PHYSICAL REVIEW E 79, 365 29 World-rade web: Topological properies, dynamics, and evoluion Giorgio Fagiolo* Laboraory of Economics and Managemen, San Anna School of Advanced Sudies, Piazza Mariri della Liberà 33, I-5627 Pisa, Ialy Javier Reyes Deparmen of Economics, Sam M. Walon College of Business, Universiy of Arkansas, Fayeeville, Arkansas 727-2, USA Sefano Schiavo Deparmen of Economics and School of Inernaional Sudies, Universiy of Treno, Via Inama 5, 38 Treno, Ialy Received 3 July 28; revised manuscrip received December 28; published 27 March 29 This paper sudies he saisical properies of he web of impor-expor relaionships among world counries using a weighed-nework approach. We analyze how he disribuions of he mos imporan nework saisics measuring conneciviy, assoraiviy, clusering, and cenraliy have coevolved over ime. We show ha all node-saisic disribuions and heir correlaion srucure have remained surprisingly sable in he las 2 years and are likely o do so in he fuure. Conversely, he disribuion of posiive link weighs is slowly moving from a log-normal densiy owards a power law. We also characerize he auoregressive properies of nework-saisics dynamics. We find ha nework-saisics growh raes are well-proxied by fa-ailed densiies like he Laplace or he asymmeric exponenial power. Finally, we find ha all our resuls are reasonably robus o a few alernaive, economically meaningful, weighing schemes. DOI:.3/PhysRevE.79.365 PACS numbers: 89.75.k, 89.65.Gh, 87.23.Ge, 5.7.Ln I. INTRODUCTION In he las decade, a lo of effor has been devoed o he empirical exploraion of he archiecure of he world-rade web WTW from a complex-nework perspecive 2. The WTW, also known as inernaional rade nework ITN, is defined as he nework of impor-expor relaionships beween world counries in a given year. Undersanding he opological properies of he WTW, and heir evoluion over ime, acquires a fundamenal imporance in explaining inernaional-rade issues such as economic globalizaion and inernaionalizaion 3,4. Indeed, i is common wisdom ha rade linkages are one of he mos imporan channels of ineracion beween world counries 5. For example, hey can help o explain how economic policies affec foreign markes 6; how economic shocks are ransmied among counries 7; and how economic crises spread inernaionally 8. However, direc bilaeral-rade relaionships can only explain a small fracion of he impac ha an economic shock originaing in a given counry can have on anoher one, which is no among is direc-rade parners 9. Therefore, a complex-nework analysis 2 23 of he WTW, by characerizing in deail he opological srucure of he nework, can go far beyond he scope of sandard inernaionalrade indicaors, which insead only accoun for bilaeralrade direc linkages 7. The firs sream of conribuions ha have sudied he properies of he WTW has employed a binary-nework analysis, where a possibly direced link beween any wo counries is eiher presen or no according o wheher he *FAX: 39-5-883343; giorgio.fagiolo@sssup.i jreyes@walon.uark.edu sefano.schiavo@ofce.sciences-po.fr rade flow ha i carries is larger han a given lower hreshold 2 4. According o hese sudies, he WTW urns ou o be characerized by a high densiy and a righ-skewed bu no exacly power-law disribuion for he number of parners of a given counry i.e., he node degree. Furhermore, here seems o be evidence of bimodaliy in he node-degree disribuion. While he majoriy of counries enerain few rade parnerships, here exiss a group of counries rading wih almos everyone else,2. Also, he binary WTW is a very disassoraive nework i.e., counries holding many rade parners are on average conneced wih counries holding few parners and is characerized by some hierarchical arrangemens i.e., parners of well-conneced counries are less inerconneced among hem han hose of poorly conneced ones. Remarkably, hese properies are quie sable over ime 4. More recenly, a few conribuions have adoped a weighed-nework approach 24 26 o he sudy of he WTW, where each link is weighed by some proxy of he rade inensiy ha i carries. The moivaion is ha a binary approach canno fully exrac he wealh of informaion abou he rade inensiy flowing hrough each link and herefore migh dramaically underesimae he role of heerogeneiy in rade linkages. Indeed, Refs. 9,,2 show ha he saisical properies of he WTW viewed as a weighed nework crucially differ from hose exhibied by is unweighed counerpar. For example, he weighed version of he WTW appears o be weakly disassoraive. Moreover, wellconneced counries end o rade wih parners ha are srongly conneced beween hem. Finally, he disribuion of he oal rade inensiy carried by each counry i.e., node srengh is righ-skewed, indicaing ha a few inense rade connecions co-exis wih a majoriy of low-inensiy ones. This is confirmed, a he link level, by Refs. 6,7 who find ha he disribuion of link weighs can be approximaed by a log-normal densiy robusly across he years. The main 539-3755/29/793/3659 365-29 The American Physical Sociey
FAGIOLO, REYES, AND SCHIAVO insigh coming from hese sudies is ha a weighed-nework analysis is able o provide a more complee and ruhful picure of he WTW han a binary one 2. Addiional conribuions have insead focused on specific feaures of he srucure and dynamics of he WTW. For example, Refs. 3,8 find evidence in favor of a hidden-variable model, according o which he opological properies of he WTW in boh he binary and weighed case can be well explained by a single node-characerisic i.e., counry grossdomesic produc conrolling for he poenial abiliy of a node o be conneced. Furhermore, Ref. 5 sudies he weighed nework of bilaeral rade imbalances 7. The auhors noe ha also he inernaional rade-imbalance nework is characerized by a high level of heerogeneiy: For each counry, he profile of rade fluxes is unevenly disribued across parners. A he nework level, his promps o he presence of high-flux backbones, i.e., sparse subneworks of conneced rade fluxes carrying mos of he overall rade in he nework. The auhors hen develop a mehod o exrac for any significance level he flux backbone exising among counries and links. This urns ou o be exremely effecive in soring ou he mos relevan par of he rade-imbalance nework and can be convenienly used for visualizaion purposes. Finally, Ref. considers he formaion of rade islands, ha is conneced componens carrying a oal rade flow larger han some given hresholds. The analysis of he evoluion of he WTW communiy srucure 27 finds mixed evidence for globalizaion. In his paper we presen a more horough sudy of he opological properies of he WTW by focusing on disribuion dynamics and evoluion. More specifically, following he insighs of Ref. 2, we employ a weighed nework approach o characerize, for he period 98 2, he disribuion of he mos imporan nework saisics measuring node conneciviy, assoraiviy, clusering, and cenraliy; as well as link weighs. We ask hree main ypes of quesions: i Have and, if so, how he disribuional properies of hese saisics and heir correlaion srucure been changing wihin he sample period considered? ii Can we make any predicion on he ou-of-sample evoluion of such disribuions? iii Do he answers o he previous quesions change if we play wih a number of alernaive, economically meaningful weighing schemes i.e., if we allow for differen rules o weigh exising links? The res of he paper is organized as follows. Secion II presens he daa ses and defines he saisics sudied in he paper. Secion III inroduces he main resuls. Finally, Sec. IV concludes and discusses fuure exensions. II. DATA AND DEFINITIONS We employ inernaional-rade daa provided by 28 o build a ime-sequence of weighed direced neworks. Our balanced panel refers o T=2 years 98 2 and N = 59 counries. For each counry and year, daa repor rade flows in curren U.S. dollars. To build adjacency and weigh marices, we followed he flow of goods. This means ha rows represen exporing counries, whereas columns sand for imporing counries. We define a rade relaionship by PHYSICAL REVIEW E 79, 365 29 seing he generic enry of he binary adjacency marix ã ij = if and only if expors from counry i o counry j e ij are sricly posiive in year. Following Refs.,6 8, he weigh of a link from i o j in year is defined as w ij =e ij 72. Thus, he sequence of N N adjacency and weigh marices Ã,W, =98,...,2 fully describes he wihin-sample dynamics of he WTW. A preliminary saisical analysis of boh binary and weighed marices suggess ha Ã,W are sufficienly symmeric o jusify an undireced analysis for all. From a binary perspecive, on average abou 93% of WTW direced links are reciprocaed in each given year. This means ha, almos always, if counry i s expors o counry j are posiive ã ij =, hen ã ji =, i.e., counry j s expors o counry i are also posiive. To check more formally his evidence from a weighed perspecive, we have compued he weighed symmery index defined in Ref. 29. The index ranges in he sample period beween.7 and.43, signalling a relaively srong and sable symmery of WTW weigh marices 73. We have herefore symmerized he nework by defining he enries of he new adjacency marix A so ha a ij = if and only if eiher ã ij = or ã ji =, and zero oherwise. Accordingly, he generic enry of he new weigh marix W is defined as w ij = 2 w ij+w ji. This means ha he symmerized weigh of link ij is proporional o he oal rade impors plus expors flowing hrough ha link in a given year. Finally, in order o have w ij, for all i, j and, we have renormalized all enries in W by heir maximum value w N * =max i,j= w ij. For each Ã,W, we sudy he disribuions of he following node saisics: i Node degree 2,3, defined as ND i =A i, where A i is he ih row of A and is a uniary vecor. ND is a measure of binary conneciviy, as i couns he number of rade parners of any given node. Alhough we mainly focus here on a weighed-nework approach, we sudy ND because of is naural inerpreaion in erms of number of rade parnerships and bilaeral rade agreemens. ii Node srengh 3, defined as NS i =W i, where again W i is he ih row of W. While ND ells us how many parners a node holds, NS is a measure of weighed conneciviy, as i gives us an idea of how inense exising rade relaionships of counry i are. iii Node average neares-neighbor srengh 3, ha is ANNS i =A i W /A i. ANNS measures how inense are rade relaionships mainained by he parners of a given node. Therefore, he correlaion beween ANNS and NS is a measure of nework assoraiviy if posiive or disassoraiviy if negaive. I is easy o see ha ANNS boils down o average neares-neighbor degree ANND if W is replaced by A. iv Weighed clusering coefficien 9,32, defined as WCC i =W /3 3 ii /ND i ND i. Here Z 3 ii is he ih enry on he main diagonal of Z Z Z and Z /3 sands for he marix obained from Z afer raising each enry o /3. WCC measures how much clusered a node i is from a weighed perspecive, i.e., how much inense are he linkages of rade 365-2
WORLD-TRADE WEB: TOPOLOGICAL PROPERTIES, PHYSICAL REVIEW E 79, 365 29 3 x 3 4 x 3 2.5 2 Mean 2.5 Sd Dev 8 (a) 6 (b).5 98 985 99 995 2 35 4 98 985 99 995 2 2 3 25 8 Skewness 2 Kurosis 6 5 4 (c) 2 (d) 5 98 985 99 995 2 98 985 99 995 2 FIG.. Firs four sample momens of he link-weigh disribuion vs years. Panels a mean; b sandard deviaion; c skewness; d kurosis. riangles having counry i as a verex 74. Again, replacing W wih A, one obains he sandard binary clusering coefficien BCC, which couns he fracion of riangles exising in he neighborhood of any give node 33. v Random-walk beweenness cenraliy 34,35, which is a measure of how much a given counry is globally cenral in he WTW. A node has a higher random-walk beweenness cenraliy RWBC he more i has a posiion of sraegic significance in he overall srucure of he nework. In oher words, RWBC is he exension of node beweenness cenraliy 36 o weighed neworks and measures he probabiliy ha a random signal can find is way hrough he nework and reach he arge node where he links o follow are chosen wih a probabiliy proporional o heir weighs. The above saisics allow one o address he sudy of node characerisics in erms of four dimensions: conneciviy ND and NS, assoraiviy ANND and ANNS, when correlaed wih ND and NS, clusering BCC and WCC and cenraliy RWBC. In wha follows, we will mainly concenrae he analysis on ND and he oher weighed saisics NS, ANNS, WCC, RWBC, bu we occasionally discuss, when necessary, also he behavior of ANND and BCC. We furher explore he nework-conneciviy dimension by sudying he ime-evoluion of he link-weigh disribuion w =w ij,i j=,...,n. In paricular, we are ineresed in assessing he fracion of links ha are zero in a given year and becomes posiive in year +, =,2,... and he percenage of links ha are sricly posiive a and disappear in year +. This allows one o keep rack of rade relaionships ha emerge or become exinc during he sample period 75. III. RESULTS A. Shape, momens, and correlaion srucure of nework saisics We begin by sudying he shape of he disribuions of node and link saisics and heir dynamics wihin he sample period under analysis. As already found in Refs. 6,7, link weigh disribuions display relaively sable momens see Fig. and are well proxied by log-normal densiies in each year cf. Fig. 2 for an example. This means ha he majoriy of rade linkages are relaively weak and coexis wih few high-inensiy rade parnerships. The fac ha he firs four momens of he disribuion do no display remarkable srucural changes in he sample period hins o a relaively srong sabiliy of he underlying disribuional shapes 76. We shall sudy his issue in more deails below. A similar sable paern is deeced also for he momens of he disribuions of all node saisics under analysis, see Fig. 3 for he case of NS disribuions. To see ha his applies in general for node saisics, we have compued he ime average across 9 observaions of he absolue value of -year growh raes of he firs four ineresing momens of ND, NS, ANNS, WCC, and RWBC saisics, namely mean, sandard deviaion, skewness, and kurosis 77. Table 365-3
FAGIOLO, REYES, AND SCHIAVO ln(rank) 9 8 7 6 5 4 3 2 Empirical Log Normal Fi 5 5 ln(w) FIG. 2. Size-rank log-log plo of he link-weigh disribuion in year 2. X axis: Naural log of link weigh w. Y axis: Naural log of he rank of link-weigh observaion w. PHYSICAL REVIEW E 79, 365 29 TABLE I. Average over ime of absolue-valued -year growh raes of he firs four momens of node saisics. Given he value of he node saisic X a ime for counry i X i and E k he momen operaor ha for k=,2,3,4 reurns, respecively, he mean, sandard deviaion, skewness, and kurosis, he ime average of absolue-valued -year growh raes of he kh momen saisic is defined as T =2 T E k X i /E k X i. Average Absolue Growh Raes Mean Sandard Deviaion Skewness Kurosis ND.43.47.375.86 ANND.79.279.6.26 BCC.43.97.65.25 NS.379.42.263.37 ANNS.479.52.223.452 WCC.544.97.274.543 RWBC.49.7.25.556 I shows ha hese average absolue growh raes range in our sample beween.43 and.65, hus indicaing ha he shape of hese disribuions seem o be quie sable over ime. Bu wha does he shape of node- and link-saisic disribuions look like? To invesigae his issue we have begun by running normaliy ess on he naural logs of posiivevalued node and link saisics. As Table II suggess 78, binary-nework saisic disribuions are never log normal i.e., heir naural logs are never normal, whereas all weighed-nework saisics bu RWBC seem o be well proxied by log-normal densiies. To see why his happens, Fig. 4 shows he rank-size plo of ND in 2 wih a kernel densiy esimae in he inse 79. I is easy o see ha ND exhibis some bimodaliy, wih he majoriy of counries feauring low degrees and a bunch of counries rading wih almos everyone else. Figure 5 shows insead, for year 2, how NS is nicely proxied by a log-normal disribuion. This is no so for RWBC, whose disribuion seems insead power law in all years, wih slopes oscillaing around.5, see Figs. 6 and 7. Therefore, being more cenral is more likely.4.6.35.3.25.5.4 Mean.2.5..5 (a) Sd Dev.3.2. (b) 98 985 99 995 2 98 985 99 995 2 5 8 4 Skewness 6 4 Kurosis 3 2 2 (c) (d) 98 985 99 995 2 98 985 99 995 2 FIG. 3. Sample momens of node srengh NS disribuion vs years. Panels a mean; b sandard deviaion; c skewness; d kurosis. 365-4
WORLD-TRADE WEB: TOPOLOGICAL PROPERTIES, PHYSICAL REVIEW E 79, 365 29 TABLE II. P values for Jarque-Bera normaliy es 55,56. Null hypohesis: Naural logs of posiive-valued disribuion are normally disribued wih unknown parameers. Aserisks: null hypohesis rejeced a %; null hypohesis rejeced a 5%; null hypohesis rejeced a %. 98 982 983 984 985 986 987 988 989 99 ND.***.***.***.***.***.***.***.***.***.*** ANND.277**.26**.4**.43**.525*.39**.333**.97**.254.2 BCC.6***.4***.4***.4***.5***.2***.5***.3***.8***.4*** NS.2925.246.42.287.4344.684.6238.53.3496.5343 ANNS.8.25.2724.2463.286.2532.3243.633.666.65 WCC.5673.2525.282.2874.2867.26.3564.235.25.22 RWBC.***.***.***.***.***.***.***.***.***.*** 99 992 993 994 995 996 997 998 999 2 ND.***.2***.***.***.2***.***.***.***.***.*** ANND.8*.63*.9*.7*.58*.4**.48**.46**.4**.26** BCC.9***.4***.4***.2***.2***.6***.5***.5***.5***.5*** NS.5367.2398.297.26.3685.4693.6.632.598.526 ANNS.245.95*.42.33.269.574*.734*.899*.668*.385 WCC.266.66.562.356.358.638*.26.95.72.583 RWBC.***.***.***.***.***.***.***.***.***.*** han having high NS, ANNS, or WCC i.e., he laer disribuions feaure upper ails hinner han ha of RWBC disribuions; we shall reurn o complexiy issues relaed o his poin when discussing ou-of-sample evoluion of he disribuions of node and link saisics. The foregoing qualiaive saemens can be made quaniaive by running comparaive goodness-of-fi GoF ess o check wheher he disribuions under sudy come from predefined densiy families. To do so, we have run Kolmogorov-Smirnov GoF ess 57,58 agains hree null hypoheses, namely ha our daa can be well described by log-normal, sreched exponenial, or power-law disribuions. The sreched-exponenial disribuion SED has been employed because of is abiliy o saisfacorily describe he ail behavior of many real-world variables and nework-relaed measures 59,6. Table III repors resuls ln(rank) 5.5 5 4.5 4 3.5 3 2.5 2.5 Kernel Densiy..5 Node Degree Empirical LogNormal 3.5 4 4.5 5 ln(nd) FIG. 4. Size-rank log-log plo of node-degree disribuion in year 2. X axis: Naural log of node degree ND. Y axis: Naural log of he rank of node-degree observaion ND. Inse: Kernel densiy esimae of ND disribuion. for year 2 in order o faciliae a comparison wih Figs. 4 6, bu again he main insighs are confirmed in he enire sample. I is easy o see ha he SED does no successfully describe he disribuions of our main indicaors. On he conrary, i clearly emerges ha NS, ANNS, and RWBC seem o be well described by log-normal densiies, whereas he null of power-law RWBC canno be rejeced. For ND, neiher of he hree null appears o be a saisfacory hypohesis for he KS es. We now discuss in more deail he evoluion over ime of he momens of he disribuions of node saisics. As already noed in Refs. 3,6,7,2, he binary WTW is characerized by an exremely high nework densiy d = NN i j a ij, ranging from.5385 o.644. Figure 8 plos he normalized by N populaion average of ND, which is equal o nework densiy up o a N N facor, ogeher wih populaion average of NS. While he average ln(rank) 5 4.5 4 3.5 3 2.5 2.5.5 Empirical LogNormal 2 8 6 4 2 2 ln(ns) FIG. 5. Size-rank log-log plo of node-srengh disribuion in year 2. X axis: Naural log of node srengh NS. Y axis: Naural log of he rank of node-srengh observaion NS. 365-5
FAGIOLO, REYES, AND SCHIAVO ln(rank) 6 5 4 3 2 Empirical Power Law Fi y =.285.6x 5 4 3 2 ln(rwbc) FIG. 6. Size-rank log-log plo of node random-walk beweenness cenraliy RWBC disribuion in year 2. Solid line: Power-law fi equaion of he regression line in he inse. X axis: Naural log of RWBC. Y axis: Naural log of he rank of RWBC observaion. number of rade parnerships is very high and slighly increases over he years, heir average inensiy is raher low a leas as compared o NS conceivable range, i.e.,,n and ends o decline 8. As far as ANND/ANNS, clusering and cenraliy are concerned, a more meaningful saisical assessmen of he acual magniude of empirical populaion-average saisics requires comparing hem wih expeced values compued afer reshuffling links and/or weighs. In wha follows, we consider wo reshuffling schemes RSs. For binary saisics, we compue expeced values afer reshuffling exising links by keeping fixed he observed densiy d hereafer, B-RS. For weighed ones, we keep fixed he observed adjacency marix A and redisribue weighs a random by reshuffling he empirical link-weigh disribuion w =w ij,i j=,...,n hereafer, W-RS 8. Figure 9 shows empirical averages vs expeced values over ime. Noice ha empirical averages of ANND, BCC, ANNS, WCC, and RWBC are larger han expeced, meaning ha he WTW is on average more clusered; feaures a larger nearesneighbor conneciviy, and counries are on average more cenral han expeced in comparable random graphs. The relaively high clusering level deeced in he WTW hins o a nework archiecure ha, especially in he binary case, feaures a peculiar clique srucure. To furher explore he clique srucure of he WTW we compued, in he binary case, he node k-clique degree NkCD saisic 62. The NkCD for node i is defined as he number of k-size fully conneced subgraphs conaining i. Since NkCD for k =2 equals node degree, and for k=3 is closely relaed o he binary clusering coefficien, exploring he properies of NkCD for k3 can ell us somehing abou higher-order clique srucure of he WTW. Figure repors for year 2 an example of he cumulaive disribuion funcion CDF of NkCDs for k=4,5 very similar resuls hold also for he case k=6. In he inses, a kernel esimae of he corresponding probabiliy disribuions are also provided. We also compare empirical CDFs wih heir expeced shape in random nes where, his ime, links are reshuffled so as o preserve he iniial degree sequence i.e., we employ he edge-crossing algorihm, cf. Refs. 62,63 for deails. The plos indicae ha he majoriy of nodes in he WTW are involved in a very large number of higher-degree cliques, bu ha he observed paern is no ha far from wha would have been expeced in neworks wih he same degree sequence if any, observed NkCDs place relaively more mass on smaller NkCDs and less on medium-large values of NkCDs. Noice ha hese findings are a odds wih wha is commonly observed in many oher real-world neworks, where NkCD disribuions are ypically power law. However, his peculiar feaure of he WTW does no come as a surprise, given is very high average degree, and he large number of counries rading wih almos everyone else in he sample. Noe also ha he high conneciviy of he binary WTW makes i very expensive o compue NkCD disribuions for k6. In order o shed some ligh on he clique srucure for larger values of k, we have employed sandard search algorihms o find all Luce and Perry LP k cliques 64, ha is maximally conneced k-size subgraphs 82. Noice ha in general he WTW does no exhibi LP cliques wih size smaller han 2 his is rue in any year. Furhermore, a large number of LP cliques are of a size beween 5 and 6 see Fig., lef-hand panel, for a kernel-densiy esimae of LP clique size disribuion in 2. Hence, he binary WTW, due o is exremely dense conneciviy paern, seems o display a very inricae clique srucure. Addiional suppor o his conclusion is provided by Fig. righ-hand panel, where we plo a kernel-densiy esimae of he disribuion of he number of agens belonging o LP cliques of any size. Alhough a large majoriy of counries belong o less han 2 LP cliques, a second peak in he upper-ail of he disribuion emerges, indicaing ha a nonnegligible number of nodes are acually involved in a leas differen LP cliques of any size greaer han 2. B. Correlaion srucure and node characerisics To furher explore he opological properies of he WTW, we urn now o examine he correlaion srucure exising beween binary- and weighed-nework saisics 83. As expeced 2 4, Figure 2 shows ha he binary version of he WTW is srongly disassoraive in he enire sample period. Furhermore, counries holding many rade parners do no.3 Exponen.2. PHYSICAL REVIEW E 79, 365 29.95 Exponen Adjused R 2 98 985 99 995.9 2 s FIG. 7. Lef-hand side Y-axis scale: Esimaed power-law exponen for node random-walk beweenness cenraliy RWBC disribuions vs years. Righ-hand side Y-axis scale: Adjused R 2 associaed o he power-law fi. Adjused R 2 365-6
WORLD-TRADE WEB: TOPOLOGICAL PROPERTIES, PHYSICAL REVIEW E 79, 365 29 TABLE III. Kolmogorov-Smirnov goodness-of-fi es resuls for year 2 disribuions. The hree null hypoheses esed are ha he observed disribuions come from, respecively, log-normal, sreched exponenial, or power-law pdfs. Aserisks: null hypohesis rejeced a %; null hypohesis rejeced a 5%; null hypohesis rejeced a %. Log normal Sreched exponenial Power law KS Tes Saisic p value Saisic p value Saisic p value ND.262.4**.5326.***.485.*** NS.647.559.3563.***.2862.*** ANNS.9.35.54.***.336.6*** WCC.528.7658.6493.***.2569.*** RWBC.23.***.483.***.65.564 ypically form rade riangles. Conversely, he weighed WTW urns ou o be a weakly disassoraive nework. Moreover, counries ha are inensively conneced high NS are also more clusered high WCC. This mismach beween binary and weighed represenaions can be parly raionalized by noicing ha he correlaion beween NS and ND is posiive bu no very large on average abou.45, hus hining o a opological srucure where having more rade connecions does no auomaically imply o be more inensively conneced o oher counries in erms of oal rade conrolled. As o cenraliy, RWBC appears o be posiively correlaed wih NS, signalling ha in he WTW here is lile disincion beween global and local cenraliy 84. Anoher ineresing issue o explore concerns he exen o which counry specific characerisics relae o nework properies. We focus here on he correlaion paerns beween nework saisics and counry per capia GDP pcgdp in order o see wheher counries wih a higher income are more or less conneced, cenral, and clusered. The oucomes are very clear and end o mimic hose obained above for he correlaion srucure among nework saisics. Figure 3 shows ha high-income counries end o hold more, and more inense, rade relaionships and o occupy a more cenral posiion. However, hey rade wih few and weakly conneced parners, a paern suggesing he presence of a sor of rich-club phenomenon 85. To furher explore his evidence, we have firs considered he binary version of he WTW and we have compued he rich-club coefficien R k, defined, for each ime period Populaion Average.8.7.6.5.4.3.2 Normalized Node Degree Node Srengh. 98 985 99 995 2 FIG. 8. Populaion averages of node degree normalized by populaion size N=59 and node srengh vs years. and degree k, as he percenage of edges in place among he nodes having degree higher han k see, e.g., Ref. 65. Since a monoonic relaion beween k and R k is o be expeced in many neworks, due o he inrinsic endency of hubs o exhibi a larger probabiliy of being more inerconneced han low-degree nodes, R k mus be correced for is version in random uncorrelaed neworks see Ref. 66 for deails. If he resuling correced rich-club index R k, especially for large values of k, hen he corresponding graph will exhibi saisically significan evidence for richclub behavior. In our case, he binary WTW does no seem o show any clear rich-club ordering, as Fig. 4 shows for year 2. This conrass wih, e.g., he case of scienific collaboraions neworks, bu is well in line wih he absence of any rich-club paern in proein ineracion and Inerne neworks 66. This may be inuiively due o he very high densiy of he underlying binary nework, bu also o he fac ha, as suggesed by Ref. 66 and discussed above, any binary srucure underesimaes he imporance of inensiy of ineracions carried by he edges. In fac, if sudied from a weighed-nework perspecive, he WTW exhibis indeed much more rich-club ordering han is binary version. To see why, for any given year we have sored in a descending order he nodes counries according o heir srengh, aken as anoher measure of richness. We have hen compued he percenage of he oal rade flows in he nework ha can be impued o he rade exchanges occurring only among he firs k nodes of he NS year- ranking, i.e., a k-sized rich club. More precisely, le i,i 2,...,i N be he labels of he N nodes sored in a descending order according o heir year- NS. The coefficien for a given rich-club size k is compued as he raio beween j= h= w ij k k ih and he sum of all enries of he marix W. Figure 5 shows how our crude weighed rich-club coefficien behaves in year 2 for an increasing rich-club size. I is easy o see ha he riches counries in erms of NS are responsible of abou 4% of he oal rade flows see doed verical and horizonal lines, a quie srong indicaion in favor of he exisence of a rich club in he weighed WTW. This is furher confirmed if one compares he empirical observaions very high wih heir expeced values under a W-RS reshuffling scheme much lower, also afer 95% confidence inervals have been considered 86. In summary, he overall picure ha our correlaion analysis suggess is one where counries holding many rade par- 365-7
FAGIOLO, REYES, AND SCHIAVO PHYSICAL REVIEW E 79, 365 29.9.8-2 Average.7.6 ln(average) -4-6 -8.5 ANND (Empirical) BCC (Empirical) ANND (Expeced) BCC (Expeced).4 98 985 99 995 2 (a) s - ANNS (Empirical) ANNS (Expeced) WCC (Empirical) WCC (Expeced) RWBC (Empirical) RWBC (Expeced) -2 98 985 99 995 2 (b) s FIG. 9. Populaion average vs expeced values of node saisics. Panel a: Expeced values for binary-nework saisics are compued by reshuffling binary links by keeping observed densiy fixed. Panel b: Expeced values for weighed-nework saisics are compued by reshuffling observed link weighs while keeping he binary sequence fixed edge-crossing algorihm. ners and/or very inense rade relaionships are also he riches and mos globally cenral; ypically rade wih many counries, bu very inensively wih only a few veryconneced ones; and form few, bu inensive, rade clusers riangles. Furhermore, our correlaion analysis provides furher evidence o he disribuional sabiliy argumen discussed above. Indeed, we have already noiced ha he firs four momens of he disribuions of saisics under sudy ND, NS, ANNS, WCC, RWBC display a marked sabiliy over ime. Figure 2 shows ha also heir correlaion srucure is only weakly changing during he sample period. This suggess ha he whole archiecure of he WTW has remained fairly sable beween 98 and 2. To furher explore he implicaions of his resul, also in he ligh of he ongoing processes of inernaionalizaion and globalizaion, we urn now o a more in-deph analysis of he in-sample dynamics and ou-of-sample evoluion of WTW opological srucure. C. Wihin-sample disribuion dynamics The foregoing evidence suggess ha he shape of he disribuions concerning he mos imporan opological properies of WTW displays a raher srong sabiliy in he 98 2 period. However, disribuional sabiliy does no auomaically rule ou he possibiliy ha beween any wo consecuive ime seps, say and, a lo of shapepreserving urbulence was acually going on a he node and link level, wih many counries and/or link weighs moving back and forh across he quaniles of he disribuions. In order o check wheher his is he case or no, we have compued sochasic-kernel esimaes 38,39 for he disribuion dynamics concerning node and link saisics. More formally, consider a real-valued node or link saisic X. Le, be he join disribuion of X,X and be he marginal disribuion of X. We esimae he -year sochasic kernel, defined as he condiional densiy s xy = x,y/ y 87. Figures 6 and 7 presen he conour plos of he esimaes of he -year kernel densiy of logged NS and logged posiive link weighs. Noice ha he bulk of he probabiliy mass is concenraed close o he main diagonal displayed as a solid 45 line. Similar resuls are found for all oher realvalued node saisics ANNS, WCC, and RWBC also a larger ime lags. The kernel densiy of logged posiive link weighs, conrary o he logged NS one, is insead exremely CDF.8.6.4 Kernel Densiy.6.4.2 6 8 2 CDF.8.6.4 Kernel Densiy.4.2 8 2 4.2 Empirical Random 95% Conf In.2 Empirical Random 95% Conf In 5 6 7 8 9 2 3 (a) ln(cd ) 4 (b) 6 8 2 4 6 ln(cd ) 5 FIG.. Clique-degree disribuions in year 2. Panel a, clique degree of order k=4; panel b, clique degree of order k=5. Main plos: Cumulaive disribuion funcion CDF, filled circles ogeher wih expeced shape of CDF in randomly reshuffled graphs wih he same degree disribuions as he observed one. Doed lines: 95% confidence inervals for he CDF esimae based on samples. X axis: Log of he number of nodes belonging o k-order cliques. Y axis: Cumulaive disribuion funcion. 365-8