Ising-like agent-based technology diffusion model: adoption patterns vs. seeding strategies

Transcription

1 Isng-lke agent-based technology dffuson model: adopton patterns vs. seedng strateges Carlos E. Lacana ab,1 and Santago L. Rovere b a Facultad de Cencas Fscomatemátcas e Ingenería, Unversdad Católca Argentna, Avenda Alca M. de Justo 1400, Cudad Autónoma de Buenos Ares, C1107AAR, Argentna. b Laboratoro de Modelacón Matemátca, Facultad de Ingenería, Unversdad de Buenos Ares, Avenda Las Heras 2214 Cudad Autónoma de Buenos Ares, C1127AAR, Argentna. Abstract The well-known Isng model used n statstcal physcs was adapted to a socal dynamcs context to smulate the adopton of a technologcal nnovaton. The model explctly combnes (a) an ndvdual s percepton of the advantages of an nnovaton and (b) socal nfluence from members of the decson-maker s socal network. The mcro-level adopton dynamcs are embedded nto an agent-based model that allows exploraton of macro-level patterns of technology dffuson throughout systems wth dfferent confguratons (number and dstrbutons of early adopters, socal network topologes). In the present work we carry out many numercal smulatons. We fnd that when the gap between the ndvdual s percepton of the optons s hgh, the adopton speed ncreases f the dsperson of early adopters grows. Another test was based on changng the network topology by means of stochastc connectons to a common opnon reference (hub), whch resulted n an ncrement n the adopton speed. Fnally, we performed a smulaton of competton between optons for both regular and small world networks. Keywords: Collectve decson; Technology adopton; Isng model; Early adopters; Networks. Pacs codes: s; Gh. 1 Correspondng author. Tel./Fax: (54-11) E-mal address: clacan@f.uba.ar (Carlos Lacana). 1

2 1. Introducton The model for statstcal physcs known as Isng model was orgnally developed by Ernst Isng n 1925 to explan phase transtons n ferromagnetc materals [1]. It has been recently used n the smulaton of several socal processes [2], such as collectve opnon formaton [3][4][5, 6] or adopton of new technologes [7]. The versatlty of the Isng model lays on the fact that the nteracton effects for any gven object wth ts neghbors s consdered proportonal to the number of neghbors n each state. Those objects can be spns (n up or down states), ndvduals wth poltcal postons (A or B), adopters and nonadopters of a new technology, populaton members nfected and not nfected wth a contagous dsease, etc. Socal networks are the man channels for the nteracton n socal models[2].in order to adapt the Isng model to a socal context, we must add them to the orgnal model [8].Network nodes represent ndvduals and lnks represent the communcaton channels between them. The topologcal characterstcs of socal networks have consderable nfluence on nteracton dynamcs n ths case, the dffuson of nnovatons. Dfferent topologes have been dscussed n the lterature, such as the small world [9][10], scale-free [11][12][13], modular [14] and regular [7]. Any of these topologes maybe used wth the Isng model n a straghtforward way [15]. However, when physcal proxmty among nodes s mportant, a regular lattce provdes a good approach. In our analyses, we wll mostly consder regular two-dmensonal lattce and n some cases, small worlds networks. In the orgnal Isng model, the change n the spn orentaton occurs when a threshold s reached n the mean feld of the node. In a smlar way, a threshold of decson must be reached n order to change the decson-maker agent state. In most models of technology adopton (Delre, Jager and Janssen 2006) there are two basc terms whch determne the threshold of decson: (a) socal nfluence from a decson-maker s socal network; and (b) the ndvdual percepton of a decson-maker agent about the benefts (or utlty) of the new opton. These two factors are combned nto an effectve utlty that reflect the effects of both ndvdual utlty and socal nfluence. The comparson of effectve utltes (.e., the relatve effectve utlty) of both optons leads to the selecton of one opton or another. The relatve 2

3 weght of the ndvdual percepton and socal nfluence depends on the choce type[16]. For example, n fashonable markets (clothes, electronc gadgets) socal nfluence has a strong weght, whereas on other choces (e.g., groceres), socal nfluences are weaker. As n physcal systems, ntal condtons have a strong nfluence on the evoluton of a socal system. In ths partcular case, technology adopton patterns are senstve to the dstrbuton of ntal adopters (referred to as seedng ) n the network. Ths effect was llustrated by Lba et al. [17], who showed that marketng strateges leadng to dfferent spatal dstrbutons of early adopters ntroduce dfferences n the speed of adopton of a new product. The same ssue was addressed by Delre et al. [18], who explored dffuson patterns resultng from alternatve (spread out or concentrated) dstrbutons of early adopters. In ths paper, we explore systematcally the nfluence of spatal dsperson of early adopters on the subsequent adopton dynamcs. A close relaton exsts between the dstrbuton of ntal adopters and the take-off tme of the new product. In another way, Delre et al. [18] concentrate on the targetng and the tmng of the promotons n relaton to the take-off. That s prevous to the generaton of the dstrbuton of ntal adopters. Moreover, they use an agent-based model wth a slghtly dfferent decson algorthm(both thendvdual percepton and the socal nfluence must reach dfferent thresholds ndependently, whle n our approach the decson results from the effectve utltes assocated to each opton). That decson s made n each tme step, determnng whch opton s adopted. Therefore the possblty of dsadopton s ntroduced. Ths mechansm s useful when the adopton of a new product does not mply any nvestment (learnng, technology or any other resource). If the last assumpton s not satsfed, a model wth no dsadopton would be more approprate. As our approach allows dsadopton, t can be consdered as a smple competton process. In reference [7], the basc mcro-structure of two or three ntal adopters necessary to keep up dffuson was studed. In the present paper, we propose an extenson by ntroducng many dstrbutons of ntal adopters wth dfferent dsperson degrees, n order to understand how the clusterng of ntal adopters affects the adopton speed. In ths paper, the smulatons were performed usng an agent-based model. Agent-based modelng s a way of dong thought experments, obtanng, n many cases, non-obvous results and emergent patterns of the system [19]. The orgnalty of ths work does not lay n the ntroducton of a new statstcal model (snce the well-known Isng model has already been studed), but n the analyss of emergng evolutonary patterns assocated to the adopton of a new product. 3

4 The paper s organzed as follows. Secton 2 presents a bref revew of the Isng model and ts applcaton to modelng of nnovaton adopton n a socal context. In Secton 3 our specfc mplementaton of the agent-based model s presented. Secton 4 nvolves varous experments of technology dffuson ncludng the study of adopton rate due to seedng effects, changes n the ndvdual preference (both n space and tme) and connecton to a hub. Secton 5 presents conclusons. 2. A model of technology dffuson 2.1 The Isng model n the physcal context The Isng model was orgnally developed to explan phase transtons n ferromagnetc materals. For example, suppose we are nterested on descrbng a phase transton process n a ferromagnetc materal. We can envson the materal as consttuted by a lattce of mcro-magnets called spns that can nteract wth ther nearest neghbors and wth an external feld. We wll dentfy the state of spn n the th poston of the lattce by dscrete varable s that can take the values +1 or -1. If the system s consttuted by N spns, ts total energy s N N N N E E w s h s m s, (1) k k 1 1 k 1 1 Where w k s the couplng strength between nearest neghbor spns, and h s a constant external magnetc feld. E s the energy assocated wth spn, where m s the magnetc feld around spn. From Eq. (1) t follows that E ( s 1) m. The probablty of fndng spn n state s {+1;-1} s gven by the Boltzmann-Gbbs dstrbuton: 1 Ps ( 1) 2 m 1 e, (2) 4

5 Wth = 1/kT, where k s Boltzmann s constant, and T s the system temperature. Moreover, usng detaled balance condton, that ensures convergence to equlbrum, Eq. (2) can be used to calculate the transton probablty between the two states.in the followng secton, an nterpretaton for each term of Eq. 2 wll be gven n the context of a socal system. 2.2 Usng the Isng model to smulate the dffuson of nnovatons The orgnal Isng model can be adapted to a socal context n order to smulate the adopton (or dsadopton) of an nnovaton. In a socal system, the spns n the Isng model can be nterpreted as N ndvduals, households or frms hereafter referred to as agents who must choose between two optons: A (s = +1) or B (s = -1); optons A and B may represent, for example, new and exstng technologes respectvely. The magntude m can be nterpreted as the relatve effectve utlty between optons A and B. In order to use a more famlar notaton, n the socal context we wll denote the relatve utlty as U nstead of m. As n Ref. [7] and by analogy wth Eq. (1), we can wrte U as: N J U s u k 2 2 (1 ) k k 1( k ) Nv (3) The r.h.s. of Eq. (3) has two terms. The frst one (analogous to the nteracton term between spns n the orgnal Isng model) descrbes the contrbuton of socal nfluence from decson-maker s socal network, whereas the second term (analogous to the external magnetc feld n the physcal model) descrbes the contrbuton of s ndvdual preference for optons A or B, rrespectve of other agents [3, 6, 20]. N s the total number of agents, N v the number of ndvduals connected to agent by a frst-order lnk n the socal network (.e., those agents assumed to havea socal nfluence on s decsons),j k quantfes the socal nfluence of agent k on agent s decsons (n all smulatons we assume J k = 1,k, that s, all agents have the same nfluence over other agents). Factor weghs the relatve mportance of socal nfluence and ndvdual preference on the overall utlty of a gven opton/product: f >0.5,socal nfluence s more mportant than the ndvdual preferenceand vce versa. In all smulatons we wll use = 0.5 (.e., both components of effectve utlty are assumed to have the same weght). 5

6 The ndvdual preference component reflects an agent s dosyncratc preferences for optons A or B, rrespectve of other agents [3, 6, 20]. Indvdual preference u n Eq. 3 can be defned as u u(a) u(b), (4) max[ u (A); u (B)] Where u (A) and u (B) represent the utltes experenced by an ndvdual f he chooses opton A or B respectvely.here, functon u(x) reflects a broad measure of desrablty, even ncludng noneconomc factors, and can take dfferent forms: t may represent, for example, the expected value of economc profts from a gven opton. Values of u n Eq. 4 are dmensonless and range wthn nterval [-1; 1]. Note that many quanttes n Eq. 3 are ndexed by agent. Unless otherwse specfed, all our smulatons assume that these quanttes are the same for all agents. That s, possble dfferent personal characterstcs of each agent (e.g., rsk averson that may nfluence valuaton of a gven opton) are not taken nto account. Ths s analogous to consderng a set of agents wth a mean value for each parameter. 2.3 Decson algorthm The orgnal Isng model assumes an equlbrum temperature that defnes the probablty of permanence n each spn state. In a socal context, ncluson of a system temperature ntroduces global uncertanty n a decson (affectng both socal and personal components), turnng t nto a stochastc event. In such context, the temperature T (Eq. 2) can be nterpreted as random nose, due to erratc crcumstances that nfluence the opnon of all agents about the advantages of selectng one of the two optons [21, 22]. For example, f the agents are farmers decdng on adopton of a new crop varety, temperature may represent fluctuatons related to epdemcs, annual weather fluctuatons, poltcal events. These events change the percepton of farmers and mght make them take decsons that they would not have taken under normal crcumstances [7]. The effects of temperature on the probablty of occurrence of a gven event (adopton or nonadopton) wll not be analyzed here; nstead, n all subsequent smulatons we only consder the case of T=0. From Eq. (2) t follows that, when T = 0 (.e., no random nose), the probablty of a gven event s fully determned by the sgn of U : 6

7 1 f U 0 1 lm P( s 1) f U 0 T f U 0 (5) and conversely lm P( s 1) 1 lm P( s 1). (6) T 0 T 0 In ths lmt, the model s determnstc except for U = 0, when both events have equal chance. 3. Agent-based model of technology dffuson Many recent studes of dffuson and adopton rely on agent-based modelng, a powerful smulaton technque that s very promsng for developng new dffuson theory[19, 23, 24]. Zenoba et al. [25] assess the strengths, opportuntes, weaknesses, and threats facng agent-based modelng n technologcal nnovaton research. An agent-based model (ABM) conssts of a collecton of autonomous decson-makng enttes ( agents ), an envronment, and rules that determne sequencng of actons n the model. Each agent has sensory capabltes and makes decsons on the bass of a set of rules[26]. Agents can nteract ether ndrectly through a shared envronment and/or drectly wth each other through markets and, especally, through socal networks [27, 28]. We mplement here an agent-based model of technology dffuson based on the Isng model descrbed n the prevous secton. Many software frameworks are avalable that facltate ABM development; we use REPAST Smphony, an open-source framework mantaned by Argonne Natonal Laboratory[29]. The envronment of the ABM s a 2-D lattce wth 10,000 nodes(arranged n a 100 X 100 grd). Each node represents an agent that makes decsons about the adopton (or dsadopton) of a technology.agents located on the edges of the lattce have fewer neghbors because we do not assume perodc boundares (.e., a torus). 7

8 At each tme step of the smulaton, an agent decdes f he/she adopts a technologcal nnovaton based on the overall relatve utlty of the new technology (Eq. 3). The decson algorthm s gven n Eq. 5. That s, the sgn of overall relatve utlty U determnes f the agent goes from state -1 to +1 (adopton) or vceversa (dsadopton) or, alternatvely, stays n hs current state. Each smulaton contnues untl only small fluctuatons are observed n the adopton pattern, or a gven opton prevals completely. 3.1 Adopton threshold As shown n a prevous secton, the adopton of an nnovaton by agent at tme t depends on the sgn of the overall relatve utlty U of old and new products or technologes. It follows that, for each value of the ndvdual preference between optons ( u n Eq. 3), there s a threshold of socal nfluence for agent (expressed as the number of adopter neghbors) below whch adopton does not take place. We denote the number of neghbors of agent n states s = +1 and s = -1 as v + and v -, respectvely. It follows from Eq. (3) that k N 1( k ) J s v v (7) k k The assumpton used n ths calculaton s that all components J k are zeroes or ones; ths means that there are no agents wth hgher nfluence than others,.e., all socal lnks have the same weght. Therefore, the condton U > 0 for the adopton of opton +1 by agent (assumng α = 0.5) can be rewrtten more explctly as: 1 ( v v ) u 0 N v (8) Then, takng nto account that v + + v - = N v, the condton for adopton s: 1 mn v Nv(1 u ) v (9) 2 where v + mn s the mnmum number of frst-order socal contacts necessary for adopton. 8

9 4. Results 4.1 Effect of seedng on dffuson speed Prevous work has shown that technology adopton patterns are senstve to the seedng or dstrbuton of early adopters (.e., those who already have adopted the nnovaton when smulatons are started). For example, [17] showed that marketng strateges leadng to dfferent spatal dstrbutons of early adopters can, n turn, ntroduce dfferences n the speed of adopton of a product. Smlarly, [18] explored dffuson patterns resultng from spread-out or concentrated dstrbutons of early adopters. Ths secton explores the effects of the spatal dstrbuton of ntal adopters on the speed of dffuson of a new technology. We smulate dffuson patterns resultng from ncreasng spatal dsperson n the dstrbutons of a constant number (N = 100) of ntal adopters. To defne the postons of the 100 ntal adopters on a 100 x 100 lattce, we draw random values from two Gaussan dstrbutons representng, respectvely, two uncorrelated varables: the x- and y- coordnates on the grd. The two dstrbutons are centered on the grd. To quantfy the spatal dsperson of adopters, we ntroduce an ad-hoc, computatonally convenent defnton of dsperson. By analogy wth the unvarate standard devaton, we defne a square area, centered n the mddle of the grd, that ncludes about 68% of the ntal adopters (n ths case, 68 agents), and defne "dsperson", as half the sde of ths square, measured n number of postons n the lattce (Fg. 1a). [Insert Fgure 1 about here] We wll consder the lattce space as the potental market for a new product or technology. In the begnnng, the market s saturated wth an earler product whch wll be n competton wth the new one. Possble marketng strateges for the new product are to concentrate advertsng resources n a small area or, alternatvely, target a broader area. The queston that arses s: whch of these strateges s more effcent?. To address ths queston, we wll assume a drect correspondence between the pattern of advertsng and the dstrbuton of ntal adopters. That s, f advertsng s concentrated n a small regon, early adopters wll be concentrated n the targeted area. Conversely, early adopters wll be more dspersed when the advertsng campagn targets an extended area. The spatal dsperson of ntal adopters s quantfed by (see dscusson above). 9

10 [Insert Fgure 2 about here] Fg. 2 shows the assocaton between saturaton tme (.e., the tme at whch the space s completely occuped by the new product) and a broad range of values of spatal dsperson of early adopters rangng from 4 (a very tght dstrbuton) to 41 (whch approxmates a unform dstrbuton). Results are smulated for two alternatve network topologes: (a) a regular network (tes are defned by frst-order neghborhood n the lattce) and (b) a small world network generated usng a rewrng procedure smlar to that descrbed by Watts and Strogatz[9]. The rewrng readjusts the edges for each node, movng an edge to another randomly selected node wth probablty p=0.005; ths rewrng probablty s nsde the nterval [0.003,0.02] that characterzes small-world networks n a 2-D network [30]. Smulatons also are carred out for three dfferent values of u (0.8, 0.6 and 0.4), ndcatng respectvely large, ntermedate and small dfferences n the relatve utlty of the two products; as all u values are postve, we assume that the nnovaton has advantages over the exstng technology or product. As u decreases, adopton wll occur only f socal nfluence effects are hgher:.e., a larger number of adopter neghbors s needed. The three u values used n smulatons requre 1, 2 and 3 adoptng neghbors, respectvely, as shown n Eq 9. For each combnaton of u and network topology, 100 smulatons are run wth dfferent dstrbutons of early adopters; results plotted n Fg. 2 represent the average of the 100 runs. For u = 0.4, saturaton s reached n most experments up to a certan dsperson of ntal adopters ( values up to about 15). In ths range, dffuson proceeds faster for lower values (.e., tme to saturaton s lower). Ths pattern s qualtatvely smlar for regular and small-world networks, although dffuson s faster for the latter (the offset between curves for both topologes seems farly constant). As dsperson ncreases ( )most experments do not lead to saturaton, and therefore saturaton tme s nfnte (thus, no lne s shown). When the adopton threshold s lower ( u = 0.6 or 0.8), saturaton s reached n every experment. Unlke the prevously descrbed stuaton, dffuson proceeds faster when the dsperson of ntal adopton ncreases (as grows). Smallworld networks show faster dffuson than regular networks because of the exstence of shortcuts (weak tes). 10

11 4.2 Changes n the spatal dstrbuton of ndvdual preference In real stuatons, ndvdual preferences for competng technologcal optons can be dfferent for each decson-maker. In some cases, ndvdual preference may be a functon of the spatal locaton of an agent. For example, suppose we are comparng two crop varetes, one of whch has a hgher tolerance to stresses assocated wth water shortages. In ths case, t s lkely that the droughttolerant varety wll have a hgher utlty n drer locatons where water stresses are more lkely, and lower utlty n places where ranfall s plentful. To explore ths effect, we perform smulatons n whch u decreases radally from the center of the lattce. The value of u at any node of the lattce s gven by 0 d l 2 u u e (12) where u = 0.8 s the utlty at the centre of the lattce, ls a length scale assocated wth the grd (n ths case, l N /2, where Ns the total number of agents) and s a dmensonless parameter that descrbes how quckly u changes wth dstance (n ths case = 3). We also carry out experments preservng the u gradent but varyng the network topology through a rewrng process smlar to the one descrbed above. Both sets of experments are based on 100 early adopters wth a unform spatal dstrbuton. From Equatons 9 and 12, t can be nferred that there are concentrc regons n the lattce wth dfferent numbers of adopter neghbors requred for adopton. These adopton thresholds are ndcated n Fgs. 3a and b for neghborhoods of sze 4 and 8 respectvely. [Insert Fgure 3 about here] Fgure 3 dsplays adopton patterns after the dffuson dynamcs have been completed (.e., when no further changes occur). In most cases, total adopton does not occur (an excepton s Fg. 3g). For regular lattces, the fnal dstrbutons nvolve a central sland of adopters wth compact and regular geometrc shapes: a square when four neghbors (a von Neumann neghborhood) are consdered (Fg. 3c), and an octagon when eght neghbors (a Moore neghborhood)are used (Fg. 3d). These patterns can be easly nterpreted. Gven the radally decreasng u pattern, as we move away from the center of the lattce, more adopter neghbors are requred for adopton to occur. At a gven dstance from the center, three adopter neghbors are needed. Ths threshold, however, s 11

12 dffcult to reach gven the scattered dstrbuton of ntal adopters therefore adopton does not proceed beyond ths boundary. For a regular lattce (rewrng probablty p = 0), socal nfluence nvolves only geographcal vcnty therefore the sland of adopters grows outwards regularly, n crystal-lke fashon. In contrast, when randomness n tes s ntroduced through a rewrng probablty p > 0, fnal adopton slands no longer have regular shapes. Ths s because neghbors no longer nclude only adjacent agents n the lattce. The edges of the adopton slands correspond to adopters who are farthest from the center of the space, and thus are most lkely to become non-adopters f they lose an adopter neghbor due to rewrng. For p = 0.25, adopton slands are approxmately crcular, wth a few adopters outsde (Fg. 3e-f). The sze of the adopton sland s larger for neghborhoods of sze 8. The larger sze s due to the regon n ths neghborhood where the adopton threshold s 3; ths regon does not exst for neghborhoods of sze 4. To explan ths behavor, one must remember that that socal nfluence s defned by the proporton of adopters n the neghborhood, not ther absolute number. For an 8- agent neghborhood, n the zone where the adopton threshold s 3 an agent needs 37.5 % of neghbors (3 out 8) to be adopters. In the same zone, for a 4-agent neghborhood an agent needs 50% of adopter neghbors (2 out of 4) to adopt. When rewrng probablty s ncreased from p = 0.25 to p = 0.50, the adopton patterns are very dfferent for neghborhoods of sze 4 and 8 (Fgs. 3g and 3h). For neghborhoods of sze 4, complete adopton s observed (Fg. 3g), that s, an ncrease n adopton wth respect to the case n whch p = 0.25 (Fg. 3e). In contrast, for an 8-agent neghborhood there s a lower number of adopters (Fg. 3f); furthermore the ncrease n rewrng probablty results n a decrease n adopton for the same neghborhood (Fg. 3h vs 3f). 4.3 A partcular change n the topology of the socal network: connecton to a hub Dffuson of nnovatons s thought to be strongly nfluenced by people who have a large number of tes to other people [31]. In the socal network lterature, these ndvduals are referred to as hubs, nfluentals, opnon leaders, or members of a royal famly, watched by many others n the network[32]. In ths secton we study the dynamcs of adopton when we ntroduce a hub agent connected to a large number of agents n the lattce. The lterature suggests that hubs may have a strong nfluence 12

13 on other agents opnons or actons thus the te to the hub would have a stronger nfluence than other tes. Nevertheless, here we assgn to the hub the same nfluence as any other agent, so we can observe the senstvty of the model to the changed network topology. A smlar model s used n [17], where each ndvdual s nfluenced by two groups of agents: hs or her nearest neghbors ("neghborhood effect") and agents from other regons, denoted as "relatves" and connected through weaker tes. However, the Lba et al. [17]model dffers slghtly from the one used here: the term correspondng to ndvdual preference for an nnovaton does not appear (.e., only socal nfluences are consdered) and rreversblty of transton s assumed consumers cannot dsadopt once they have adopted. Our model, n contrast, s symmetrcal (adopton and dsadopton are both allowed), as n models used foropnon formaton. To perform smulatons n ths secton, the orgnal regular network was modfed by stochastcally selectng 100 agents;these agents were then connected to a hub arbtrarly located at the center of the lattce by replacng stochastcally wth a probablty p = 0.1 one of ther ncomng lnks from a neghbor. Although ndvduals wth many socal tes are not necessarly nnovators [33], our smulated hub s ntalzed as an adopter. [Insert Fgure 4 about here] Fgures 4a and 4d dsplay the proporton of adopters as a functon of tme for experments wth and wthout a hub, and for u = 0.6 and u = 0.4, respectvely. For both u values, adopton proceeds faster when a hub s present. Fgures 4b, c, e and f show snapshots of adopton patterns for comparable stages of the dffuson. These plots confrm the faster spread of an nnovaton when a hub s present, as suggested by the much larger number of adopters at the same step of a smulaton. Furthermore, the growth of adopton slands s dfferent wth and wthout the hub. When there s no hub and only local neghborhood nfluences dffuson, patterns show a crystal-lke growth as shown prevously. The hub ntroduces a random component n the network topology, and adopton can spread n any drecton. 4.4 A smple model of competton between two optons In all prevous experments, the relatve utltes of old and new technologes or products were defned at the begnnng of an experment and dd not change throughout the smulaton. In ths secton, n contrast,we smulate dynamc competton between two products by allowng changes n u through out a smulaton. 13

14 We assume that a new product s ntroduced nto a market and starts competng wth a pre-exstent product. At ts ntroducton, the new product s better than the older one (.e., u > 0) and thus gans market share. However, once the manufacturers of the older product notce that the new product has reached a certan proporton of the market (whch we denote as the crtcal market share or CMS), they react by ntroducng mprovements such that the older product matches the utlty of the new product (.e., u becomes zero).we explore how the dynamcs of competton evolve n response to ths change n relatve utltes. Frst, we study the probablty that the new product wll contnue to preval (.e., mantan or ncrease ts market share) after the enhancement of the older product. We denote the probablty of prevalence as, and we estmate t as the relatve frequency wth whch the new product contnues to preval n a set of 100 smulatons for a gven set of condtons. [Insert Fgures 5 (a) and 5 (b) about here] Fgure 5a shows how vares as a functon of CMS for a regular network. When CMS < 0.5, there s a hgher probablty of prevalence when the seedng of ntal adopters s spatally dsperse.in contrast, when CMS 0.5, concentrated dstrbutons of ntal adopters have a hgher probablty of prevalence. Ths transton happens more abruptly when the early adopters are more concentrated because n ths case the amount of non-equvalent confguraton s smaller. Fgure 5b shows results for a small-world network (wth a rewrng probablty of 0.005). In ths case, changes n the probablty of prevalence are much less senstve to the ntal dsperson of early adopters. In ths case, the probablty of prevalence evolves much more regularly as a functon of CMS: the hgher the share ganed by a new, superor product before t s matched by a prevous product, the hgher are the chances that the new product wll retan or enhance the market share durng the perod when t was better than the older product. 5. Conclusons The man conclusons of our experments are as follows: When a new product or technology has clear advantages over exstng products (.e., u > 0.4), the nnovaton s adopted more quckly when early adopters are spatally dsperse. In contrast, when the new product s only slghtly better ( u = 0.4) than the exstng opton, market saturaton s not reached by the new product f early adopters are 14

15 dspersed. Therefore, n the case of new products wthout clear advantages, marketng strateges should am to develop a concentrated set of early adopters. In contrast, when the new product s clearly superor, the best marketng strategy seems to be to attan a broader spatal dstrbuton of early adopters. These conclusons are applcable to both regular and small-world networks. We explored a stuaton n whch the advantages of an nnovaton ( u) decrease regularly as a functon of dstance from the center of the lattce. Adopton thresholds have a regular pattern of concentrc crcles. However, the functonal form of u constrans most adopters to a compact central regon. Therefore, equlbrum adopton patterns look lke slands of adopton (octagonal for neghborhoods of sze 8, square for sze 4 neghborhoods). These symmetrcal patterns are reached ndependently of the degree of dsperson of the ntal adopters. The symmetrcal patterns are broken when other topologes are used. The presence of agents who have a large number of tes to other agents (referred to as hubs, nfluentals, or opnon leaders) accelerates the adopton of a new technology or product. Moreover, geometrc patterns n the dffuson of a new product are observed, whch are very dfferent to those obtaned when only spatal neghbors are consdered. Fnally, we performed an experment n whch the ntal advantage of a new product s subsequently matched by enhancements n the older competng product. For a regular network wth spatally concentrated early adopters, a marketng strategy should am to acheve quckly at least half of the market share. In ths case, reacton by the competton does not decrease the market penetraton ntally ganed by the new product durng the perod when t was superor to the alternatve. If the dstrbuton of ntal adopters s dspersed, the chances of the new product retanng ts market share decrease. For smallworld networks, spatal dstrbuton of early adopters does not nfluence sgnfcantly the probablty of retanng market share. Acknowledgements Useful comments by two anonymous revewers are gratefully acknowledged. Ths research was supported by two U.S. Natonal Scence Foundaton (NSF) Coupled Natural and Human Systems 15

16 grants ( and ). Addtonal support for one of the authors (S. R.) was provded by grant CRN-2031 from the Inter-Amercan Insttute for Global Change Research (IAI), whch s funded by NSF Grant GEO , and by the Unversty of Buenos Ares. References [1] E. Isng, "Betrag zur Theore des Ferromagnetsmus," Zetschrft für Physk, vol. 31, pp , [2] F. Vega-Redondo, Complex Socal Networks, Cambrdge Unversty Press. ed.: Cambrdge Unversty Press., [3] W. Wedlch, "The use of stochastc models n socology " Collectve Phenomena, vol. 1, pp , 1972 [4] H. Haken, Synergetcs. An Introducton, 2da ed.: Sprnger-Verlag, [5] A. Grabowsk and R. A. Kosnsk, "Isng-based model of opnon formaton n a complex network of nterpersonal nteractons," Physca A, vol. 361, pp , [6] S. Galam, "Ratonal group decson makng: A random feld Isng model at T = 0," Physca A, vol. 238, pp , [7] G. Wesbuch and G. Boudjema, "Dynamcal aspects n the adopton of agrenvronmental measures," Advances n Complex Systems, vol. 2, pp , [8] M. Newman, et al., The Structure and Dynamcs of NETWORKS: Prnceton Unv. Press [9] D. J. Watts and S. H. Strogatz, "Collectve dynamcs of small-world networks," Nature, vol. 393, pp , [10] D. J. Watts, Small Worlds, The Dynamcs of Networks between Order and Randomness: Prnceton Studes n Complexty, 2004 [11] R. Albert, et al., "Dameter of the world-wde web," Nature, vol. 401, pp , [12] J. M. Klenberg, et al., "The Web as a graph: Measurements, models and methods " n Internatonal Conference on Combnatores and Computng, 1999, pp [13] R. Pastor-Santorras and A. Vespgnan, "Epdemc dynamcs n fnte sze scale-free networks," Phys. Rev. E vol. 65, pp. ( )-( ) [14] R. K. Pan and S. Snha. The small world of modular networks [Onlne]. [15] A. Pekalsk, "Isng model on a small world network," Phys. Rev. E, vol. 64, p , [16] S. A. Delre, et al., "Dffuson dynamcs n small-world networks wth heterogeneous consumers," Comput. Math. Organz. Theor., vol. 13, pp , [17] B. Lba, et al., "The role of seedng n mult-market entry " Inter. J. of Research n Marketng, vol. 22 pp ,

17 [18] S. A. S.A. Delre, et al., "Targetng and tmng promotonal actvtes: An agentbased model for the takeoff of new products," Journal of Busness Research vol. 60, pp , [19] R. Axelrod, The complexty of Cooperaton: Prnceton Unv. Press [20] A. Grabowsk, "Opnon formaton n a socal network: The role of human actvty," Physca A, vol. 388, pp , [21] F. Schwetzer and J. A. Holyst, "Modellng collectve opnon formaton by means of actve Brownan partcles," European Physcal Journal B, vol. 15, No 4, pp , [22] A. Grabowsk, "Opnon formaton n a socal network: The rolo of human actvty," Physca A, vol. 388, pp , [23] D. Parker, Manson, S.M., Janssen, M.A., Hoffmann, M.J. and Deadman, P.., "Mult-agent systems for the smulaton of land-use and land-cover change," Annals of the Assocaton of Amercan Geographers vol. 94, pp , [24] V. a. R. Grmm, S.F., Indvdual-based modelng and Ecology Prnceton Unversty Press, Prnceton. [25] B. Zenoba, et al., "Artfcal markets: A revew and assessment of a new venue for nnovaton research," Technovaton, vol. 29, pp , [26] M. W. a. W. Macy, R., ". From Factors to Actors: Computatonal Socology and Agent-Based Modelng " Annual Revew of Socology vol. 28 (1), pp , [27] N. Glbert, Agent-based models vol Los Angeles: SAGE Publcatons, [28] M. J. North and C. M. Macal, Managng busness complexty: dscoverng strategc solutons wth agent-based modelng and smulaton. Oxford: Oxford Unversty Press, [29] M. J. North and C. M. Macal, Managng Busness Complexty. Dscoverng Strategc Solutons wth Agent-Based Modelng and Smulaton: Oxford Unv. Press, [30] Z. Xu and D. Z. Su, "Effect of Small-World Networks on Epdemc Propagaton and Interventon," Geographcal Analyss vol. 41, pp , [31] M. A. Janssen and W. Jager, "Smulatng market dynamcs: Interactons between consumer psychology and socal networks," Artfcal Lfe, vol. 9, pp , [32] V. Bala and S. Goyal, "Learnng from neghbours," The Revew of Economc Studes, vol. 65, No 3, pp , [33] J. Goldenberg, et al., "The role of hubs n the adopton process," Journal of Marketng, vol. 73, pp. 1-13,

18 Fgure 1. (a) Defnton of the ad-hoc metrc of spatal dsperson of early adopters. a) s defned n a square regon of 2 x 2 such as about 68.2% of early adopters are ncluded there. b) Intal dstrbutonfor σ 4. c) Intal dstrbuton for σ d) Intal dstrbuton for a unform dstrbuton 18

19 Fgure 2: Saturaton tme ( ) versus dsperson parameter ( ) n the dstrbuton of early adopters for 2 dfferent values of u. Black lnes represent results for regular networks (RN) and grey lnes, results for small world networks (SWN). 19

20 Fgure 3: Adopton patterns assocated to changes n the topology; a) and b) shows the adopton thresholds for a neghborhood of 4 and 8 agents respectvely; c) and d) the slands of adopters at the end of the process, wth rewrng probablty p = 0; e) and f) wth p = 0.25 and g) and h) wth p =

21 Fgure 4: Comparson of adopton process for u = 0.6 (top fgures), u = 0.4 (bottom fgures) and rewrng probabltes p = 0 (wthout HUB) and p = 0.1 (wth HUB). For u = 0.6, we use a unform ntal dstrbuton of early adopters, whereas for u = 0.4 we use a dstrbuton wth = 29. The left chartsshow a comparson of the adopton curves for u = 0.6 (upper-left chart) and u = 0.4 (bottom-left chart). Fgures a), b), c) and d) are snapshots of the adopton pattern for dfferent combnatons of u, p and tme. 21

22 5a) Regular network 22

23 5b) Small world network Fgure 5: Probablty of prevalence vs. the crtcal market share (CMS). 23