Leaders and obstacles raise cultural boundaries
aa r X i v : . [ phy s i c s . s o c - ph ] S e p Leaders and obstacles raise cultural boundaries
M. G. Cosenza, O. Alvarez-Llamoza, C. Echeverria, and K. Tucci School of Physical Sciences & Nanotechnology, Universidad Yachay Tech, Urcuqu´ı, Ecuador Grupo de Simulaci´on, Modelado, An´alisis y Accesabilidad,Universidad Cat´olica de Cuenca, Cuenca, Ecuador CeSiMo, Facultad de Ingenier´ıa, Universidad de Los Andes, M´erida, Venezuela (Dated: October 2020)We employ an agent-based model for cultural dynamics to investigate the effects of spatial het-erogeneities on the collective behavior of a social system. We introduce heterogeneity as a randomdistribution of defects or imperfections in a two-dimensional lattice. Two types of defects are con-sidered separately: obstacles that represent geographic features, and opinion leaders, described asagents exerting unidirectional influence on other agents. In both cases, we characterize two collec-tive phases on the space of parameters of the system, given by the density of defects and a quantityexpressing the number of available states: one ordered phase, consisting of a large homogeneousgroup; and a disordered phase, where many small cultural groups coexist. In the case of leaders,the homogeneous state corresponds to their state. We find that a high enough density of obstaclescontributes to cultural diversity in the system. On the other hand, we find a nontrivial effect whenopinion leaders are distributed in the system: if their density is greater than some threshold value,leaders are no longer efficient in imposing their state to the population, but they actually promotemulticulturality. In this situation, we uncover that leaders, as well as obstacles, serve as locations forthe formation of boundaries and segregation between different cultural groups. Moreover, a lowerdensity of leaders than obstacles is needed to induce multiculturality in the system.
PACS numbers: 89.75.Fb, 87.23.Ge, 05.50.+q
INTRODUCTION
The study of dynamical processes on nonuniform orheterogeneous media is a topic of wide interest. Thenonuniformity may arise from the intrinsic heterogeneousnature of the substratum, such as porous or fractal me-dia, or it may be due to random defects or vacancies inthe medium at some length scales [1], or it may consistof added impurities in a material, as in semiconductordoping [2]. Heterogeneities can have significant effects inthe behavior of a system; for example, imperfections inthe crystal lattice of a solid can produce changes in prop-erties that open up new applications [3], or they can af-fect the formation of spatial patterns, by inducing phasetransitions in excitable media [4], and defects can serveas nucleation sites for growth processes [5].In the context of Social Sciences, the relation betweenspatial heterogeneities, in the form of geographic fea-tures, and the development and dissemination of cul-tures, has been an issue of long standing interest [6, 7].Recently, technological resources have allowed the studyof geographically embedded networks [8], or the relation-ship between geographic constraints and social networkcommunities [9]. Spatial inhomogeneities in social dy-namics may also occur by the presence of elements suchas influential agents, distributed mass-media messages,or outdoor advertising [10–12].In this article we employ an agent-based model to in-vestigate the effects of spatial heterogeneities on the col-lective behavior of a social system. We introduce hetero-geneity as defects or imperfections in a two-dimensional lattice. In Section II, we present a general model for cul-tural dynamics on a lattice with distributed defects. Ina first application, defects correspond to vacancies thatsimulate landscape obstacles. In a second model, we as-sociate defects to influential agents distributed on the lat-tice. Influentials are considered as a minority of agentswho can influence a large number of people, but are lesssusceptible to influences from other agents [13, 14]. Inopinion formation models, these agents are named opin-ion leaders or spreaders [15–18]; they are also called in-flexibles, zealots, or committed agents [19–22]. We em-ploy the spatial density of defects –either obstacles oropinion leaders– as a parameter of the system. The in-teraction dynamics is based on Axelrod’s rules for the dis-semination of culture among social agents [23], a paradig-matic model of much interest in Sociophysics [24–34].Section III contains our main results. We find that ahigh enough density of obstacles induces cultural diver-sity in the system. On the other hand, we encounter anontrivial effect when opinion leaders are distributed inthe system: if their density is greater than some thresholdvalue, leaders are not efficient in imposing their messageto the population, but they actually promote multicul-turality. We find that leaders, as well as obstacles, serveas locations for the formation of boundaries and segre-gation between different cultural groups. Moreover, ittakes a lower density of leaders than obstacles to inducemulticulturality in the system. Section IV presents theConclusions of this work.
CULTURAL DYNAMICS ON LATTICES WITHDEFECTS
An inhomogeneous lattice can be generated as fol-lows. Start from a two-dimensional array of sites of size N = L × L with periodic boundary conditions, and setas defects a given fraction ρ of sites at random. The dis-tribution of defects on the lattice can be characterized interms of the minimum Euclidean distance d between de-fects, as shown in Fig. 1. With this setting, the densityof defects scales with d as ρ = 0 . d − [35, 36]. FIG. 1. Two-dimensional spatial support of the system, show-ing active sites ( ◦ ) and defects ( • ). Defects are randomlyplaced in such a way that their density distribution is maxi-mum for a given minimum distance d between them. There are ρN randomly distributed defects in the lat-tice. The remaining N (1 − ρ ) sites consist of active agents, susceptible of changing their states. Follow-ing Axelrod’s model [23], the state of an active agent i ( i = 1 , , . . . , N (1 − ρ )) is given by the F -component vec-tor c i = ( c i , . . . , c fi , . . . , c Fi ), where each component c fi represents a cultural feature that can take any of q dif-ferent traits or options in the set { , , ..., q − } . Thereare q F possible equivalent states.In a first model, the ρN randomly distributed defectscorrespond to obstacles or vacancies in the lattice. Wedefine obstacles as empty sites that possess no dynamics;they may represent dispersed geographic features. In thesecond model, we consider that the ρN randomly dis-tributed defects are occupied by opinion leaders or influ-ential agents. We assume that opinion leaders share thesame fixed state, denoted by ( y , . . . , y f , . . . , y F ), whereeach component y f ∈ { , , ..., q − } remains unchangedduring the evolution of the system. Additionally, we as-sume that the interaction of opinion leaders with activeagents is unidirectional; i.e., opinion leaders can affect thestate of other agents, but their state does not change.This simplifying assumption reflects the basic role at-tributed to opinion leaders and influentials in a society[10–22]. This also comprises the notion of cultural sta-tus proposed by Axelrod [23]. With these conditions,opinion leaders can be seen as a spatially nonuniformmass media field acting on the system, or as distributed sources-transmitters of mass media messages.As initial condition in both models, each active agentis randomly assigned one of the q F possible states witha uniform probability. In this article, we fix F = 10 asthe number of cultural features. We define the dynamicsof the system for both models, in the presence of leaders(obstacles), by iterating the following steps:1. Select at random an active agent i .2. Select at random an agent j among the eight near-est neighbors of i in a Moore’s neighborhood (suchas j is not an obstacle).3. Calculate the overlap between the state c i of activeagent i and the state c j of its selected neighbor j , given by the number of shared cultural featuresbetween their respective vector states, as follows l ( i, j ) = ( P Ff =1 δ c fi c fj , if j is an active agent , P Ff =1 δ c fi y f , if j is a leader . (1)The delta Kronecker function employed is δ x,y = 1,if x = y ; δ x,y = 0, if x = y .4. If 0 < l ( i, j ) < F , with probability l ( i, j ) /F choose h randomly such that c hi = c hj and set c hi = c hj if j is an active agent, or c hi = y h if j is a leader.If l ( i, j ) = 0 or l ( i, j ) = 1, the state c i does notchange.In the absence of defects ( ρ = 0), a system subject toAxelrod’s dynamics reaches a stationary configuration inany finite network, where the agents constitute domainsof different sizes. A domain is a set of connected agentsthat share the same state. A homogeneous or orderedphase in the system is characterized by the presence ofone domain. The coexistence of several domains corre-sponds to an inhomogeneous or disordered phase in thesystem. On different networks, the system undergoes anonequilibrium transition between an ordered phase forvalues q < q o , and a disordered phase for q > q o , where q o is a critical value [25, 26]. The critical value for F = 10on a two-dimensional lattice has been numerically esti-mated at q o ≈
55 [27].
INFLUENCE OF OPINION LEADERS ANDOBSTACLES IN CULTURAL DYNAMICS
When defects in the form of leaders or obstacles arepresent in the lattice, the order-disorder transition per-sists, but the critical value q c for which the transitionoccurs decreases as the density ρ is increased. Figure 2shows the spatial configurations of the asymptotic sta-tionary states of the system with different densities ofleaders or obstacles, and fixed q = 44 < q o .The patterns corresponding to the model with obsta-cles are exhibited in the top panels of Fig. 2. We observethat the system reaches a homogeneous state when thedensity of obstacles is small. The homogeneous statecan be any of the possible q F states, depending on ini-tial conditions. Increasing the density of obstacles abovesome threshold value leads to the formation of multi-ple domains or multiculturality. The bottom panels ofFig. 2 show the behavior of the model with opinion lead-ers. When the density of opinion leaders is small, thesystem is driven towards a homogeneous state equal tothe state of the leaders. However, as the density of lead-ers is increased, the system no longer converges to thestate of the leaders, but reaches a disordered state. Do-mains having a cultural state equal to that of the opinionleaders survive, but they become smaller in size as ρ in-creases. Thus, we have the counterintuitive effect that,above some threshold value of their density, opinion lead-ers actually induce cultural diversity in the system. FIG. 2. Stationary spatial patterns for a two-dimensionallattice with different densities of obstacles (top panels) andleaders (bottom panels). Fixed parameters: N = 50 × F = 10, q = 44 < q o . Different domains are represented bydifferent colors. Top: obstacles are indicated as black siteswith densities ρ = 0 .
04 (left) and ρ = 0 .
14 (right). Bottom:leaders are identified as black sites for visualization, but theirfixed state corresponds to the light blue predominant color onthe left panel, ρ = 0 .
005 (left); ρ = 0 .
04 (right).
A useful order parameter to characterize the collectivebehavior is the average normalized size of the largest do-main in the system [25], denoted by S max . Here, wecalculate the quantity S max over the set of active agentsin the system, of size N (1 − ρ ). Figure 3 shows S max as a function of the number of options q for both models,with leaders or obstacles, for the values of density em-ployed in Fig 2. For each value of the density of leaders ρ , we observe a transition at a critical value q c ( ρ ), fromthe homogeneous cultural state imposed by the leaders,where S max = 1, to a disordered or multicultural statefor which S max →
0. The critical value q c ( ρ ) decreaseswith increasing ρ . Similarly, in the obstacle model, anorder-disorder transition takes place at a critical value of q that decreases as the density of obstacles is increased. ♠ ❛ ① ✼ ✵✻ ✵✺ ✵✹ ✵✸ ✵✷ ✵✶✵ ✿ ✽✵ ✿ ✻✵ ✿ ✹✵ ✿ ✷✵ FIG. 3. Order parameter S max as a function of q , for differ-ent values of ρ , with F = 10 and size N = 100 × S max are averaged over 50 realizations of initial condi-tions for each value of q . Curves for the model with obstaclescorrespond to ρ = 0 .
04 (empty squares) and ρ = 0 .
14 (solidsquares), while curves for the model with leaders are plottedfor ρ = 0 .
005 (empty circles) and ρ = 0 .
04 (solid circles). Thedotted vertical line signals the value q = 44 used in Fig. 2. Figure 4 shows the phase diagram of the system forboth models, with leaders and with obstacles, on thespace of parameters ( ρ, q ). There is a critical curve q c ( ρ )in each case, signaled by a continuous line for the modelwith leaders and by a dotted line for the system withobstacles. Each critical curve separates two well definedcollective phases: an ordered or homogeneous phase be-low the curve, characterized by S max = 1; and a dis-ordered or multicultural phase above the curve, where S max →
0. For the model with leaders, these phasesare indicated in Fig. 4 by the regions in white and graycolors, respectively. In both models, when ρ = 0, we re-cover the critical value q o corresponding to the transitionin Axelrod’s model.Below the critical boundary indicated by the dottedline in Fig. 4, the homogeneous phase reached in themodel with obstacles can be any of the possible statesin the system. A high enough density of obstacles leadsto multiculturality or a disordered phase, a result thatone may expect, since geographic obstacles usually con-tribute to the separation of cultures [7]. Counterintu-itively, if the density of opinion leaders is greater than q✚ ✻ ✵(cid:0) ♦✺ ✵✹ ✹✹ ✵✸ ✺✸ ✵✵ ✿ ✶ ✹✵ ✿ ✶ ✷✵ ✿ ✶✵ ✿ ✵ ✽✵ ✿ ✵ ✻✵ ✿ ✵ ✹✵ ✿ ✵ ✷✵ FIG. 4. Phase diagram for the system with leaders and withobstacles. Fixed parameters are F = 10, N = 100 × q = 44 and the density values used inFig. 3: obstacles with ρ = 0 .
04 (empty square); obstacles with ρ = 0 .
14 (solid square); leaders with ρ = 0 .
005 (empty circle);leaders with ρ = 0 .
04 (solid circle). some critical value, leaders are not longer efficient in im-posing their message to the population of active agents,and disorder ensues. Thus, leaders act as obstacles onthe collective behavior of the system when their densityexceeds a threshold value. Moreover, for a given numberof options q , a lower density of leaders than obstacles isrequired to produce multiculturality in the system.To elucidate the mechanism by which leaders and ob-stacles induce multiculturality, we consider the subsetof sites that constitute the interface between differentcultural domains in the stationary configuration of thesystem. We define that a site belong to the interfacesubset if at least three of its neighbors share its state,and at least three other share a different state. Then,we calculate the fraction of those sites in the interfacethat are leaders (or obstacles), denoted by ρ f . A ratiovalue ρ f /ρ < ρ f /ρ >
1, the opposite situation takesplace: leaders (obstacles) tend to lie on the interface be-tween domains rather than inside the domains.Figure 5 shows the mean ratio ρ f /ρ as a function of thedensity of leaders in the system ρ , for different values of q ,calculated over several realizations of initial conditions.We observe that the value of the ratio ρ f /ρ increases asthe density of leaders ρ is incremented. For q = 30, theratio ρ f /ρ < ρ displayed. ✚✚ ❢ ❂ ✚ ✵ ✿ ✶ ✻✵ ✿ ✶ ✹✵ ✿ ✶ ✷✵ ✿ ✶✵ ✿ ✵ ✽✵ ✿ ✵ ✻✵ ✿ ✵ ✹✵ ✿ ✵ ✷✵✶ ✿ ✷✶ ✿ ✶✶✵ ✿ ✾✵ ✿ ✽ FIG. 5. Mean ratio ρ f /ρ as a function of the density of lead-ers ρ , for different values of q , calculated over 30 realizations.Errors bars indicate ± ρ f /ρ = 1. Curves shown for: q = 30 (triangles); q = 40 (circles); q = 50 (squares). On the other hand, the phase diagram in Figure 4 showsthat, for that range of values of ρ and q = 30, the systemalways reaches the homogeneous state imposed by theleaders. For both values q = 40 and q = 50 in Fig. 5, wesee a change in the ratio, from ρ f /ρ < ρ f /ρ >
1, as ρ varies. Correspondingly, for the parameter values q = 40and q = 50, Fig. 4 reveals a change of the collectivephase of the system, from homogeneous to disordered, as ρ increases across the critical boundary.These results suggest that the ordering properties ofthe system under the influence of leaders are related tothe ratio ρ f /ρ . To investigate this relation, we plot inFig. 6 the value of ρ for which ρ f /ρ = 1 as a function of q . Because of the error bars in the quantity ρ f /ρ shown inFig. 5, this value of ρ is determined within an error inter-val, displayed in Fig. 6 as the vertical width in gray colorfor each value of q . The critical boundary q c ( ρ ), that sep-arates the homogeneous from the disorder phases for thesystem with leaders, is also shown. This boundary coin-cides, within statistical fluctuations, with the condition ρ f /ρ = 1 as a function of q . Thus, we find that the ho-mogeneous phase is associated to the condition ρ f /ρ < ρ f /ρ > q✚ ❍ ❡ t ❡ r ♦ ❣ ❡ ♥ ❡ ♦ ✉ s❍ ♦ ♠ ♦ ❣ ❡ ♥ ❡ ♦ ✉ s ✻ ✵(cid:0) ✁✺ ✵✹ ✺✹ ✵✸ ✺✸ ✵✵ ✿ ✶ ✹✵ ✿ ✶ ✷✵ ✿ ✶✵ ✿ ✵ ✽✵ ✿ ✵ ✻✵ ✿ ✵ ✹✵ ✿ ✵ ✷✵ FIG. 6. Values of ρ for which ρ f /ρ = 1 as a function of q ,within statistical deviations indicated by a gray band. Thecontinuous black curve corresponds to the critical bound-ary that separates the homogeneous from the heterogeneousphases for the system with leaders in Fig. 4. Figure 7 shows the ratio ρ f /ρ as a function of ρ , forseveral values of q , on a lattice with obstacles. Again, thecondition ρ f /ρ >
1, that characterize the prevalence ofobstacles on the interface rather than inside domains, isrelated to the multicultural phase. We have verified thatthe critical curve (dotted line) on the phase diagram ofFig. 4 corresponds to the value of the density of obstacles ρ , for which ρ f /ρ = 1, as a function of q within statisticaldeviations. Then, obstacles induce multiculturality byfacilitating the occurrence of boundaries along their sitesthat separate different domains. ✚✚ ❢ ❂ ✚ ✵ ✿ ✶ ✻✵ ✿ ✶ ✹✵ ✿ ✶ ✷✵ ✿ ✶✵ ✿ ✵ ✽✵ ✿ ✵ ✻✵ ✿ ✵ ✹✵ ✿ ✵ ✷✵✶ ✿ ✷✶✵ ✿ ✽✵ ✿ ✻✵ ✿ ✹✵ ✿ ✷ FIG. 7. Mean ratio ρ f /ρ as a function of the density of obsta-cles ρ , for different values of q , calculated over 30 realizations.Errors bars indicate ± ρ f /ρ = 1. Curves shown for: q = 40 (circles); q = 44 (diamonds); q = 50 (squares). Our results reveal that, when the concentration ofopinion leaders is sufficiently large, they actually be-have as obstacles: leaders favor the formation of culturalboundaries in a social system on a two-dimensional space.We have verified that employing a square lattice with freeboundary conditions as spatial support or increasing thesystem size do not significantly change these results.
CONCLUSIONS
We have employed an agent-based model for culturaldynamics to investigated the effects of spatial hetero-geneities on the collective behavior of a social system.Heterogeneities are introduced as a random distributionof defects or impurities in a two-dimensional lattice. Wehave employed the spatial density of defects as a rele-vant parameter for the collective behavior of the system.We have considered two types of defects: (i) obstaclesor vacancies that represent geographic features; and (ii)opinion leaders as mediators-originators of mass mediamessages or advertisement.A high enough density of obstacles contributes to cul-tural diversity in the system. This result should beexpected, since geographic features such as mountains,lakes, and deserts, have traditionally restricted culturalexchanges [6, 7]. In the contemporary globalized world,landscape obstacles have become irrelevant. However,some ecosystems with natural barriers such as tropi-cal jungles still harbor uncontacted cultures [37]. Ourmodel with opinion leaders shows a counterintuitive ef-fect: a sufficiently large concentration of opinion leaders,far from favoring the spreading of their message to theentire system, actually promotes multiculturality. Thiseffect is similar to that produced by a global uniformmass media field acting on a regular lattice when theintensity of the field is increased [33]. The model withdistributed opinion leaders can be considered as a systemsubject to a spatially nonuniform field. Thus, the densityof leaders is analogous the intensity of an external globalfield when considering the collective dynamics of a socialsystem.We have investigated the mechanism by which obsta-cles and opinion leaders induce cultural diversity whentheir density increases. Above some respective criticalvalue of their density, obstacles as well as leaders, serveas locations for the formation of the interface or bound-aries between different cultural domains. A lower densityof leaders than obstacles is needed to induce multicultur-ality in the system. In particular, leaders change theirrole when their density crosses over a critical value, frommostly being efficient transmitters of their message in-side a domain, to become sites that rather facilitate theemergence of cultural boundaries.Our results should be relevant for the diffusion of infor-mation, advertising, and marketing strategies in a socialsystem extended on a surface. Future research problemsinclude the competition of opinion leaders in differentstates, the consideration of diverse interaction dynam-ics among agents, and the effects of opinion leaders incomplex network topologies and in real social media.
ACKNOWLEDGMENTS
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