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1 This rticle ppered in journl pulished y Elsevier. The ttched copy is furnished to the uthor for internl non-commercil reserch nd eduction use, including for instruction t the uthors institution nd shring with collegues. Other uses, including reproduction nd distriution, or selling or licensing copies, or posting to personl, institutionl or third prty wesites re prohiited. In most cses uthors re permitted to post their version of the rticle (e.g. in Word or Tex form) to their personl wesite or institutionl repository. Authors requiring further informtion regrding Elsevier s rchiving nd mnuscript policies re encourged to visit:

2 Physic A 390 (2011) Contents lists ville t ScienceDirect Physic A journl homepge: Strtified economic exchnge on networks J.L. Herrer, M.G. Cosenz,, K. Tucci c Escuel Básic de Ingenierí, Universidd de Los Andes, Mérid, Venezuel Centro de Físic Fundmentl, Universidd de Los Andes, Mérid, Venezuel c SUMA-CeSiMo, Universidd de Los Andes, Mérid, Venezuel r t i c l e i n f o s t r c t Article history: Received 4 Octoer 2010 Received in revised form 18 Decemer 2010 Aville online 16 Jnury 2011 Keywords: Economic models Networks Economic clsses Welth distriution We investigte model of strtified economic interctions etween gents when the notion of sptil loction is introduced. The gents re plced on network with nerneighor connections. Interctions etween neighors cn occur only if the difference in their welth is less thn threshold vlue tht defines the width of the economic clsses. By employing concepts from sptiotemporl dynmicl systems, three types of ptterns cn e identified in the system s prmeters re vried: lminr, intermittent nd turulent sttes. The trnsition from the lminr stte to the turulent stte is chrcterized y the ctivity of the system, quntity tht mesures the verge exchnge of welth over long times. The degree of inequlity in the welth distriution for different prmeter vlues is chrcterized y the Gini coefficient. High levels of ctivity re ssocited to low vlues of the Gini coefficient. It is found tht the topologicl properties of the network hve little effect on the ctivity of the system, ut the Gini coefficient increses when the clustering coefficient of the network is incresed Elsevier B.V. All rights reserved. 1. Introduction Socil strtifiction refers to the clssifiction of individuls into groups or clsses sed on shred socio-economic or power conditions within society [1]. A chrcteristic feture of strtified societies is tht individuls tend to interct more strongly with others in their own group. This tendency hs een oserved in clss endogmy [2], scientific communities nd cittions [3], popultion iology [4], humn cpitl [5], opinion formtion [6], epidemic dynmics [7], nd economic exchnges etween nks [8]. Recently, the effects of socil strtifiction on the welth distriution of system of intercting economic gents hve een studied [9]. In this model, gents ehve s prticles in gs nd they cn interct with ech other t rndom, s in most models tht hve een proposed for economic exchnge [10 12]. However, mny rel socil nd economic systems cn e descried s complex networks, such s smll-world networks nd scle-free networks [13 15]. Some models hve considered economic dynmics on networks; for exmple, Refs. [16,17] studied the effects of the network s topology on welth distriutions; while Ref. [18] proposed model of closed mrket on fixed network with free flow of goods nd money. In this pper, we study the effects of the topology of network on the collective ehvior of system suject to strtified economic exchnges. Our model, sed on the interction dynmics in strtified society proposed y Lgun et l. [9], is presented in Section 2. The inclusion of sptil support llows one to employ concepts from the dynmics of sptiotemporl systems in economic systems. Our results indicte tht the size of the locl neighorhood plys n importnt role for chieving n equitle distriution of welth in systems possessing strtified economic exchnge. Conclusions re presented in Section 3. Corresponding uthor. E-mil ddress: mcosenz@ul.ve (M.G. Cosenz) /$ see front mtter 2011 Elsevier B.V. All rights reserved. doi: /j.phys

3 1454 J.L. Herrer et l. / Physic A 390 (2011) The model We consider network defined y following the lgorithm of construction of smll-world networks originlly proposed y Wtts nd Strogtz [13]. We strt from regulr ring with N nodes, where ech node is connected to its k nerest neighors, k eing n even numer. Then, ech connection is rewired t rndom with proility p to ny other node in the network. After the rewiring process, the numer of elements coupled to ech node which we cll neighors of tht node my vry, ut the totl numer of links in the network is constnt nd equl to Nk/2. The condition log N k N is employed to ensure tht no node is isolted fter the rewiring process, which results in connected grph. For p = 0, the network corresponds to regulr ring, while for p = 1 the resulting network is completely rndom. With this lgorithm, smll-world network is formed for vlues of the proility in the intermedite rnge [13]. A smll-world network is chrcterized y high degree of clustering, s in regulr lttice, nd smll chrcteristic pth length compred to the size of the system. We consider popultion of N intercting gents plced t the nodes of this network. At discrete time t, n gent i (i = 1,..., N), is chrcterized y welth w i (t) 0 nd fixed risk version fctor β i, where the vlues β i re rndomly nd uniformly distriuted in the intervl [0, 1]. The quntity (1 β i ) mesures the frction of welth tht gent i is willing to risk in n economic interction [11,19,20]. The initil vlues w i (0) re uniformly distriuted t rndom in the intervl w i (0) [0, W]. We ssume tht the totl welth of the system, W T = i w t(i), is conserved. For simplicity, we ssume tht the strtifiction of economic clsses is uniform, i.e., ll clsses hve the sme width, denoted y prmeter u. Thus, gents i nd j elong to the sme economic clss if they stisfy the condition w i (t) w j (t) < u. Strtified economic exchnge mens tht only gents elonging to the sme economic clss my interct. As consequence of these interctions, the welth of the gents in the system will chnge. At ech time step t, the dynmics of the system is defined y iterting the following steps: (1) Choose n gent i t rndom. (2) Choose rndomly n gent j i from the set of neighors of gent i, i.e., j [i k/2, i + k/2]. (3) Check if they elong to the sme economic clss, i.e., w i (t) w j (t) < u. Repet steps (1) nd (2) until condition (3) is chieved. (4) Compute the mount of welth w(t) to e exchnged etween gents i nd j, defined s w(t) = min[(1 β i )w i (t); (1 β j )w j (t)]. (2) (5) Clculte the proility r of fvoring the gent tht hs less welth etween i nd j t time t, defined s [9,20] r = f w i(t) w j (t) w i (t) + w j (t), where the prmeter f [0, 1/2]. (6) Assign the quntity w(t) with proility r to the gent hving less welth nd with proility (1 r) to the gent with greter welth etween i nd j. The prmeter f descries the proility of fvoring the poorer of the two gents when they interct. For f = 0 oth gents hve equl proility of receiving the mount w(t) in the exchnge, while for f = 1/2 the gent with less welth hs the highest proility of receiving this mount. In typicl simultion following these dynmicl rules, nd fter trnsient time, this dynmicl network reches sttionry stte where the totl welth W T hs een redistriuted etween the gents. The sptil locliztion of the intercting economic gents llows one to see this system s sptiotemporl dynmicl system. Fig. 1 shows the sptiotemporl ptterns of welth rising in network with k = 2 nd p = 0, corresponding to regulr one-dimensionl lttice with periodic oundry conditions, for different vlues of the prmeters. In nlogy to mny nonliner sptiotemporl dynmicl systems [21], this network of economic gents cn exhiit three sic sttes depending on prmeter vlues: sttionry, coherent or lminr stte (left pnel), where the welth of ech gent i mintins constnt vlue; n intermittent stte (center pnel), chrcterized y the coexistence of coherent nd irregulr domins evolving in spce nd time; nd turulent stte (right pnel) where the welth vlues chnge irregulrly in oth spce nd time. To chrcterize the trnsition from the lminr to the turulent stte, vi sptiotemporl intermittency, we employ the verge welth exchnge for long times, quntity tht we cll the ctivity of the system nd define s A = 1 T τ T w(t), t=τ where τ is trnsient numer of steps tht re discrded efore tking the verge. The lminr phse is ssocited to vlues A = 0, where no trnsctions tke plce in the symptotic stte of system, while the turulent phse is chrcterized y A > 0. (1) (3) (4)

4 J.L. Herrer et l. / Physic A 390 (2011) Fig. 1. Sptiotemporl ptterns in one-dimensionl lttice with k = 2, size N = 50 nd W = 1, fter discrding 5000 time steps. The verticl xis descries the ordered position i of the gents in the lttice, incresing from ottom to top. Horizontl xis represents time, incresing from left to right. The welths w i (t) evolving in time re represented y color code. The color plette goes from light gry (the poorest gent) to drk gry (the richest gent). Left: lminr stte; u = 10, f = Center: sptiotemporl intermittent stte; u = 3, f = 0.4. Right: turulent stte; u = 30, f = 0.4. Fig. 2. () Activity s function of u in regulr lttice (p = 0) with fixed k = 4, for different vlues of f. The curves correspond to f = 0.5 (dimonds); f = 0.3 (circles); nd f = 0.1 (squres). () Activity s function of f in regulr lttice with k = 4, for u = 1 (squres), nd u = 30 (circles). In our clcultions, we hve fixed these vlues of prmeters: size N = 10 4, τ = 10 8, T = , nd W = 1. Ech vlue of the sttisticl quntities shown hs een verged over 100 reliztions of initil conditions. Fig. 2() shows the ctivity in the system s function of the width of the economic clsses u for different vlues of the prmeter f. The trnsition from the lminr phse to the turulent stte occurs out the vlue u W = 1 in ll cses. When the vlue of the width u reches the vlue of the mximum initil welth of the gents, exchnges my tke plce in every neighorhood, nd this is reflected in the increse in the ctivity in the system. For u > W, interctions continue to occur in the entire system nd the totl welth exchnged reches the mximum mount llowed y the fvoring prmeter f. Thus, the ctivity in the system reches n lmost constnt vlue in this region, for given vlue of f. On the other hnd, Fig. 2() shows the ctivity in the system s function of f. The increment in f enhnces the trnsfer of welth from richer to poorer gents. Therefore, the proility tht neighoring gents elong to the sme economic clss increses, nd so does the proility tht they exchnge welth. As consequence, the ctivity in the system increses with incresing f. To explore the effects of the network s topology on the collective properties of the system, we show in Fig. 3() the ctivity s function of the size of the neighorhood k in the network, for different vlues of u. The rnge of the locl interction, given y k, hs little effect on the ctivity. Similrly, Fig. 3() shows the ctivity s function of the rewiring proility in the network, for fixed k = 4. We see tht the exchnge ctivity in the system is prcticlly unffected y the topologicl properties of the network, represented y k nd p. Thus, the prmeters of the dynmics, f nd u, re more relevnt for the increse in the ctivity in the system thn the topologicl prmeters of the underlying network. An importnt vrile in economic dynmics is the Gini coefficient, sttisticl quntity tht mesures the degree of inequlity in the welth distriution in system, defined s [22] G(t) = 1 2N N w i (t) w j (t) i,j=1 N w i (t) i=1. A perfectly equitle distriution of welth t time t, where w i (t) = w j (t), i, j, yields vlue G(t) = 0. The other extreme, where one gent hs the totl welth N i=1 w i(t), corresponds to vlue G(t) = 1. The rndom, uniform distriution of (5)

5 1456 J.L. Herrer et l. / Physic A 390 (2011) Fig. 3. () Activity s function of k, on regulr lttice with p = 0, fixed f = 0.5, nd for different vlues u = 1 (squres) nd u = 10 (circles). () Activity s function of p, on network with k = 4, nd for different vlues f = 0.5 (tringles), f = 0.3 (circles) nd f = 0.1 (squres). Fig. 4. Gini coefficient t t = 10 8 s function of u with fixed k = 4 for different f. The curves correspond to f = 0.5 (circles); f = 0.3 (squres); nd f = 0.1 (tringles). () Gini coefficient t t = 10 8 s function of prmeter f for different vlues of k nd fixed u = 30. The curves correspond to k = 2 (squres); k = 4 (circles); nd k = N 1 (tringles). welth used s initil condition hs G(0) 0, nd the verge initil welth per gent is w i (0) = 0.5. Fig. 4() shows the symptotic, sttisticlly sttionry Gini coefficient s function of the width of the socil clsses u, for different vlues of the prmeter f. For smll vlues of u, there is smll proility of interction etween neighors, nd therefore the initil rndom, uniform distriution of welth with G 0 is mintined in the system, mnifested in low vlue of G. As u increses, the trnsfer of welth etween neighors lso increses, producing redistriution of welth reflected in the increse of the Gini coefficient. A mximum of G occurs round u W = 1, when ech gent cn initilly interct with his neighors, nd therefore greter vrition with respect to the initil uniform distriution of welth occurs in the system. For lrger vlues of u, ll locl interctions re llowed initilly. In this regime, redistriution of welth should occur s the proility f of fvoring the poorest gents is incremented. This cn e seen in Fig. 4() s decrese in the vlues of G, for u > W, s f increses. Fig. 4() shows the Gini coefficient s function of the proility f, for different sizes of the neighorhood k. The vlues of G re lmost constnt for smll vlues of f, ut they decrese for lrger vlues of f. In order to study the influence of the topology of the network on the distriution of welth, Fig. 5() displys G s function of k, on regulr lttice with p = 0, for different vlues of u. Incresing the numer of neighors k contriutes to n increse in the inequlity of the welth distriution, s mesured y G. Note tht G tends to n symptotic, lrge vlue s k N 1, corresponding to fully connected network, i.e., ny gent cn interct with ny other in the system, losing the notion of sptil loction. This corresponds to the most commonly studied situtions in models of economic exchnge [9]. Incresing the sptil rnge of the interctions, represented y k, implies oth n increment in the clustering coefficient nd decrese in the chrcteristic pth length of the network. To see which of these two topologicl properties of the network is more relevnt for the vrition of the Gini coefficient oserved in Fig. 5(), we plot in Fig. 5() G s function of the rewiring proility p, for different vlues of the prmeter f. Note tht there is little chnge in the vlues of G s p increses, in comprison to the lrger vrition experienced y G when k is ugmented in Fig. 5(). The chrcteristic pth length in the network decreses in oth cses, ut the clustering coefficient does not increse on the rnge of vlues of p shown in Fig. 5() [13]. Thus, the increment in the Gini coefficient oserved in Fig. 5() cn e minly ttriuted to the increse in the clustering coefficient of the network when k is vried. In other words, the size of the neighorhood is more relevnt for the occurrence of n equitle distriution of welth thn the presence of long rnge connections in system suject to strtified economic exchnge.

6 J.L. Herrer et l. / Physic A 390 (2011) Fig. 5. () Gini coefficient t t = 10 8 s function of k on regulr lttice with p = 0, for f = 0.1, u = 1 (squres) nd u = 10 (circles). () Gini coefficient t t = 10 8 s function of the rewiring proility p on network with k = 4, with fixed u = 10 nd f = 0.5 (dimonds), f = 0.3 (circles), nd f = 0.1 (squres). 3. Conclusions The inclusion of network or sptil loction for intercting economic gents llows the use of concepts from sptiotemporl dynmicl systems in economic models. We hve considered model of strtified economic exchnge defined on network nd hve shown tht different sptiotemporl ptterns cn occur s the prmeters of the system re vried. We hve chrcterized these ptterns s lminr, intermittent nd turulent, employing nlogies from sptiotemporl dynmicl systems. We hve chrcterized the trnsition from lminr stte to turulent stte through the ctivity of the system, tht mesures the verge welth exchnged in the symptotic regime of the system. This quntity depends minly on the dynmicl prmeters u nd f. Similrly, the Gini coefficient, tht chrcterizes the inequlity in the distriution of welth, depends on the prmeters u nd f. For lrge vlues of u, incresing f increses the ctivity ut decreses the Gini coefficient. Thus, high levels of economic exchnge ctivity re ssocited to low vlues of the Gini coefficient, i.e., to more equitle distriutions of welth in the system. The topology of the underlying network hs little effect on the ctivity of the system A. In contrst, the Gini coefficient G increses when the rnge of the interctions, represented y k, is incresed. We hve shown tht the relevnt topologicl property of the network tht influences the ehvior of G, is the clustering coefficient, insted of the chrcteristic pth length of the network. Fig. 5 shows tht reduction of the Gini coefficient in system suject to dynmics of strtified economic exchnge my e chieved y reducing the size of the neighorhood of the intercting gents. Our model could e representtive of economic exchnges of non-trdle goods nd services; in prticulr the concept of neighorhood my e pplicle to locl mrkets. Our results dd support to the view of locl interctions s relevnt ingredient tht cn hve importnt consequences in the collective ehvior of economic models. Acknowledgements This work ws supported y grnt No. C B from Consejo de Desrrollo Científico, Humnístico y Tecnológico of Universidd de Los Andes, Mérid, Venezuel. M. G. C. cknowledges support from project /2007-0, CNPq-PROSUL, Brzil. References [1] D. Grusky, Socil Strtifiction: Clss, Rce, nd Gender in Sociologicl Perspective, 3rd ed., Westview Press, [2] T.C. Belding, rxiv:nlin/ v3. [3] S. Lehmnn, A.D. Jckson, B. Lutrup, Europhys. Lett. 69 (2005) 298. [4] A. Vzquez, Phys. Rev. E 77 (2008) [5] T.V. Mrtins, T. Arujo, M.A. Sntos, M.St. Auyn, Physic A 388 (2009) [6] A.C.R. Mrtin, Phys. Rev. E 78 (2008) [7] N. Msud, N. Konn, J. Theoret. Biol. 243 (2006) 64. [8] H. Inok, H. Tkysu, T. Shimizu, T. Ninomiy, K. Tniguchi, Physic A 339 (2004) 621. [9] M.F. Lgun, S. Risu Gusmn, J.R. Iglesis, J. Veg, Physic A 342 (2004) 186. [10] V.M. Ykovenko, in: R.A. Meyers (Ed.), Encyclopedi of Complexity nd System Science, Springer, [11] A. Chtterjee, B.K. Chkrrti, S.S. Mnn, Physic A 335 (2004) 155. [12] F. Slnin, Phys. Rev. E 69 (2004) [13] D.J. Wtts, S.H. Strogtz, Nture 393 (1998) 440. [14] A.L. Brási, R. Alert, Science 286 (1999) 509. [15] M.E.J. Newmn, A.L. Brsi, D.J. Wtts, The Structure nd Dynmics of Networks, Princeton University Press, Princeton, NJ, [16] J.R. Iglesis, S. Gonçlves, S. Pinegond, J.L. Veg, G. Armson, Physic A 327 (2003) 12. [17] D. Grlschelli, M.I. Loffredo, J. Phys. A 41 (2008) 22. [18] M. Ausloos, A. Peklski, Physic A 373 (2007) 560. [19] A. Chkrorti, B.K. Chkrrti, Eur. Phys. J. B 17 (2000) 167. [20] J.R. Iglesis, S. Gonçlves, G. Armson, J.L. Veg, Physic A 342 (2004) 186. [21] P. Mnneville, Instilities, Chos nd Turulence, 2nd ed., Imperil College Press, London, [22] M. Rodríguez-Achch, R. Huert-Quintnill, Physic A 361 (2006) 309.