the output degree distributions over the nodesare set to power laws and the input degree distributions resultin power laws too Each agent has a different belief about the119870 assets depending on hisher rank Agent 119894 is characterizedby a sentiment 119878119896119894 (ie real number in the interval [minus1 1]) thatrepresents a propensity to invest in asset 119896 A positive averagesentiment denotes a propensity to buy whereas a negativeaverage sentiment corresponds to a propensity to sell Thegraphs are responsible for the changes in agentrsquos sentimentsAt each time step ℎ information is propagated through themarket and sentiments 119878119896119894 of agent 119894 are updated

Let I119896119894 be the set of agents that influences the behaviorof trader i-th for the asset 119896 and 119901119896 the market price of therisky asset 119896Thenew sentiments 119878119896119894 of agent i-th for each asset119896 are functions of the previous sentiments of the log return(market feedback) of the influence of interacting agents andof average sentiment of the agent about the market behaviorThe expression is

119878119896119894 (ℎ) = 119865 (120572119878119894119878119896119894 (ℎ minus 1) + 120572119872119894119903119896 (ℎ minus 1)+ 120572119873119894119878119896119894 (ℎ minus 1) + 120572119875119894119878119894 (ℎ minus 1)) (1)

where

119865 (119909) = tanh (119909) (2)

Complexity 3


Furthermore



119878119896119894 (ℎ minus 1) = sum119895isinI119896119894

119892119896119895119894119878119896119895 (ℎ minus 1)sum119895isinI119896

119894

119892119896119895119894 (4)


119878119894 (ℎ minus 1) = sum119896 119878119896119894119870 (5)


10038161003816100381610038161205721198731198941003816100381610038161003816 = (120578 minus 10038161003816100381610038161205721198781198941003816100381610038161003816) (6)






119882119903119894 (ℎ minus 1) = sum119896

119902119896119894 (ℎ minus 1) 119901119896 (ℎ minus 1) (7)



119903119894 (ℎ) = 120574119903 (ℎ minus 1) 119882119894 (ℎ minus 1) (8)




120574119896119886 = 119878119896119894sum119896isin119860119894 119878119896119894 (10)



Δ119896119894 (ℎ) = 119902119896119894 (ℎ) minus 119902119896119894 (ℎ minus 1) (11)


119889119896119894 (ℎ) = 119901119896 (ℎ minus 1) sdot 119873119894 (120583119896119894 120590119896119894 ) (12)


120583119896119894 = (1 + sgn (Δ119896119894) 10038161003816100381610038161003816119878119896119894 10038161003816100381610038161003816) (13)


4 Complexity

20

21

22

23

24Pr

ices

(eur

o)

0 100 200 300 400 500 600 700(Days)

(a)

0 100 200 300 400 500 600 700(Days)

minus001

minus0005

0

0005

001

Retu

rns

(b)
















Complexity 5

Returns

Surv

ival

func

tion

GASMGaussian data

100

10minus1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus2





minus5 0 10

10

15

15

20

20

25

25

30

30

35Eigenvalues

0

02

04

06

08

1

12

PDF

0

005

01


5

5






6 Complexity

Eigenvalue number

minus25

minus20

minus15

minus10

minus5

0

AD

F



100 1009080706050403020












Complexity 7









8 Complexity



6 Conclusions




Acknowledgments


References





















Complexity 9



































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3


Furthermore



119878119896119894 (ℎ minus 1) = sum119895isinI119896119894

119892119896119895119894119878119896119895 (ℎ minus 1)sum119895isinI119896

119894

119892119896119895119894 (4)


119878119894 (ℎ minus 1) = sum119896 119878119896119894119870 (5)


10038161003816100381610038161205721198731198941003816100381610038161003816 = (120578 minus 10038161003816100381610038161205721198781198941003816100381610038161003816) (6)






119882119903119894 (ℎ minus 1) = sum119896

119902119896119894 (ℎ minus 1) 119901119896 (ℎ minus 1) (7)



119903119894 (ℎ) = 120574119903 (ℎ minus 1) 119882119894 (ℎ minus 1) (8)




120574119896119886 = 119878119896119894sum119896isin119860119894 119878119896119894 (10)



Δ119896119894 (ℎ) = 119902119896119894 (ℎ) minus 119902119896119894 (ℎ minus 1) (11)


119889119896119894 (ℎ) = 119901119896 (ℎ minus 1) sdot 119873119894 (120583119896119894 120590119896119894 ) (12)


120583119896119894 = (1 + sgn (Δ119896119894) 10038161003816100381610038161003816119878119896119894 10038161003816100381610038161003816) (13)


4 Complexity

20

21

22

23

24Pr

ices

(eur

o)

0 100 200 300 400 500 600 700(Days)

(a)

0 100 200 300 400 500 600 700(Days)

minus001

minus0005

0

0005

001

Retu

rns

(b)
















Complexity 5

Returns

Surv

ival

func

tion

GASMGaussian data

100

10minus1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus2





minus5 0 10

10

15

15

20

20

25

25

30

30

35Eigenvalues

0

02

04

06

08

1

12

PDF

0

005

01


5

5






6 Complexity

Eigenvalue number

minus25

minus20

minus15

minus10

minus5

0

AD

F



100 1009080706050403020












Complexity 7









8 Complexity



6 Conclusions




Acknowledgments


References





















Complexity 9



































Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity

20

21

22

23

24Pr

ices

(eur

o)

0 100 200 300 400 500 600 700(Days)

(a)

0 100 200 300 400 500 600 700(Days)

minus001

minus0005

0

0005

001

Retu

rns

(b)
















Complexity 5

Returns

Surv

ival

func

tion

GASMGaussian data

100

10minus1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus2





minus5 0 10

10

15

15

20

20

25

25

30

30

35Eigenvalues

0

02

04

06

08

1

12

PDF

0

005

01


5

5






6 Complexity

Eigenvalue number

minus25

minus20

minus15

minus10

minus5

0

AD

F



100 1009080706050403020












Complexity 7









8 Complexity



6 Conclusions




Acknowledgments


References





















Complexity 9



































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

Returns

Surv

ival

func

tion

GASMGaussian data

100

10minus1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus2





minus5 0 10

10

15

15

20

20

25

25

30

30

35Eigenvalues

0

02

04

06

08

1

12

PDF

0

005

01


5

5






6 Complexity

Eigenvalue number

minus25

minus20

minus15

minus10

minus5

0

AD

F



100 1009080706050403020












Complexity 7









8 Complexity



6 Conclusions




Acknowledgments


References





















Complexity 9



































Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

Eigenvalue number

minus25

minus20

minus15

minus10

minus5

0

AD

F



100 1009080706050403020












Complexity 7









8 Complexity



6 Conclusions




Acknowledgments


References





















Complexity 9



































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7









8 Complexity



6 Conclusions




Acknowledgments


References





















Complexity 9



































Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity



6 Conclusions




Acknowledgments


References





















Complexity 9



































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9



































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








traders’ networks of interactions and structural...

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