Taking into Account the Variations of Neighbourhood Sizes in the Mean-Field Approximation of the Threshold Model on a Random Network

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1 Takng nto ccount the Varatons of Neghbourhood Szes n the Mean-Feld pproxmaton of the Threshold Model on a Random Network Sylve Huet 1, Margaret Edwards 1, Gullaume Deffuant 1 1 Laboratore d Ingénere des Systèmes Complexes Cemagref - 24, av. des Landas, F63 ubère sylve.huet@cemagref.fr margaret.edwards@clermont.cemagref.fr gullaume.deffuant@cemagref.fr bstract. We compare the ndvdual-based threshold model of nnovaton dffuson n the verson whch has been studed by Young [22], wth an aggregate model we derved from t. The classcal threshold model supposes that an ndvdual adopts a behavour accordng to a trade-off between a socal pressure and a personal nterest. Our study consders only the case where all have the same threshold. We present an aggregated model, whch takes nto account varatons of the neghbourhood szes, whereas prevous work assumed ths sze fxed [6]. The comparson between the aggregated models (the frst one assumng a neghbourhood sze and the second one, a varable one) ponts out an mprovement of the approxmaton n most of the value of parameter space. Remanng dfferences between both models gve us some clues about the specfc ablty of ndvdual-based model to mantan a mnorty behavour whch becomes a majorty by an addton of stochastc effects. 1 Introducton Indvdual-based models are more and more commonly used n ecologcal modellng [14], or to model socal and economcal processes [1][8][9][12]. Ths work ams at studyng the nterest of an ndvdual based model compared to an aggregated one. In both cases, we are nterested n aggregated varables but the aggregated modellng requres lower computer capacty and tme. Therefore, t s nterestng to dentfy the condtons n whch an ndvdual based model can be replaced by ts aggregated approxmaton. Recent studes have sought to compare ndvdual-based to aggregate models. Dengels and Rose [4] study the nfluence of transformng contnuous varables nto dscrete dstrbutons n models of ecologcal dynamcs; Pcard and Franc [18] show that space-dependent ndvdual based models and aggregated models (regardng ether spatal nfluence or descrpton of the populaton) of forest dynamcs lead to dfferent results. Fahse et al. [1] and Duboz et al. [5] use ndvdual-based models to extract parameters for populaton-level dynamcs. Ths work follows that presented n Edwards et al. [6][7] n whch an ndvdual based model of nnovaton dffuson [19][17] studed by Blume [2][3] and Young 1/15

2 [22][23] s aggregated followng the prncples of socodynamcs [2][21]. Ths model represents ndvduals who choose between two behavours ( or B) accordng to a utlty functon based on a socal beneft (dependng on the behavour of ther neghbours and a payoff related to each behavour) and an ndvdual beneft. Followng Blume and Young, we consder that the ndvdual beneft s the same for all the ndvduals, and t s put to zero for convenence. The choce s randomsed by a Gbbs functon wth a gven temperature [15]. Edwards et al. approxmate the ndvdual based threshold model on a random network by an aggregate model based on a network wth a fxed number of neghbours for each ndvdual (number of lnks at each node of the graph). Ths strong approxmaton yelds pretty good results. The asymptotc behavours of the two models are dentcal: % of behavour or 1% of behavour. Both models have a bfurcaton: there exsts n the space of parameters an unstable zone and a zone wth two attractors. The attracton basn s lmts depend essentally on the randomness of the ndvdual decson functon (temperature of the Gbbs functon), on the payoff of and the mean number of neghbours for an ndvdual for the rest. However, we could observe two types of dfferences between the ndvdual based model and ts aggregated approxmaton: n some condtons the models can exhbt dfferent lmts of attracton basns, n terms of ntal proporton of. Therefore, the condton of basn change could be totally ll predcted by the aggregated model; n some cases the fnal state of the ndvdual based model s between % and 1% of. Moreover, for some ntal condtons, ths value of the fnal state vares from one run to the other, leadng to a bmodal dstrbuton of fnal states. The aggregated approxmaton wth a fxed neghbourhood sze never shows such a behavour. We call the frst type of dfference "strong dfference" because fnal states are located on opposte attractors (correspondng respectvely to % of behavours and 1% of behavours ). We wll show ths strong dfference s related to the fxed neghbourhood sze smplfcaton. Indeed, when the neghbourhood has a strong mpact on the decson functon, dfferences between the models can appear. To mprove the approxmaton, we propose an extended verson of the aggregated model n whch we used a Posson dstrbuton to approxmate the varable neghbourhood sze n a random network. 2 ggregatng an ndvdual based threshold model on a random network 2.1 The ndvdual based model of nnovaton dffuson The threshold model s ntally nspred from the socology of nnovaton dffuson [13]. Its prncple s that the decson of an ndvdual to change ts behavour reles on a trade-off between a socal pressure (consderng the behavour of ts neghbours), 2/15

3 and an ntrnsc, personal nterest. Studyng ths mode, Blume and Young have only consdered the case where the personal nterest s zero for all. We do the same here, and we consder here the stochastc verson of the decson model, whch s ntroduced by Luce (1959) and uses a Gbbs functon. We recall hereafter the man features of the model. We consder N ndvduals, havng the choce between two behavours: and B. N s constant over tme. The ndvduals are connected to each other through a random graph (socal network) : we defne a pror a number of lnks for the whole populaton, and pck pars of ndvduals at random to defne the lnks. fterwards, each ndvdual has a neghbourhood V() (ndvdual socal network), whch s the set of ndvduals he s connected wth through the graph. The socal network s therefore charactersed by an average number of lnks v for an ndvdual (average neghbourhood sze). To take an example, let's consder a vllage, the waterng practces of whch we would lke to follow; represents scarce waterng and B generous waterng. The choce to adopt ether the or the B behavour ensues from utlty computaton. The utlty U () and U (B) of ndvdual to choose or B s expressed as: U (1) ( ) = g V (, ) (2) U ( B) = gb V (, B) where V(,) and V(,B) are the number of neghbours of ndvdual respectvely of behavour or B and the g, gb are parameters of the model. One gets a postve socal utlty when followng the behavour of one's neghbours, because t strengthens lnks and provdes nterestng subjects for conversaton (ths corresponds to the terms gv(,) and gbv(,b)). The order between the utltes depends on g and gb : f g=gb, the greater utlty wll correspond to the domnant behavour n the neghbourhood. If g=2gb, t wll suffcent that every thrd neghbour follows to get U ()>U (B): ths could correspond to a case where the behavour spreads easly because t appears attractve. Nevertheless, f all neghbours follow B, U (B) > U (). In order to take nto account ndvdual varablty and uncertanty n the decson, a stochastc response s often ntroduced. The probablty for an ndvdual of adoptng or B s defned by a Gbbs functon: exp( β U ( )) (3) P( chooses ) = exp( β U ( )) + exp( β U ( B)) exp( β U ( B)) (4) P( chooses B) = exp( β U ( )) + exp( β U ( B)) Parameter β plays the role of the nverse of a temperature: for β= (hgh temperature), the functon s purely random. Both behavours have the same probablty of.5. In ths case the choce of ndvduals doesn't depend on 3/15

4 ther socal context traced by the computaton of utltes. The greater β, the more the dfference between the utltes appears n the probabltes, by favourng the behavour wth the greater utlty. for β (low temperature), ths functon becomes quas determnstc: f U()>U(B), P() becomes very close to 1, whereas P(B) nears. 2.2 ggregatng and Mean Feld pproxmaton We propose a new aggregate model whch takes nto account the varable neghbourhood sze n a random network. Let us descrbe the method by whch we approxmate the dstrbuton of a random network before presentng the new aggregated model pproxmatng the Dstrbuton of Neghbourhood Szes n a Random Network The random network, presented by Paul Erdös and lfred Rény n 1959, s now well known and the neghbourhood szes can be approxmated as follows: let a node n a random network be lnked wth the probablty p to each of the (N 1) other nodes of the graph. The probablty p that ts neghbourhood sze s s gven by the Bnomal dstrbuton: N 1 (5) N 1 p = p ( 1 p) Note that the mean number of lnks for one node s z = (N-1)p. Therefore, we can wrte: N 1 z z p = N z 1 N 1 1 z z e! N 1 (6) where the last approxmate equalty expresses that a bnomal law tends to a Posson dstrbuton when N tends to nfnty. Snce our smulatons nvolve 1. ndvduals, we shall consder that ths approxmaton s correct. Now we shall use p n a new aggregated model n ggregated Model Takng ccount Dstrbuton of Neghbourhood Szes We buld the aggregate model wthn the soco-dynamcs framework [2][11] approach whch s nspred from physcs. The prncple s to consder a set of aggregate states, and the probablty ndvduals go from one aggregated state to the other. The flows between the aggregated states yelds the master equaton whch rules the dynamcs of the model. 4/15

5 We are nterested n behavour versus behavour B evoluton n the populaton. Therefore, we only consder the flow between the two sub-populatons of ndvduals and «B». The varaton dp of the populaton of behavour s gven by: dp = (proporton of ndvduals B adoptng ) (proporton of ndvduals adoptng B) or: dp = (probablty of B ) (probablty of B) In the ndvdual-based model, the probablty of an ndvdual to choose or B s calculated by the classcal decson functon called Perturbed Best Reply. For ths functon, the behavour chosen ensues from the proporton of and B n the neghbourhood of an ndvdual and from the value of the g parameter. The calculaton of respectve probablty of choosng or B s a functon of the number of neghbours over the neghbourhood sze n the ndvdual-based model, for >: P( B, k P(, k / ) = exp( β ( k / ) = exp( β ( k (7) exp( β (( k ) / ). gb) (8) / ). g) + exp( β (( k ) / ). gb) exp( β ( k / ). g) (9) / ). g) + exp( β (( k ) /. gb) where P(,k /) and P(B,k /) are the respectve probablty of: choosng (respectvely B) gven the number k of neghbours over the neghbourhood sze. For =, the probabltes of behavour and B are both equal to.5. Therefore, n order to compute dp, we have to defne the probablty,k ) of ndvduals of behavour k neghbours of behavour over neghbours, and B,k ), the probablty to have ndvduals of behavour B and k neghbours of behavour over neghbours. The ncrease or the decrease of the populaton of behavour can be now wrtten: dp = N 1 = k = S N 1 v ( B, k / ) P(, k / ), k / ) P( B, k / ) = k = The possblty to calculate,k /) and B,k /) probabltes s the key pont of ths t model. It s possble, wth p the proporton of behavour n the populaton at t, to calculate the ndvdual probablty of havng k neghbours wth behavour over the neghbourhood sze supposng t follows a bnomal law: k k k ( k ) = C p.(1 p ) (11) P / Therefore, the probablty of states,k /) t and B,k /) t at tme t s: S t t ( k / ) p. p. P( k ) / (1), = (12) 5/15

6 S t t ( B k / ) p. p. P( k ), = (13) B / The aggregated model can now be terated untl t reaches ts statonary state by calculatng the proporton of behavour at t+1 : t t +1 p p + dp (14) The proporton of B behavour s the complementary of the proporton because the t+ 1 t+ 1 populaton sze s constant : p B = 1 p. In fne, the state of the system s defned by a vector of probabltes to have behavour wth, 1 neghbours havng, varyng from to N-1, and the probablty havng B behavour wth, 1 neghbours havng, varyng from to N-1. The system state vector C, k /) t at tme t has a of (²+3+2)/2 wth C {,B}, k {,,} and {,,N-1}: C, k t,),,1);,),,1),,2);,),...,, v); / ) = B,), B,1); B,), B,1), B,2); B,),..., B, v) Ths gves us the new value of n, and we can compute the value of s(t+1), and terate agan. In practce, we do not sum the second member of (6) for = to N-1. We stop when the frst neghbourhood sze appears whch has a probablty equals to. (15) 3 Expermental Comparsons 3.1 Expermental Desgn bsolute value of dfference between attracton basn lmt values predcted by both aggregated models,5 nb of nteger ntal part of values,4,3,2, β,56,58,62,66,7,78,84 g - beta g 3 neghbours 5 neghbours 8 neghbours 15 neghbours 25 neghbours 6/15

7 Fg. 1. bsolute value of dfference between attracton basn lmt values predcted by both aggregated models for a part of tested value (all values tested are: for g; all values from.5 to 1 by step of.2; for β, 52 ncrease values from 5 to 92; for mean sze neghbourhood: 3, 5, 8, 15 and 25) We compare the lmts of attracton basns value predcted by the fxed neghbourhood sze aggregated model wth those predcted by the varable neghbourhood sze aggregated model. We consder the results for fve mean neghbourhood szes (3, 5, 8, 15 and 25), for g parameter values startng from.5 to 1 by step of.2 and for 52 ncrease values of β parameter varyng from 5 to 92. Fgure n 1 llustrates ths comparson. The comparson ndcator s lmt of attracton basn value predcted by the fxed neghbourhood sze aggregated model - lmt of attracton basn value predcted by the varable neghbourhood aggregated model. We observe that dfferences ncrease n value and quantty when the average number of neghbours decreases. Very few dfferences can be found wth 25 neghbours on average, whereas they are very mportant for 3 neghbours on average. Values of mean sze neghbourhood, g and β chosen for the expermental desgn are lsted n table n 1. We chose to focus on small average neghbourhood szes. The tested values of parameter ntal proporton of behavour are all nteger values located n a range of 5% to 7% around the value correspondng to the attracton basn lmt predcted by the varable neghbourhood sze aggregated model. We add some values f necessary,.e. n the case where the aggregated model does not predct the value of attracton basn lmt observed n the ndvdual-based model. Table 1. Expermental desgn for g, β and average neghbourhood sze values of expermental desgn for ntal proporton of n the populaton are obtaned as explaned n the paragraph above Mean sze of neghbourhood g β (level of randomness n decson functon) (behavour payoff) , , ,59 8, ,51 5,55,6,61 9,46-9, ,65 13, , , , ,7 12 8, /15

8 , , , , , s Edwards et al. explaned, the equlbrum states of the aggregated models can be found by consderng the values of p for whch dp =. We use the same method here. We compare these results wth the ndvdual-based model smulatons for 2 tme steps wth 1. ndvduals. One tme step corresponds to the asynchronously updatng of all ndvduals of the populatons. 5 runs are performed for each combnaton of parameter settngs. 3.2 Results of Smulatons The varable neghbourhood sze model does better n most cases than the fxed neghbourhood sze one. However, some dfferences reman between the ndvdualbased model and the varable neghbourhood sze aggregated model Global Improvement of the pproxmaton We defne the followng thresholds correspondng to attracton basn lmts for the models: Let IBM 5 be the lowest ntal proporton of behavour whch allows to obtan at least 5% of behavours as fnal state (for the ndvdual based model); Let IBM 99 be the lowest ntal proporton of behavours whch allows us to obtan at least 99% of behavour as fnal state (for the ndvdual based model) or the lowest ntal proporton of behavour for whch the model converged to a value close to 1% of (notce that for three neghbours and g =.51 and.57, the model never gves 1% of ; t stops to a value comprsed between 9% of for low value of β and 96% of for hgh value of β because of the large proporton of ndvduals whch has neghbours when the average of the connectvty s low (ths part has a totally stochastc decson) ; b the most mportant ntal proporton of behavour whch allows us to obtan % of behavour as fnal state (for both aggregated models). Values from IBM 5 to IBM 99 correspond to ntal proportons of behavours for whch ndvdual-based model fnal states are unstable: ths ndcates that the model s changng ts attracton basn. We consder models as dfferent when the value of attracton basn lmt of the aggregated model ( b ) s dfferent from all values correspondng to the zone of changng of attracton basn for the ndvdual-based model. We measure ths dfference by a dstance D. Ths dstance permts us to evaluate f the varable neghbourhood sze aggregated model has really mproved the 8/15

9 approxmaton of the ndvdual based model. Ths dstance between the aggregated model and the ndvdual-based model s: D = b IBM 99 f b > IBM 99 (16) = IBM 5 b f b < IBM 5 = " No error" n all other cases The modalty No error corresponds to an attracton basn lmt of the aggregated model stuated n the zone of attractor change for ndvdual-based model (cf. fgures n 3 for example). Results are presented n fgure n 2 below n whch D s grouped nto three ntervals plus the modalty no error and 1. The value 1 s comprsed n no error because the dervatve result has a precson of ± 1% of necessary ntal to change of equlbrum. Dstrbuton of dstances between aggregated models and ndvdual based model bfurcaton zone and more 5 to less than 1 2 to less than 5 No error and 1 Regular N. Model n % of the sample Random N. Model n % of the sample Fg. 2. Global mprovement of approxmaton We notce that the varable neghbourhood sze aggregated model approxmates well the ndvdual based model n 85% of the cases whle the fxed neghbourhood aggregated model does so n only 55% of the cases. Moreover, when an error stll exsts, the varable neghbourhood sze model always makes t n less cases than the regular network model. fter ths brd s eye vew, let us get a closer vew of the results and the error produced by the new aggregated model Better pproxmaton But Can Be Stll Locally Sgnfcantly Wrong Where s the resdual error of approxmaton? Let us look at the graphs below, whch present a part of the result. For all these graphs, we dstngush three zones whch corresponds to three ndvdual-based model fnal states: The dark zone represents all the ntal condtons whch gve % of behavour for fnal state; 9/15

10 The grey zone represents all the ntal condtons for whch the model s changng attractor (.e. unstable zone); The whte zone represents all the ntal condtons, whch gve 1% of behavour for fnal state; a commented llustraton s n fgure 3. We compare the condtons of change of attractor for the three models for a network wth 5 neghbours on average (fgure n 3). We see that n most cases, the varable neghbourhood aggregated model predcts the attracton basn s change n the nterval of ndvdual based model attractor s change (wth a precson of ± 1% of ntal behavour ). However, for hgh value of g (.7) and low value of β (28, 29), both aggregated models predct two attractors whle the ndvdual based model has only one. The same s true for g equal to.65 and.67. Notce that, smlarly, for an average neghbourhood sze of 8 and g equal to.74,.75 and.76, two attractors are predcted by the aggregated models when there exsts only one for the ndvdual based model whle no dfference s observed for g=.66. % ntal Equlbrum zones for 5 neghbours, g =.59 Beta 8, IBM end of transton agregate wth regular network % ntal IBM % agregate wth random network % ntal Beta Equlbrum zones for 5 neghbours, g =.61 IBM end of transton agregate wth regular network Equlbrum zones for 5 neghbours, g =,7 1% of % of 8 1 IBM % agregate wth random network ttractor change zone for IBM Beta 18, det IBM end of transton agregate wth regular network IBM % agregate wth random network Fg models bfurcatons condtons for 5 neghbours on average For 3 neghbours on average, we observe another type of error (see fgure n 4). For g equal to.57 and low value of β (16), the varable neghbourhood aggregated model predcts one attractor whle the ndvdual based model has two. Moreover, always for low values of β (21, 23, 35), the approxmaton of the ntal condton needed for attractor s transton s bad. For g equal to.51, no dfference s observed. 1/15

11 55 5 Equlbrum zones for 3 neghbours, g =, Beta % ntal IBM end of transton agregate wth regular network IBM % agregate wth random network Fg models bfurcatons condtons for 3 neghbours mean Fgure n 5 confrms the approxmaton qualty of aggregated models for large average neghbourhood szes, as well as the better accuracy of the varable neghbourhood approxmaton Beta % ntal Equlbrum zones for 25 neghbours, g = IBM end of transton agregate wth regular network IBM % agregate wth random network Fg models bfurcatons condtons for 25 neghbours mean It seems n general that the dfferences appear for low value of β and hgh values of g relatvely to the neghbourhood sze: the varable neghbourhood aggregated model foresees two attractors for g =.59 maxmum when the average neghbourhood s 3 ; for g =.74 maxmum when the average neghbourhood s 5 ; for g =.8 maxmum when the average neghbourhood s 8. Moreover, we keep n mnd that there s no dfference for a bg sze of mean neghbourhood whle they exst for small average neghbourhood szes (3, 5, 8 neghbours). We notce that probablty to have 3, 5 or 8 neghbours n the network wth a average neghbourhood sze of 25 s null. The varable neghbourhood aggregated model predcts often the correct value for the attractor change of ndvdual-based, more often than the regular neghbourhood aggregated model. However, there remans some fnal state dfferences between aggregated and ndvdual-based model when: β s weak (lowest values of β whch allow the ndvdual-based model to jon the stable attractors zone); and 11/15

12 g takes the hghest values n the parameter zone where the model has two stable attractors (for superor g values, the fnal state model s always 1% behavours); and the average neghbourhood sze s small (less than 1 neghbours). We dstngush two types of dfference: 1. the varable neghbourhood aggregated model has two fnal states; t s senstve to the ntal proporton of whle the ndvdual-based model has only one fnal state (cases 5 and 8 neghbours on average). 2. the varable neghbourhood aggregated model s not senstve to the ntal proporton of and always has a 1% behavours fnal state whle the ndvdual-based model has two fnal states (case 3 neghbours on average). 4 Possble Explanaton of the Dfference Between the IBM and the Varable Neghbourhood ggregated Model In the frst case, of remanng dfference between the ndvdual-based and the varable neghbourhood aggregated model can be explaned by the ndvdual-based model s ablty to mantan a mnorty behavour, whch becomes a majorty when a set of favourable random events occurs. On fgure n 7 and 8, we observe that the ndvdual-based model always goes to 1% behavor even f a long tme s necessary to (see 7 on left). It generally begns by decreasng, then stablses for a varable length of tme, then ncreases slowly and fnally dramatcally (see 8). The aggregated model goes to a plateau of 1.73% of behavours after a decrease from 4% to 1.73%. Indvdual-based model evoluton for 2 runs ggregated model evoluton % behavours n the populaton % behavours n the populaton 4,5 4 3,5 3 2,5 2 1,5 1,5 ter ter5 ter1 ter15 ter2 ter25 ter3 ter35 ter4 ter45 ter5 ter55 ter6 ter65 ter7 ter75 ter8 ter85 ter9 ter95 ter1 ter15 ter11 ter115 ter12 ter125 ter13 ter135 ter14 ter145 ter15 ter Tme Tme 12/15

13 1 Fg. 6. Evoluton of % behavours for the ndvdual-based model for 2 runs (on left) and for the varable neghbourhood aggregated model (on rght) for β = 29, g=.7, 5 neghbours and 4% ntal behavours Indvdual-based model evoluton for 2 runs 1 % behavors n the populaton ter ter5 ter1 ter15 ter2 ter25 ter3 ter35 ter4 ter45 ter5 ter55 ter6 ter65 ter7 ter75 ter8 ter85 ter9 ter95 ter1 ter15 ter11 ter115 ter12 ter125 ter13 ter135 ter14 ter145 ter15 ter155 Tme Fg. 7. Detals on the evoluton of % behavours on tme for the ndvdual-based model for 2 runs for β = 29, g=.7, 5 neghbours and 4% ntal behavours We have notced that both models can mantan a mnorty, at least for a certan tme for the ndvdual based model. The dfference s due to the fact that the aggregated model has no stochastcty to allow the mnorty to spread. Let s see fgure n 9 whch presents the aggregated model potental functon. ggregated model potental functon for g=.7, beta=29 and 5 neghbours ntal percentage of "" ggregated model potental functon for g=.7, beta=29 and 5 neghbours ntal percentage of "" 2,499 2, , , , , , , Fg. 8. ggregated model potental functon for β = 29, g=.7, 5 neghbours The potental functon shows that the aggregated model cannot ncrease to 1 % of behavour when t begns wth less than 5% of behavour (see the zoom on rght), because t s caught nto the potental mnmum whch appears n the zoomed fgure. The stochastc events takng place n the ndvdual based model allow t to clmb out and to jon 1% of behavour. In the second case for whch the aggregated model goes to 1% of behavours whle the ndvdual based model converges to a percentage of behavours close to 13/15

14 , the dfference s easer to understand. It can be totally explaned by the fact that behavours obtaned by the ndvdual wth neghbours (who has n consequence a random decson) are, at the next step, dstrbuted nto each neghbourhood sze. The behavour s n consequence artfcally favoured and can be spread to the entre populaton whereas t s not the case n the ndvdual based model. 5 Dscusson Concluson We worked on a smplfed verson of a classcal threshold model. We have just consdered the case where all have the same personal nterest fxed to zero. prevous work compared ths model to an aggregated approxmaton, based on a smplfcaton assumng a fxed neghbourhood sze. In ths paper, we propose a refned verson of aggregated approxmaton, based on a Posson dstrbuton of the neghbourhood szes. The comparson between the three models ponts out that the varable neghbourhood aggregated model better predcts the fnal state of the ndvdual based model. However, there reman some dfferences between ndvdual based and the refned aggregated model. We showed that ths happens n partcular when the aggregated model s caught n a very small local mnmum of ts potental. Ths mnmum s not stable for the ndvdual based model, because the stochastc events are strong enough to extract t from such a mnmum, and send t to a much deeper one. 6 cknowledgment Ths work has been led partly under the responsblty of Laurent Derouss and Mchel Gourgand from LIMOS (Unversty Blase Pascal of Clermont-Ferrand). I thank them for ther support. 7 References 1. xelrod, R.: The convergence and stablty of cultures: local convergence and global polarzaton. nn rbor, Insttute of Publc Polces Study, Unversty of Mchgan, 34 (1995). 2. Blume, L.: The statstcal mechancs of strategc nteracton. Games and economc behavour 4: (1993). 3. Blume, L.: The statstcal mechancs of best-response strategy revson. Games and economc behavour 11: (1995). 4. De ngels, D.L., Gross, L.J. Edtors: Indvdual-based models and approaches n ecology, Chapmann & Hall (1992). 5. Duboz, R., Ramat, E., Preux, P.: Scale transfer modellng : usng emergent computaton for couplng an ordnary dfferental equaton system wth a reactve ndvdual model (22). 14/15

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