We follow up on the study of correlations between GDP's of rich countries. We
analyze web-downloaded data on GDP that we use as individual wealth signatures
of the country economical state. We calculate the yearly fluctuations of the
GDP. We look for forward and backward correlations between such fluctuations.
The system is represented by an evolving network, nodes being the GDP
fluctuations (or countries) at different times.
  In order to extract structures from the network, we focus on filtering the
time delayed correlations by removing the least correlated links. This
percolation idea-based method reveals the emergence of connections, that are
visualized by a branching representation. Note that the network is made of
weighted and directed links when taking into account a delay time. Such a
measure of collective habits does not fit the usual expectations defined by
politicians or economists.Comment: 9 pages, 3 figure

Basu

Di Matteo

Garlaschelli

Gligor

Lambiotte

Long

M. Ausloos

Miskiewicz

Mora

R. Lambiotte

Serrano

English

arXiv

We study correlations between web-downloaded gross domestic product (GDP)'s of rich countries. GDP is used as wealth signatures of the country economical state. We calculate the yearly fluctuations of the GDP. We look for forward and backward correlations between such fluctuations. The correlation measure is based on the Theil index. The system is represented by an evolving weighted network, nodes being the GDP fluctuations (or countries) at different times. In order to extract structures from the network, we focus on filtering the time delayed correlations by removing the least correlated links. This percolation idea-based method reveals the emergence of connections, that are visualized by a branching representation. Note that the network is made of weighted and directed links when taking into account a delay time. Such a measure of collective habits does not readily fit the usual expectations, except if an economy globalization framework is accepted

Ausloos, M.

Lambiotte, R.

Repository of the University of Namur

RESEARCH OUTPUTS / RÉSULTATS DE RECHERCHEAuthor(s) - Auteur(s) :Publication date - Date de publication :Permanent link - Permalien :Rights / License - Licence de droit d’auteur :Bibliothèque Universitaire Moretus PlantinInstitutional Repository - Research PortalDépôt Institutionnel - Portail de la Rechercheresearchportal.unamur.beClusters or networks of economies?Ausloos, M.; Lambiotte, R.Published in:Physica A: Statistical Mechanics and its ApplicationsDOI:10.1016/j.physa.2007.02.005Publication date:2007Document VersionPublisher's PDF, also known as Version of recordLink to publicationCitation for pulished version (HARVARD):Ausloos, M & Lambiotte, R 2007, 'Clusters or networks of economies? A macroeconomy study through GrossDomestic Product', Physica A: Statistical Mechanics and its Applications, vol. 382, no. 1, pp. 16-21.https://doi.org/10.1016/j.physa.2007.02.005General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 23. Jun. 2020ARTICLE IN PRESS0378-4371/$ - sedoi:10.1016/j.phCorrespondE-mail addrPhysica A 382 (2007) 16–21www.elsevier.com/locate/physaClusters or networks of economies? A macroeconomy studythrough Gross Domestic ProductM. Ausloos, R. LambiotteGRAPES, Institute of Physics, B5, University of Lie´ge, B-4000 Lie´ge, Belgium, EurolandAvailable online 1 March 2007AbstractWe study correlations between web-downloaded gross domestic product (GDP)’s of rich countries. GDP is used aswealth signatures of the country economical state. We calculate the yearly fluctuations of the GDP. We look for forwardand backward correlations between such fluctuations. The correlation measure is based on the Theil index. The system isrepresented by an evolving weighted network, nodes being the GDP fluctuations (or countries) at different times.In order to extract structures from the network, we focus on filtering the time delayed correlations by removing the leastcorrelated links. This percolation idea-based method reveals the emergence of connections, that are visualized by abranching representation. Note that the network is made of weighted and directed links when taking into account a delaytime. Such a measure of collective habits does not readily fit the usual expectations, except if an economy globalizationframework is accepted.r 2007 Published by Elsevier B.V.Keywords: Directed network; Time delay; Correlations; Theil index; Econophysics; GDP; Distance; Patterns1. IntroductionSome time ago, the question of economy globalization was studied along the lines of macro-econophysicsresearch [1–3]. Other considerations along related lines can be found in contributions by Miskiewicz [4] andGligor [5]. The question pertains to well-known observations: there are political, cultural, scientific, economiccooperations all over the world, even though there are even anti-globalization organizations.In our case, we consider that a globalization process in economy should be understood as an increase ofsimilarities within (macroeconomy) development patterns. In so doing we have proposed to study whether thepattern economy of countries can be represented by and discussed through the distance between each countrynormalized gross domestic product (GDP) fluctuations. The notion of distance is somewhat arbitrary, as hasbeen already discussed in Ref. [1], but theoretical considerations are left for further investigation elsewhere.We have chosen to present here a study based on the statistical distance, in Section 2, for the 23 mostdeveloped countries for which annual GDP data is available in the interval [1950–2003]. Previous studies usinge front matter r 2007 Published by Elsevier B.V.ysa.2007.02.005ing author. Tel.: +32 4 366 37 52; fax: +324 366 29 90.esses: marcel.ausloos@ulg.ac.be (M. Ausloos), renaud.lambiotte@ulg.ac.be (R. Lambiotte).ARTICLE IN PRESSM. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–21 17Network analysis tools to understand the dynamics of GDP’s have been already published, for a few examplessee Refs. [6–9].One might argue [10] that some data detrending is necessary to take into account cycle-like effects [11,12]. Inprevious studies [1] it has also been observed that there is much noise in the data. We have shownthat some average should be taken over optimal time windows to remove such a noise. We will use foraveraging a time window of T ¼ 4 years which is a priori optimizing the statistics but is clearly related to mostelectoral periods.One usual political question is whether one has to follow another country policy in order to improve one’scountry economy. The question of followers and leaders can be tackled by considering time delayedcorrelations. In this first attempt we will only consider a one year time lag t, being aware that other time lagsshould be involved. Note that positive or negative ones should be considered. Nevertheless it is worthwhile tocheck whether such an approach is already meaningful for t ¼ 1 and 1.The system can be displayed as an evolving network, nodes being the GDP fluctuations or, in a short way,countries at different times. In Section 3, we extract structures from the network through filtering the timedelayed correlations (or distance matrix) by removing the least correlated links. This percolation idea-basedmethod reveals the emergence of connections, that are visualized by a branching representation. However, thesystem is pretty unstable. Yet, expected features (connections) are reproduced, though some others aresurprising. This should imply some further consideration in the line of Glansdorff and Prigogine [13] aboutstructure and stability of world economic systems and the intrinsic role of fluctuations. Notice also that thenetwork is much more complex when taking into account delay time. In such a case, the correlation matrix isNOT symmetrical, whence the network links are not only weighted but also directed. The number of distancesto be considered is also NðN  1Þ, where N is the number of countries, in contrast to NðN  1Þ=2, when t ¼ 0.In Section 4, we conclude that such a measure of collective habits does not fit the usual expectations, definedby politicians, but that does not imply that they are wrong or right, nor are we.We stress that the following relevant time instances must be considered, i.e.,1. Initial observation time t0,2. Data acquisition time t,3. Increment or fluctuation time span D,4. Delay time (between two countries) t,5. Window observation (averaging) time T , i.e., [tmin; tmax],6. Conclusion time tN .Thus the general correlation matrix can be of quite high dimension (i; j; t0; t; t; ..).2. MethodologyWe use data from http://www.ggdc.net/index-series.html for recording the Normalized GDP of the 23 mostdeveloped countries. Most of these countries have natural time scales associated to the time lags betweenelections. Whence we use an average time of T ¼ 4 years. For each of the countries, let us define pi;t ¼lnGi;tþT=Gi;t that is equivalent to the average variation of the GDP over T years.Let us now introduce a correlation measure and associated distance measure for the countries. Thecorrelation measure is based on the Theil index and is defined as follows:Ci;j;t;t ¼hpi;t0pj;t0þtit  hpi;t0 ithpj;t0þtitffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihp2i;t0  hpi;t0 i2ithp2j;t0þt  hpj;t0þti2itq , (1)where the averages are defined as follows:hF ðt0Þit ¼1DXtþDt0¼tF ðt0Þ. (2)ARTICLE IN PRESSM. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–2118The ‘‘averaging time’’ D has been hereby chosen to be D ¼ 4 in the following. The distance is obtained fromthe correlation from the definition (dropping indices here):d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð1 CÞq. (3)Let us stress that this short hand writing accounts for time delays between the signals. Let us also note thatother measures of distance between different countries can be defined, e.g.Di;j;t;t ¼ jpi;t  pj;tþtj. (4)The comparison of results obtained for different measures will be considered elsewhere.When t ¼ 0, the above quantities are at fixed time. When ta0, these quantities measure the correlationswith a delay time between the economies of countries. If the evolutions were completely identical, one wouldfind D ¼ 0. Let us stress again that Di;j;t;t is symmetric in i; j, when t ¼ 0. For each pair of countries, thisquantity defines therefore a surface whose statistical properties can be studied.The first relevant quantity to report is the rank analysis of Di;j;t;t, at different values of t (1974, 1984, 1994)and t ¼ 0; 1;1 (Figs. 1–3), like on a Zipf plot. Notice the non-trivial behavior, with various crossing of thecurves, on one hand; moreover, the data does not seem to lead to a fine straight line fit. Because the GDPincrement distance between countries is rather uniform, the results suggest non-trivial correlations in evolving00.10.20.30.40.50.60.70.80.910 50 100 150 200 250 300Drank197419841994Fig. 1. Rank analysis of Di;j;t;t for 1974, 1984, 1994 years when t ¼ 0.00.10.20.30.40.50.60.70.80.910 100 200 300 400 500 600Drank197419841994Fig. 2. Rank analysis of Di;j;t;t for 1974, 1984, 1994 years, when t ¼ 1.ARTICLE IN PRESS00.10.20.30.40.50.60.70.80.910 100 200 300 400 500 600Drank197419841994Fig. 3. Rank analysis of Di;j;t;t for 1974, 1984, 1994 years, when t ¼ 1.M. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–21 19economy schemes, which likely result from complex interactions of country economies. A globalizationframework is hardly proved from the above. However, it is clear that the ranking pattern does not changeduring the last part of the century, when t ¼ 0, indicating a rather common economy philosophy and similarresults for all countries. The drastic change between the 1974 epoch and the later ones (1984,1994) if a smalltime delay is introduced in order to study the economy patterns seems also to indicate a tendency towardglobalization and a similar economic policy for all.3. Clustering analysisIn order to extract structures from the distance matrix, we can build country networks through filtering thetime delayed correlation matrix by removing the least correlated links, e.g. moving a threshold down. To doso, we define the filtered matrix DF , whereDFij ¼ 0 if Dij4f ,DFij ¼ 1 if Dijof . ð5ÞThe resulting matrix therefore connects nodes that are very close to each other, and removes the weak links.By decreasing the values of f, one observes therefore the breaking of the continent into several islands thatcorrespond to very connected ‘‘communities of countries’’. Let us stress that this method has already beensuccessfully applied in order to reveal structures in online communities [14]. Contrary to this previous study,the distance matrix may be asymmetric, i.e., when ta0. Consequently, the network representation will bemade of directed links from i to j, if the matrix element Dij verifies Dijof . This percolation idea-based methodreveals the emergence of connections, that are interestingly visualized by a branching representation. It seemsbest due to lack of space in these proceedings (other displays are available form the authors) to illustrate thestudy through the case t ¼ 1984, t ¼ 1 (Fig. 4); the four levels allow clusters to appear, but the figureobviously indicates correlations at odd with geography or trivial political economy minded expectations.4. ConclusionsIn this short paper, we have introduced techniques in order to study the development of structures in macro-economical systems. To do so, we have considered the GDP’s of the 23 most developed countries, and definedstatistical distances between their time evolution. A possible delay between economies has been accounted forthrough the parameter t, thereby allowing for some possible distinction between leading and followingcountries.ARTICLE IN PRESSFranceFranceIrelandIrelandU.K.SwitzerlandSwitzerlandFinlandNorwayNorwayDenmarkBelgium CanadaIcelandNewZealandNew ZealandU.S.ASpainNetherlandNetherlandAustriaSwedenPortugalItalyJapan AustraliaTurkeyLuxemburgLuxemburgGreeceBelgiumAustriaItalyU.K.Japan U.S.ASwedenIcelandSpainFinlandDenmarkPortugalCanadaAustraliaTurkey GreeceFranceDenmarkFinlandSwedenIcelandSpainU.K.New ZealandAustriaJapanItalyU.S.AAustriaU.S.AFinlandFranceDenmarkPortugalTurkeyAustraliaSwedenJapanCanadaItalyU.K.NorwayIrelandBelgiumNew Zealand Spain IcelandIrelandPortugalAustraliaNorwayGreece Luxemburg NetherlandSwitzerlandBelgium Canada GreeceSwitzerland TurkeyLuxemburg NetherlandFig. 4. Clusters resulting from filtering the distance matrix for t ¼ 1984 and t ¼ 1. The values of the filtering f are respectively 0:45, 0:20,0:15 and 0:1.M. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–2120In the above analysis, we have observed that the rank distribution of GDP fluctuation correlations isunexpectedly giving a law different from the classical Zipf power law (with an exponent¼ 1); even after timeaveraging the data contains much noise [15]. Nevertheless, patterns are found, under the form of clusters in thecountries’ network [15–17]. These are somewhat unexpected from geography. Therefore, this structuringseems to be in line with economy globalization, that tends to homogenize the economic development ofcountries; it may be related to the influence of political considerations.Clearly, other values of the time lag should be examined, and not so rich countries as well. There is noreason that the same time lag should be considered for different countries, in order to find some stronginfluence of one on another, if any. In fact one could look for higher order time delayed correlation functionsas well.ARTICLE IN PRESSM. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–21 21Finally, let us stress that there are many possible approaches, that could help the understanding ofglobalization processes in economy. A classical way in searching for clusters is to find subgraphs with a (high)clustering coefficient or to investigate the graph topology.AcknowledgmentsThis work is partially financially supported by ARC 02-07/293. RL would like also to thank financialsupport from the European Commission Project CREEN FP6-2003-NEST-Path-012864. Thanks to J.Miskiewicz for fruitful comments.References[1] J. Miskiewicz, M. Ausloos, Correlations between the most developed (G7) countries, A moving average window size optimization,Acta Phys. Pol. B 36 (2005) 2477–2486.[2] J. Miskiewicz, M. Ausloos, An attempt to observe economy globalization: the cross correlation distance evolution of the top 19GDP’s, Int. J. Mod. Phys. C 17 (2006) 317–332.[3] J. Miskiewicz, M. Ausloos, G7 country Gross Domestic Product (GDP) time correlations—A graph network analysis, in:H. Takayasu (Ed.), Practical Fruits of Econophysics, Springer, Tokyo, 2006, pp. 312–316.[4] J. Miskiewicz, M. Ausloos, Influence of information flow in the formation of economic cycles, in: M. Ausloos, M. Dirickx (Eds.),The Logistic Map and the Route to Chaos, Springer, Berlin, 2006, pp. 223–238.[5] M. Gligor, M. Ausloos, Eur Phys J B 56 (2007) 1434–6028.[6] M.A. Serrano, M. Boguna, Topology of the world trade web, Phys. Rev. E 68 (2003) 015101(R).[7] D. Garlaschelli, M.I. Loffredo, Fitness-dependent topological properties of the world trade web, Phys. Rev. Lett. 93 (2004) 188701.[8] D. Garlaschelli, T. Di Matteo, T. Aste, G. Caldarelli, M.I. Loffredo, Interplay between topology and dynamics in the world tradeweb, Proceedings of WEHIA 2006.[9] T. Di Matteo, T. Aste, How does the Eurodollar Interest Rate behave?, Int. J. Theor. Appl. Fin. 5 (2002) 107–122.[10] G. Fagiolo, Question raised at the end of the oral presentation.[11] J.B. Long, C.I. Plosser, Real business cycles, J. Polit. Econ. 91 (1983) 39–69.[12] S. Basu, A.M. Taylor, Business cycles in international historical perspective, J. Econom. Perspect. 13 (1999) 45–68.[13] P. Glansdorff, I. Prigogine, Structure, Stabilite´ et Fluctuations, Masson, Paris, 1971.[14] R. Lambiotte, M. Ausloos, Uncovering collective listening habits and music genres in bipartite networks, Phys. Rev. E 72 (2005)066107.[15] An anonymous referee commented that the noise inherent to such small sample invalidates any possible outcome of the analysis.Moreover temporal series of fluctuations in economy are usually autocorrelated, like when two such series are correlated with acommon process, e.g. traders in the stock market all influenced by the same inputs, such as news or index markets, or countries withinthe EU who follow common european policies. Under these conditions, it is easy to see that both series are cross correlated atdifferent time lags; this fact does not imply that one series is driving the other one. We quote In fact, one finds cross correlation forpositive and negative lags and therefore it is not possible to say who is influencing who. We do not disagree; in fact we claim that thenetworks generated in this paper might not reflect the real (if any) pattern of influences among countries. However the clustering canbe accepted if one accepts economy (empirical) theoreticians that globalization exists. In summary, we believe that our possibleinterpretation of the results is meaningful.[16] T. Mora, Evidencing European regional convergence clubs with optimal grouping criteria, Appl. Econ. Lett. 12 (2005) 937–940.[17] E.C. Angelini, F. Farina, The size of redistribution in OECD countries: does it influence wave inequality? Paper for InternationalConference in Memory of Two Eminent Social Scientists: C. Gini and M. O. Lorenz, Siena, Italy, 23–26 May, 2005.

Clusters or networks of economies?:A macroeconomy study through Gross Domestic Product

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Clusters or networks of economies? A macroeconomy study through Gross Domestic Product

RESEARCH OUTPUTS / RÉSULTATS DE RECHERCHEAuthor(s) - Auteur(s) :Publication date - Date de publication :Permanent link - Permalien :Rights / License - Licence de droit d’auteur :Bibliothèque Universitaire Moretus PlantinInstitutional Repository - Research PortalDépôt Institutionnel - Portail de la Rechercheresearchportal.unamur.beClusters or networks of economies?Ausloos, M.; Lambiotte, R.Published in:Physica A: Statistical Mechanics and its ApplicationsDOI:10.1016/j.physa.2007.02.005Publication date:2007Document VersionPublisher's PDF, also known as Version of recordLink to publicationCitation for pulished version (HARVARD):Ausloos, M & Lambiotte, R 2007, 'Clusters or networks of economies? A macroeconomy study through GrossDomestic Product' Physica A: Statistical Mechanics and its Applications, vol. 382, no. 1, pp. 16-21.https://doi.org/10.1016/j.physa.2007.02.005General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 21. May. 2019ARTICLE IN PRESS0378-4371/$ - sedoi:10.1016/j.phCorrespondE-mail addrPhysica A 382 (2007) 16–21www.elsevier.com/locate/physaClusters or networks of economies? A macroeconomy studythrough Gross Domestic ProductM. Ausloos, R. LambiotteGRAPES, Institute of Physics, B5, University of Lie´ge, B-4000 Lie´ge, Belgium, EurolandAvailable online 1 March 2007AbstractWe study correlations between web-downloaded gross domestic product (GDP)’s of rich countries. GDP is used aswealth signatures of the country economical state. We calculate the yearly ﬂuctuations of the GDP. We look for forwardand backward correlations between such ﬂuctuations. The correlation measure is based on the Theil index. The system isrepresented by an evolving weighted network, nodes being the GDP ﬂuctuations (or countries) at different times.In order to extract structures from the network, we focus on ﬁltering the time delayed correlations by removing the leastcorrelated links. This percolation idea-based method reveals the emergence of connections, that are visualized by abranching representation. Note that the network is made of weighted and directed links when taking into account a delaytime. Such a measure of collective habits does not readily ﬁt the usual expectations, except if an economy globalizationframework is accepted.r 2007 Published by Elsevier B.V.Keywords: Directed network; Time delay; Correlations; Theil index; Econophysics; GDP; Distance; Patterns1. IntroductionSome time ago, the question of economy globalization was studied along the lines of macro-econophysicsresearch [1–3]. Other considerations along related lines can be found in contributions by Miskiewicz [4] andGligor [5]. The question pertains to well-known observations: there are political, cultural, scientiﬁc, economiccooperations all over the world, even though there are even anti-globalization organizations.In our case, we consider that a globalization process in economy should be understood as an increase ofsimilarities within (macroeconomy) development patterns. In so doing we have proposed to study whether thepattern economy of countries can be represented by and discussed through the distance between each countrynormalized gross domestic product (GDP) ﬂuctuations. The notion of distance is somewhat arbitrary, as hasbeen already discussed in Ref. [1], but theoretical considerations are left for further investigation elsewhere.We have chosen to present here a study based on the statistical distance, in Section 2, for the 23 mostdeveloped countries for which annual GDP data is available in the interval [1950–2003]. Previous studies usinge front matter r 2007 Published by Elsevier B.V.ysa.2007.02.005ing author. Tel.: +32 4 366 37 52; fax: +324 366 29 90.esses: marcel.ausloos@ulg.ac.be (M. Ausloos), renaud.lambiotte@ulg.ac.be (R. Lambiotte).ARTICLE IN PRESSM. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–21 17Network analysis tools to understand the dynamics of GDP’s have been already published, for a few examplessee Refs. [6–9].One might argue [10] that some data detrending is necessary to take into account cycle-like effects [11,12]. Inprevious studies [1] it has also been observed that there is much noise in the data. We have shownthat some average should be taken over optimal time windows to remove such a noise. We will use foraveraging a time window of T ¼ 4 years which is a priori optimizing the statistics but is clearly related to mostelectoral periods.One usual political question is whether one has to follow another country policy in order to improve one’scountry economy. The question of followers and leaders can be tackled by considering time delayedcorrelations. In this ﬁrst attempt we will only consider a one year time lag t, being aware that other time lagsshould be involved. Note that positive or negative ones should be considered. Nevertheless it is worthwhile tocheck whether such an approach is already meaningful for t ¼ 1 and 1.The system can be displayed as an evolving network, nodes being the GDP ﬂuctuations or, in a short way,countries at different times. In Section 3, we extract structures from the network through ﬁltering the timedelayed correlations (or distance matrix) by removing the least correlated links. This percolation idea-basedmethod reveals the emergence of connections, that are visualized by a branching representation. However, thesystem is pretty unstable. Yet, expected features (connections) are reproduced, though some others aresurprising. This should imply some further consideration in the line of Glansdorff and Prigogine [13] aboutstructure and stability of world economic systems and the intrinsic role of ﬂuctuations. Notice also that thenetwork is much more complex when taking into account delay time. In such a case, the correlation matrix isNOT symmetrical, whence the network links are not only weighted but also directed. The number of distancesto be considered is also NðN  1Þ, where N is the number of countries, in contrast to NðN  1Þ=2, when t ¼ 0.In Section 4, we conclude that such a measure of collective habits does not ﬁt the usual expectations, deﬁnedby politicians, but that does not imply that they are wrong or right, nor are we.We stress that the following relevant time instances must be considered, i.e.,1. Initial observation time t0,2. Data acquisition time t,3. Increment or ﬂuctuation time span D,4. Delay time (between two countries) t,5. Window observation (averaging) time T , i.e., [tmin; tmax],6. Conclusion time tN .Thus the general correlation matrix can be of quite high dimension (i; j; t0; t; t; ..).2. MethodologyWe use data from http://www.ggdc.net/index-series.html for recording the Normalized GDP of the 23 mostdeveloped countries. Most of these countries have natural time scales associated to the time lags betweenelections. Whence we use an average time of T ¼ 4 years. For each of the countries, let us deﬁne pi;t ¼lnGi;tþT=Gi;t that is equivalent to the average variation of the GDP over T years.Let us now introduce a correlation measure and associated distance measure for the countries. Thecorrelation measure is based on the Theil index and is deﬁned as follows:Ci;j;t;t ¼hpi;t0pj;t0þtit  hpi;t0 ithpj;t0þtitﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃhp2i;t0  hpi;t0 i2ithp2j;t0þt  hpj;t0þti2itq , (1)where the averages are deﬁned as follows:hF ðt0Þit ¼1DXtþDt0¼tF ðt0Þ. (2)ARTICLE IN PRESSM. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–2118The ‘‘averaging time’’ D has been hereby chosen to be D ¼ 4 in the following. The distance is obtained fromthe correlation from the deﬁnition (dropping indices here):d ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ12ð1 CÞq. (3)Let us stress that this short hand writing accounts for time delays between the signals. Let us also note thatother measures of distance between different countries can be deﬁned, e.g.Di;j;t;t ¼ jpi;t  pj;tþtj. (4)The comparison of results obtained for different measures will be considered elsewhere.When t ¼ 0, the above quantities are at ﬁxed time. When ta0, these quantities measure the correlationswith a delay time between the economies of countries. If the evolutions were completely identical, one wouldﬁnd D ¼ 0. Let us stress again that Di;j;t;t is symmetric in i; j, when t ¼ 0. For each pair of countries, thisquantity deﬁnes therefore a surface whose statistical properties can be studied.The ﬁrst relevant quantity to report is the rank analysis of Di;j;t;t, at different values of t (1974, 1984, 1994)and t ¼ 0; 1;1 (Figs. 1–3), like on a Zipf plot. Notice the non-trivial behavior, with various crossing of thecurves, on one hand; moreover, the data does not seem to lead to a ﬁne straight line ﬁt. Because the GDPincrement distance between countries is rather uniform, the results suggest non-trivial correlations in evolving00.10.20.30.40.50.60.70.80.910 50 100 150 200 250 300Drank197419841994Fig. 1. Rank analysis of Di;j;t;t for 1974, 1984, 1994 years when t ¼ 0.00.10.20.30.40.50.60.70.80.910 100 200 300 400 500 600Drank197419841994Fig. 2. Rank analysis of Di;j;t;t for 1974, 1984, 1994 years, when t ¼ 1.ARTICLE IN PRESS00.10.20.30.40.50.60.70.80.910 100 200 300 400 500 600Drank197419841994Fig. 3. Rank analysis of Di;j;t;t for 1974, 1984, 1994 years, when t ¼ 1.M. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–21 19economy schemes, which likely result from complex interactions of country economies. A globalizationframework is hardly proved from the above. However, it is clear that the ranking pattern does not changeduring the last part of the century, when t ¼ 0, indicating a rather common economy philosophy and similarresults for all countries. The drastic change between the 1974 epoch and the later ones (1984,1994) if a smalltime delay is introduced in order to study the economy patterns seems also to indicate a tendency towardglobalization and a similar economic policy for all.3. Clustering analysisIn order to extract structures from the distance matrix, we can build country networks through ﬁltering thetime delayed correlation matrix by removing the least correlated links, e.g. moving a threshold down. To doso, we deﬁne the ﬁltered matrix DF , whereDFij ¼ 0 if Dij4f ,DFij ¼ 1 if Dijof . ð5ÞThe resulting matrix therefore connects nodes that are very close to each other, and removes the weak links.By decreasing the values of f, one observes therefore the breaking of the continent into several islands thatcorrespond to very connected ‘‘communities of countries’’. Let us stress that this method has already beensuccessfully applied in order to reveal structures in online communities [14]. Contrary to this previous study,the distance matrix may be asymmetric, i.e., when ta0. Consequently, the network representation will bemade of directed links from i to j, if the matrix element Dij veriﬁes Dijof . This percolation idea-based methodreveals the emergence of connections, that are interestingly visualized by a branching representation. It seemsbest due to lack of space in these proceedings (other displays are available form the authors) to illustrate thestudy through the case t ¼ 1984, t ¼ 1 (Fig. 4); the four levels allow clusters to appear, but the ﬁgureobviously indicates correlations at odd with geography or trivial political economy minded expectations.4. ConclusionsIn this short paper, we have introduced techniques in order to study the development of structures in macro-economical systems. To do so, we have considered the GDP’s of the 23 most developed countries, and deﬁnedstatistical distances between their time evolution. A possible delay between economies has been accounted forthrough the parameter t, thereby allowing for some possible distinction between leading and followingcountries.ARTICLE IN PRESSFranceFranceIrelandIrelandU.K.SwitzerlandSwitzerlandFinlandNorwayNorwayDenmarkBelgium CanadaIcelandNewZealandNew ZealandU.S.ASpainNetherlandNetherlandAustriaSwedenPortugalItalyJapan AustraliaTurkeyLuxemburgLuxemburgGreeceBelgiumAustriaItalyU.K.Japan U.S.ASwedenIcelandSpainFinlandDenmarkPortugalCanadaAustraliaTurkey GreeceFranceDenmarkFinlandSwedenIcelandSpainU.K.New ZealandAustriaJapanItalyU.S.AAustriaU.S.AFinlandFranceDenmarkPortugalTurkeyAustraliaSwedenJapanCanadaItalyU.K.NorwayIrelandBelgiumNew Zealand Spain IcelandIrelandPortugalAustraliaNorwayGreece Luxemburg NetherlandSwitzerlandBelgium Canada GreeceSwitzerland TurkeyLuxemburg NetherlandFig. 4. Clusters resulting from ﬁltering the distance matrix for t ¼ 1984 and t ¼ 1. The values of the ﬁltering f are respectively 0:45, 0:20,0:15 and 0:1.M. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–2120In the above analysis, we have observed that the rank distribution of GDP fluctuation correlations isunexpectedly giving a law different from the classical Zipf power law (with an exponent¼ 1); even after timeaveraging the data contains much noise [15]. Nevertheless, patterns are found, under the form of clusters in thecountries’ network [15–17]. These are somewhat unexpected from geography. Therefore, this structuringseems to be in line with economy globalization, that tends to homogenize the economic development ofcountries; it may be related to the inﬂuence of political considerations.Clearly, other values of the time lag should be examined, and not so rich countries as well. There is noreason that the same time lag should be considered for different countries, in order to ﬁnd some stronginﬂuence of one on another, if any. In fact one could look for higher order time delayed correlation functionsas well.ARTICLE IN PRESSM. Ausloos, R. Lambiotte / Physica A 382 (2007) 16–21 21Finally, let us stress that there are many possible approaches, that could help the understanding ofglobalization processes in economy. A classical way in searching for clusters is to ﬁnd subgraphs with a (high)clustering coefﬁcient or to investigate the graph topology.AcknowledgmentsThis work is partially ﬁnancially supported by ARC 02-07/293. RL would like also to thank ﬁnancialsupport from the European Commission Project CREEN FP6-2003-NEST-Path-012864. Thanks to J.Miskiewicz for fruitful comments.References[1] J. Miskiewicz, M. Ausloos, Correlations between the most developed (G7) countries, A moving average window size optimization,Acta Phys. Pol. B 36 (2005) 2477–2486.[2] J. Miskiewicz, M. Ausloos, An attempt to observe economy globalization: the cross correlation distance evolution of the top 19GDP’s, Int. J. Mod. Phys. C 17 (2006) 317–332.[3] J. Miskiewicz, M. Ausloos, G7 country Gross Domestic Product (GDP) time correlations—A graph network analysis, in:H. Takayasu (Ed.), Practical Fruits of Econophysics, Springer, Tokyo, 2006, pp. 312–316.[4] J. Miskiewicz, M. Ausloos, Inﬂuence of information ﬂow in the formation of economic cycles, in: M. Ausloos, M. Dirickx (Eds.),The Logistic Map and the Route to Chaos, Springer, Berlin, 2006, pp. 223–238.[5] M. Gligor, M. Ausloos, Eur Phys J B 56 (2007) 1434–6028.[6] M.A. Serrano, M. Boguna, Topology of the world trade web, Phys. Rev. E 68 (2003) 015101(R).[7] D. Garlaschelli, M.I. Loffredo, Fitness-dependent topological properties of the world trade web, Phys. Rev. Lett. 93 (2004) 188701.[8] D. Garlaschelli, T. Di Matteo, T. Aste, G. Caldarelli, M.I. Loffredo, Interplay between topology and dynamics in the world tradeweb, Proceedings of WEHIA 2006.[9] T. Di Matteo, T. Aste, How does the Eurodollar Interest Rate behave?, Int. J. Theor. Appl. Fin. 5 (2002) 107–122.[10] G. Fagiolo, Question raised at the end of the oral presentation.[11] J.B. Long, C.I. Plosser, Real business cycles, J. Polit. Econ. 91 (1983) 39–69.[12] S. Basu, A.M. Taylor, Business cycles in international historical perspective, J. Econom. Perspect. 13 (1999) 45–68.[13] P. Glansdorff, I. Prigogine, Structure, Stabilite´ et Fluctuations, Masson, Paris, 1971.[14] R. Lambiotte, M. Ausloos, Uncovering collective listening habits and music genres in bipartite networks, Phys. Rev. E 72 (2005)066107.[15] An anonymous referee commented that the noise inherent to such small sample invalidates any possible outcome of the analysis.Moreover temporal series of ﬂuctuations in economy are usually autocorrelated, like when two such series are correlated with acommon process, e.g. traders in the stock market all inﬂuenced by the same inputs, such as news or index markets, or countries withinthe EU who follow common european policies. Under these conditions, it is easy to see that both series are cross correlated atdifferent time lags; this fact does not imply that one series is driving the other one. We quote In fact, one finds cross correlation forpositive and negative lags and therefore it is not possible to say who is influencing who. We do not disagree; in fact we claim that thenetworks generated in this paper might not reﬂect the real (if any) pattern of inﬂuences among countries. However the clustering canbe accepted if one accepts economy (empirical) theoreticians that globalization exists. In summary, we believe that our possibleinterpretation of the results is meaningful.[16] T. Mora, Evidencing European regional convergence clubs with optimal grouping criteria, Appl. Econ. Lett. 12 (2005) 937–940.[17] E.C. Angelini, F. Farina, The size of redistribution in OECD countries: does it inﬂuence wave inequality? Paper for InternationalConference in Memory of Two Eminent Social Scientists: C. Gini and M. O. Lorenz, Siena, Italy, 23–26 May, 2005.

peer reviewedWe study correlations between web-downloaded gross domestic product (GDP)'s of rich countries. GDP is used as wealth signatures of the country economical state. We calculate the yearly fluctuations of the GDP. We look for forward and backward correlations between such fluctuations. The correlation measure is based on the Theil index. The system is represented by an evolving weighted network, nodes being the GDP fluctuations (or countries) at different times. In order to extract structures from the network, we focus on filtering the time delayed correlations by removing the least correlated links. This percolation idea-based method reveals the emergence of connections, that are visualized by a branching representation. Note that the network is made of weighted and directed links when taking into account a delay time. Such a measure of collective habits does not readily fit the usual expectations, except if an economy globalization framework is accepted. (C) 2007 Published by Elsevier B.V

Ausloos, Marcel

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Clusters or networks of economies? A macroeconomy study through gross domestic product