Abstract
We investigate the problem of cross-cultural interactions through mass media in a model where two populations of social agents, each with its own internal dynamics, get information about each other through reciprocal global interactions. As the agent dynamics, we employ Axelrod's model for social influence. The global interaction fields correspond to the statistical mode of the states of the agents and represent mass media messages on the cultural trend originating in each population. Several phases are found in the collective behavior of either population depending on parameter values: two homogeneous phases, one having the state of the global field acting on that population, and the other consisting of a state different from that reached by the applied global field; and a disordered phase. In addition, the system displays nontrivial effects: (i) the emergence of a largest minority group of appreciable size sharing a state different from that of the applied global field; (ii) the appearance of localized ordered states for some values of parameters when the entire system is observed, consisting of one population in a homogeneous state and the other in a disordered state. This last situation can be considered as a social analogue to a chimera state arising in globally coupled populations of oscillators.
Introduction
The study of cross-cultural experiences through mass-mediated contact is a topic of much interest in the Social Sciences [1]–[5]. Many of those studies have focused on the effects of cultural product consumption on audience beliefs, emotions, and attitudes toward the group originating these cultural products. For instance, several works have investigated the process by which international audiences develop American values, norms and stereotypes about America through the experience of watching American television series [6]–[8]. Other works have explored the political impact of international television across borders [9]. The expansion of broadcasting and telecommunication industries in recent times has led to an increase in the exchange of mass media products across countries and social groups. As a consequence, people of different groups that may have had little direct contact with each other can, however, have access to their reciprocal mass media messages. For example, the growth of media channels in East Asia has brought changing patterns of cultural consumption: younger generations in China are drawn to Korean pop stars; Korean people have begun to collect Chinese films; Japanese audiences await the broadcast of non-Japanese Asian dramas [5].
In the current research in complex systems, there is also much interest in the investigation of models of social dynamics [10]. Many of these systems have provided scenarios for investigating new forms of interactions and for studying new collective phenomena in non-equilibrium systems [11]–[20]. In this context, the model introduced by Axelrod [21] to investigate the dissemination of culture among interacting agents in a society has attracted much attention from physicists [22]–[33]. In this model, the agent-agent interaction rule is such that no interaction exists for some relative values characterizing the states of the agents that compose the system. This type of interaction is common in social and biological systems where there is often some bound or restriction for the occurrence of interaction between agents, such as a similarity condition for the state variable [34]–[38].
In particular, the effects of local and global mass media on a social group have been studied by using Axelrod's model [26], [27], [39], [40]. Some different formalisms for mass media based on Axelrod's model have also been proposed [41]–[43].
In this paper we investigate the problem of cross-cultural interactions through mass media in a model where two separated social groups, each with its own internal dynamics, get information about each other solely through reciprocal global interactions. We address the question of whether two societies subject to reciprocal mass media interactions become more similar to each other or if they can mantain some diversity. Specifically, our system consists of two populations of social agents whose dynamics is described by Axelrod's model, mutually coupled through global interactions. The global interactions act as fields that can be interpreted as mass media [27], [44]. In our model, the mass media content reaching one population corresponds to the statistical mode or cultural trend originated in the other population, and viceversa.
The existence of non-interacting states in the dynamics, as well as the competition between the time scales of local agent-agent interactions and the responses of the endogenous global fields, lead to nontrivial collective behaviors, such as the emergence of a largest minority group in a population, sharing a state different from that of the applied global field, and the occurrence of localized ordered states. In this last case, one population reaches a homogeneous state while several states coexist on the other. This situation can be considered as a social analogue to a chimera state arising in globally coupled populations of oscillators [45]–[50].
In the next section we present the model for two interacting populations of social agents and characterize the collective behavior on the space of parameters of the system. The nature of the observed localized ordered states is investigated in the following section. The final section contains the conclusions of this work.
The Model
We consider a system of 
agents consisting of two populations or subsets: 
and 
, with sizes 
and 
, such that 
. The fraction of agents in subset 
is 
and that in subset 
is 
.
Each subset consists of a fully connected network, i. e., every agent can interact with any other within a subset. We employ the notation 
to indicate “or 
”. The state of agent 
is given by an 
-component vector 
, 
, where each component can take any of 
different values 
.
Let us denote by 
and 
the global fields defined as the statistical modes of the states in the subsets 
and 
, respectively, at a given time. This means that the component 
is assigned the most abundant value exhibited by the 
th component of all the state vectors 
in the subset 
. If the maximally abundant value is not unique, one of the possibilities is chosen at random with equal probability. In the context of social dynamics, these global fields can be interpreted as mass media messages about “trends” originated in each population.
Each agent in subset 
is subject to the influence of the global field 
, and each agent in subset 
is subject to the influence of the global field 
. Figure 1 shows the configuration of the two populations subject to the influence of their reciprocal global fields.
Figure 1. Representation of two populations 
and 
interacting through their reciprocal global fields 
and 
, each acting with intensity 
.
Starting from random initial conditions in each subset, at any given time, a randomly selected agent in subset 
can interact either with the global field 
or with any other agent belonging to 
. The interaction in each case takes place according to the dynamics of Axelrod's cultural model.
The dynamics of the system is defined by iterating the following steps:
Select at random an agent
and a agent 
.Select the source of interaction: with probability
, agent 
interacts with field 
and agent 
interacts with field 
, while with probability 
, 
interacts with 
selected at random and 
interacts with 
also selected at random.Calculate the overlap (number of shared components) between agent
and its source of interaction, given by 
, where 
if the source is the field 
, or 
if the source is agent 
. Similarly, calculate the overlap 
, where 
if the source is the field 
, or 
if the source is agent 
. Here we employ the delta Kronecker function, 
, if 
; 
, if 
.If
, with probability 
choose 
such that 
and set 
; if 
or 
, the state 
does not change. If 
, with probability 
choose 
such that 
and set 
; if 
or 
, the state 
does not change.If the source of interaction is
, update the fields 
and 
.
The strength of each field 
and 
is represented by the parameter 
that measures the probability for the agent-field interactions. Step 
characterizes the time scale for the updating of the global fields in our model. In general, agents in one population do not have instantaneous knowledge of the state of the global field of the other population, but only when they effectively interact with that global field. The non-instantaneous updating of the global fields expresses the delay with which a population acquires knowledge about the other through the only available communication channel between them, as described in many cross-cultural interactions through mass media [5]. In our case, as the value of the parameter 
increases, both the intensity of the global fields and the updating rate of their states increase.
Under the mutual coupling, both populations, 
and 
form domains of different sizes in the asymptotic state. A domain is a set of connected agents that share the same state. A homogeneous or ordered phase in a population corresponds to 
, 
. There are 
equivalent configurations for this ordered phase. In an inhomogeneous or disordered phase in a population several domains coexist. The sizes of these domains within each population are ranked by the index 
: 
corresponding to the largest domain, 
indicates the second largest domain, etc. To characterize the collective behavior of the system, we define the following macroscopic quantities: (i) the average normalized size (divided by 
) of the domain in 
whose size has rank 
, denoted by 
; (ii) the probability that the largest domain in 
has a state equal to 
, designed by 
.
Figure 2 shows various of these quantities as functions of the parameter 
, for different values of 
. In this paper we fix the parameter value 
. In the absence of global fields (Fig. 2(a)), i.e. 
, we have two uncoupled and independent subsets; each subset spontaneously reaches an ordered phase, characterized by 
and 
, for values 
, and a disordered phase, corresponding to 
and 
, for 
, where 
is a critical point that depends on the subset size in each case, 
[52]. Figure 3(a) shows the asymptotic pattern in this case.
Figure 2.
, 
, and 
as functions of 
, with 
, for different values of 
.
System size is 
with partition 
. Each data point is the result of averaging over 
random realizations of initial conditions. (a) 
(open circles), 
(solid circles); with 
. (b) Left vertical axis: 
(open circles); right vertical axis: 
(crosses); fixed 
. Phases I and II. (c) Left vertical axis: 
(open circles), 
(open squares); right vertical axis: 
(crosses); fixed 
. Phases I and IV. (d) Left vertical axis: 
(open circles), 
(open squares); right vertical axis: 
(crosses); fixed 
. Phase III occurs for values 
, independent of 
. The bars in (c) and (d) indicate the probability 
of finding a localized ordered state in the system as a function of 
for the given value of the intensity 
.
Figure 3. Each panel displays an asymptotic state (vertical axis) of the agents in the interacting populations 
(upper part) and 
(lower part) vs. time (horizontal axis), corresponding to a different phase in the system.
Each value of the state variable of an agent is represented by a different color. Population sizes are 
, 
, with 
. (a) 
(no coupling). (b) 
(phase I). (c) 
(phase II). (d)
(phase IV). (e) 
(phase III). (f) 
(localized ordered state).
For 
and 
, each population reaches an ordered state with 
, as shown in Fig. 2(b). However, in this situation the spontaneous order emerging in subset 
for parameter values 
due to the agent-agent interactions competes with the order being imposed by the applied global field 
. For some realizations of initial conditions, the global field 
imposes its state on subset 
and, correspondingly, the field 
imposes its state on subset 
. As a consequence, both subsets reach the same state with 
. An asymptotic state corresponding to this situation is displayed in Fig. 3(b). We refer to this state as phase I. However, the ordered state in subset 
does not always correspond to the state of the global field 
being applied to 
. This is revealed by the probability 
shown in Fig. 2(b) that measures the fraction of realizations that the largest domain in 
has a state equal to 
. We find 
for a range of values 
. Thus, in this case there is a probability that subsets 
and 
can reach ordered states different from each other, i. e., 
. Figure 3(c) illustrates the asymptotic states in this case. We denote this situation as phase II.
Figures 2(c) and 2(d) show both 
and 
as functions of 
for greater values of 
. The quantity 
in Fig. 2(c) displays a local minimum at some value of 
that depends on 
. This local minimum of 
is associated to a local maximum value of 
, such that 
for 
. Therefore, two majority domains form in subset 
for 
. Fig. 2(c) also shows that the probability 
, indicating that the state of the largest group in 
is always equal to that imposed by the field 
. But the second largest group that occupies almost the rest of subset 
reaches a state different from 
. Thus, the value of 
for which 
has a local minimum is related to the emergence of a second largest domain ordered against the global field 
. The corresponding asymptotic pattern is shown in Fig. 3(d). We call this configuration phase IV. Figure 2(d) reveals that, for larger values of 
, various local minima of 
can occur at some values of 
. This local minima of 
correspond to local maxima of 
and to the emergence of a second largest domain in 
ordered against the field 
. The raise of a largest minority group at some values of 
is a manifestation of the tendency towards the spontaneous order related to the agent-agent interactions. For values 
, both populations reach disordered states 
, characterized by 
. The disordered behavior of the system is denoted by phase III and the corresponding pattern is displayed in Fig. 3(e).
To characterize phase II, we plot in Fig. 4 the quantity 
as a function of 
, for a fixed value 
. For 
, the state of the largest domain in 
corresponds to the state of the field 
, i.e. 
and 
, indicating the presence of phase I, and thus 
. For 
, the largest domain in 
no longer possesses the state of the field 
but another state non-interacting with this field, i.e. 
and 
, and therefore 
, characterizing phase II. For 
, 
and 
.
Figure 4. The quantity 
as a function of 
for a fixed value 
, with 
.
The critical values 
and 
, as well as the regions where phases I, II, and III occur, are indicated. System size is 
with partition 
. Each data point is averaged over 
realizations of initial conditions.
We note that phase II occurs for small values of 
, where the time scale for the agent-agent interaction dynamics is smaller than the corresponding time scale for the agent-field dynamics. This means that the state of the global field does not vary much in comparison to the changes taking place in the states of the agents and, therefore, the global field behaves approximately as a fixed external field with little influence on the system. As a consequence the system can spontaneously order in a state different from that of the global field if 
is sufficiently large, giving rise to phase II. For increasing values of 
, the updating of the global fields and the agent-agent dynamics have comparable time scales and, therefore, the state of the fields corresponds to that of the largest domain in each subset, yielding regions of both phase I and phase IV.
The collective behavior of either of the two subsets coupled through their reciprocal global fields can be characterized by four phases on the space of parameters 
, as shown in Fig. 5 for subset 
: (I) a homogeneous, ordered phase, for which 
and 
; (II) an ordered phase in a state orthogonal to the applied global field, such that 
and 
; (III) a disordered phase for 
, for which 
; and (IV) a partially ordered phase, where 
and 
, 
, characterized by the emergence of a second largest domain ordered in a state different from field 
.
Figure 5. Phase diagram of population 
on the space of parameters 
, with 
.
System size is 
with partition 
. Each data point is averaged over 
realizations of initial conditions. The color code represents the value of the normalized largest domain size 
, from black (
) to white (
). The regions where the different phases occur are labeled and separated by slashed lines: phase I (both populations share same homogeneous state); phase IV (partially ordered, emergence of second group); phase III (disordered), and phase II (each population in a different homogeneous state). Localized ordered states can occur in the transitions from phase IV to phase I.
The phase diagram of Fig. 5 reveals that the interaction through reciprocal, evolving global fields can lead to nontrivial effects in certain cases. For example, for a fixed value 
, the global field can impose its state to the system (phase I) only for a range of intermediate values of the intensity 
.
We have checked the behavior of the system for different population sizes 
and 
. Figure 6 shows the quantity 
as a function of 
with fixed coupling 
, for different values of 
. We see that the critical point for the transition to phase III scales as 
, as expected [52], and that the qualitative collective behavior represented in the phase diagram of Fig. 5 is independent of the sizes of the partitions into two populations. Since 
, the collective behavior of the system is also independent of the size 
, and 
, according to Fig. 6.
Figure 6. Normalized size of largest domain 
as a function of 
with fixed intensity 
, for different population sizes: 
(squares); 
(circles); 
(diamonds).
System size is 
and 
.
Localized Ordered States
In addition to phases I and II that display homogeneous states for both subsets 
and 
, there are configurations where homogeneous states can take place in only one subset, while the other is inhomogeneous, for some values of parameters. We refer to this configuration as
localized ordered states
. These states are characterized by 
and 
. Figure 3(f) displays the asymptotic state of the system in this case. In contrast to the four phases that can be characterized in a subset, the ordered collective states can only be defined by considering both subsets simultaneously, i.e., it requires the observation of the entire system.
To elucidate the nature of these states, we calculate the probability 
of finding a localized ordered state in the system as a function of 
in Figs. 2(c) and 2(d), employing the criterion 
. In both figures, there are ranges of the parameter 
where localized ordered states can occur; the probability 
is maximum near the values of 
that correspond to local minima of 
(and local maximum values of 
). Figure 7 shows the probability distributions 
and 
, 
, of the normalized domain sizes for both subsets 
and 
, calculated over 
realizations of initial conditions, for different values of 
, and with fixed 
corresponding to Fig. 2(c). Figure 7(a) exhibits the probabilities 
and 
when either subset is in phase I with 
, characterized by the presence of one large domain whose size is of the order of the system size 
, in agreement with Fig. 3(b). Figure 7(b) shows 
and 
associated to phase IV (
), where the size of the largest domain in either subset never reaches the system size due to the appearance of a second group, as displayed in Figs. 2(c) and 3(d). Figure 7(c) shows the probabilities 
and 
for 
. In this case either subset can reach an ordered configuration, 
, or an inhomogeneous state 
. This corresponds to the appearance of localized ordered states in the system. For 
, we find again a probability distribution typical of phase I.
Figure 7. Probability distributions 
and 
, 
, of normalized domain sizes for both populations 
(black bars) and 
(grey bars), calculated over 
realizations of initial conditions, with fixed intensity 
, 
, and for different values of the number of options 
.
(a) 
(phase I); (b) 
(phase IV); (c) 
(localized ordered states); (d) 
(phase I).
The localized ordered states are analogous to chimera states observed in two populations of dynamical oscillators having global or long range interactions, where one population in a coherent state coexist with the other in a incoherent state 45–50. In a chimera state, one part of a spatially extended system presents a coherent or synchronized behavior while another part is desynchronized.
Note that the regions of parameters where localized ordered states can emerge in our system lie between phase IV and phase I states. In fact, the configuration of localized ordered states shares features of both phase I and phase IV; they can be considered as transition configurations between phase IV and phase I states.
Discussion
We have investigated the collective behavior of a system consisting of two populations of social agents, mutually coupled through global fields, as a model for cross-cultural interactions via mass media. Specifically, we have employed Axelrod's model for social influence as the interaction dynamics.
The global interaction field associated to each population corresponds to the statistical mode of the states of the agents. In the context of social dynamics, this global autonomous field can be interpreted as mass media messages about “trends” or stereotypes originated in one population that are transmitted to the other population. Thus, our system can represent cross cultural interactions between two separated social groups, each with its own internal dynamics, but getting information about each other solely through their mass media messages [5].
We have found several phases on either subset depending on parameter values: two homogeneous phases, one having the state of the global field acting on that subset (phase I), and the other consisting of a state different from that reached by the applied global field (phase II); a partially ordered phase characterized by the emergence of a second largest domain ordered in a state different from the global field (phase IV); and a disordered phase (III).
States similar to phases I, II, and III are also observed for some regions of parameters in a system of social agents subject to an external fixed field [40]. In the present model with non-instantaneous updating of the fields, for small values of 
, the global evolving field varies very slowly in comparison to the changes in the states of the agents in a subset due to their mutual interactions. In this case, the global evolving field behaves as a fixed external field acting on the population.
However, for larger values of 
, the adaptive nature of the global fields induce two new phenomena in some range of 
on each population. One is the emergence of a largest minority group of appreciable size having a state different from that of the applied field (phase IV). The other corresponds to the appearance of localized ordered states when the entire system is observed, consisting of one population in a homogeneous state and the other in an disordered state. These configurations occur with a probability that depend on both 
and 
and appear as transitions states from phase IV to phase I. These localized ordered states are analogous to the chimera states that have been found in networks of coupled oscillators having global interactions, where a subset of the system reaches a coherent state while another subset remains incoherent [47], [48]. The recent experimental discovery of such chimera states has fundamental implications as it shows that localized order and structured patterns can emerge from otherwise structureless system [50], [51]. As noted in Ref. [47], analogous symmetry breaking is observed in dolphins and other animals that have evolved to sleep with only half of their brain at a time: neurons exhibit synchronized activity in the sleeping hemisphere and desynchronized activity in the hemisphere that is awake [53].
From a social perspective, our model shows that cross cultural reciprocal interactions through mass media do not always lead to the imposition over one population of the cultural trends being transmitted by the media of another population. A group possessing a cultural state different from that of the mass media message can spontaneously emerge in the first population. Under some circumstances, such group can encompass the entire population (phase II), or it can constitute the largest minority in that population (phase IV).
The behaviors reported here should also be expected in other non-equilibrium systems possessing non-interacting states, such as social and biological systems whose dynamics usually possess a bound condition for interaction [35]. This includes models of motile elements in population dynamics, such as swarms, fish schools, bird flocks and bacteria colonies [34], [54]–[58]. Future extensions of this work involves the consideration of complex network structures within each population and the investigation of communities, where the interaction between populations occurs through a few elements rather than a global field.
Acknowledgments
The authors are grateful to V. M. Eguíluz for fruitful discussions. M. G. C. thanks the Associates' Program of the Abdus Salam International Center for Theoretical Physics, Trieste, Italy.
Funding Statement
J.C.G-A acknowledges support from project CNPq 150452/2011-0, Brazil. M.S.M. acknowledges financial support from Comunitat Autònoma de les Illes Balears, FEDER, and MINECO (Spain) under project FIS2007-60327. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
References
- 1.Elasmar M G, editor(2003) The Impact of International Television: A Paradigm Shift. New Jersey: Lawrence Erlbaum Associates Inc. 213 p.
- 2.Bryant J, Zillman D, editors(2002), Media Effects: Advances in Theory and Research, New Jersey: Lawrence Erlbaum. 648 p.
- 3.Kovács HV, editor (2011), Mass Media: Coverage, Objectivity and Changes, Hauppauge, New York: Nova Science Publishers.
- 4.Yaple P, Korzenny F (1989) Electronic mass media effects across cultures. In: Asante MK, Gudykunst BW, editors. Handbook of International and Intercultural Communication. Newbury Park, California: Sage.
- 5.Rhee JW, Lee C (2010) Cross-cultural interactions through mass media products: Cognitive and emotional impacts of Chinese peoples consumption of Korean media products. In: Black D, Epstein S, Tokita A, editors. Complicated Currents: Media Flows, Soft Power and East Asia. Melbourne: Monash University ePress. pp. 5.1–5.16.
- 6. Tan A, Li S, Simpson C (1986) American television and social stereotypes of Americans in Taiwan and Mexico. Journalism Quarterly 64: 809–814. [Google Scholar]
- 7. Weimann G (1984) Images of life in America: The impact of American TV in Israel. International Journal of Intercultural Relations 8: 185–197. [Google Scholar]
- 8. Zaharopoulos T (1997) US television and American cultural stereotypes in Greece. World Communication 26: 30–45. [Google Scholar]
- 9.Elasmar MG, Hunter JE (2003) A meta-analysis of crossborder effect studies. In: Elasmar M G, editor. The Impact of International Television: A Paradigm Shift. New Jersey: Lawrence Erlbaum Associates Inc. pp, 127–150.
- 10. Castellano C, Fortunato S, Loreto V (2009) Statistical physics of social dynamics. Rev Mod Phys 81: 591–646. [Google Scholar]
- 11.Marro J, Dickman R (1999) Nonequilibrium Phase Transitions in Lattice Models. Cambridge: Cambridge University Press. 327 p.
- 12. Weidlich W (1991) Physics and social science - The approach of synergetics. Phys Rep 204: 1–163. [Google Scholar]
- 13. Suchecki K, Eguíluz VM, San Miguel M (2005) Voter model dynamics in complex networks: Role of dimensionality, disorder, and degree distribution. Phys Rev E 72: 036132. [DOI] [PubMed] [Google Scholar]
- 14.Stauffer D, de Oliveira SM, de Oliveira PMC, Sá Martins J (2006) Biology, Sociology, Geology by Computational Physicists. Amsterdam: Elsevier.
- 15. Vilone D, Ramasco J, Sánchez A; San Miguel M (2012) Social and strategic imitation: the way to consensus. Scientific Reports 2. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 16.Zimmerman M, Eguíluz VM, San Miguel M (2001) Economics with Heterogeneous Interacting Agents. In: Kirman A, B. Zimmerman BJ, editors. Lecture Notes in Economics and Mathematical Systems 503. Berlin: Springer Verlag, pp. 73–86.
- 17. Zimmerman M, Eguíluz VM, San Miguel M (2004) Coevolution of dynamical states and interactions in dynamic networks. Phys Rev E 69: 065102 (R). [DOI] [PubMed] [Google Scholar]
- 18. Holme P, Newman MEJ (2006) Nonequilibrium phase transition in the coevolution of networks and opinions. Phys Rev E 74: 056108. [DOI] [PubMed] [Google Scholar]
- 19. Centola D, Eguíluz VM, Macy WW (2007) Cascade dynamics of complex propagation. Physica A 374: 449–456. [Google Scholar]
- 20. Bartolozzi M, Leinweber DB, Thomas AW (2005) Stochastic opinion formation in scale-free network. Phys Rev E 72: 046113. [DOI] [PubMed] [Google Scholar]
- 21. Axelrod R (1997) The dissemination of culture: A model with local convergence and global polarization. J Conflict Res 41: 203–225. [Google Scholar]
- 22. Castellano C, Marsili M, Vespignani A (2000) Nonequilibrium phase transition in a model for social influence. Phys Rev Lett 85: 3536–3539. [DOI] [PubMed] [Google Scholar]
- 23. Vilone D, Vespignani A, Castellano C (2002) Ordering phase transition in the one-dimensional Axelrod model. Eur Phys J B 30: 399–406. [Google Scholar]
- 24. Klemm K, Eguíluz VM, Toral R, San Miguel M (2003) Nonequilibrium transitions in complex networks: A model of social interaction. Phys Rev E 67: 026120. [DOI] [PubMed] [Google Scholar]
- 25. Klemm K, Eguíluz VM, Toral R, San Miguel M (2003) Global culture: A noise-induced transition in finite systems. Phys Rev E 67: 045101 (R). [DOI] [PubMed] [Google Scholar]
- 26. González-Avella JC, Cosenza MG, Tucci K (2005) Nonequilibrium transition induced by mass media in a model for social influence. Phys Rev E 72: 065102 (R). [DOI] [PubMed] [Google Scholar]
- 27. González-Avella JC, Eguíluz VM, Cosenza MG, Klemm K, Herrera JL, San Miguel M (2006) Local versus global interactions in nonequilibrium transitions: A model of social dynamics. Phys Rev E 73: 046119. [DOI] [PubMed] [Google Scholar]
- 28. Peres LR, Fontanari JF (2011) The media effect in Axelrod's model explained. EPL 96: 38004. [Google Scholar]
- 29. Kuperman MN (2006) Cultural propagation on social networks. Phys Rev E 73: 046139. [DOI] [PubMed] [Google Scholar]
- 30. Mazzitello KI, Candia J, Dossetti V (2007) Effects of mass media and cultural drift in a model for social influence. Int J Mod Phys C 18: 1475–1482. [Google Scholar]
- 31. De Sanctis L, Galla T (2009) Effects of noise and confidence thresholds in nominal and metric Axelrod dynamics of social influence. Phys Rev E 79: 046108. [DOI] [PubMed] [Google Scholar]
- 32. Gracia-Lazaro C, Floria LM, Moreno Y (2011) Selective advantage of tolerant cultural traits in the Axelrod-Schelling model. Phys Rev E 83: 056103. [DOI] [PubMed] [Google Scholar]
- 33. Zhang W, Lim C, Sreenivasan S, Xie J, Szymanski BK, Korniss G (2011) Social influencing and associated random walk models: Asymptotic consensus times on the complete graph. Chaos 21: 025115. [DOI] [PubMed] [Google Scholar]
- 34.Mikhailov AS, Calenbuhr V (2002) From Cells to Societies: Models of Complex Coherent Action. Berlin: Springer. 299 p.
- 35. Deffuant G, Neau D, Amblard F, Weisbuch G (2000) Mixing beliefs among interacting agents. Adv Complex Syst 3: 87–98. [Google Scholar]
- 36. Weisbuch G, Deffuant G, Amblard F, Nadal JP (2002) Meet, discuss, and segregate. Complexity 7: 55–63. [Google Scholar]
- 37.Hegselmann R, Krause K (2002) Opinion dynamics and bounded confidence: models, analysis and simulation. J Artif Soc Soc Simul. 5: : (3) 2. http://jasss.soc.surrey.ac.uk/5/3/2.html. [Google Scholar]
- 38. Laguna MF, Abramson G, Zanette DH (2003) Vector opinion dynamics in a model for social influence. Physica A 329: 459–472. [Google Scholar]
- 39. Shibanai Y, Yasuno S, Ishiguro I (2001) Effects of Global Information Feedback on Diversity: Extensions to Axelrod's Adaptative Culture Model. J Conflict Res 45: 80–96. [Google Scholar]
- 40. González-Avella JC, Cosenza MG, Eguíluz VM, San Miguel M (2010) Spontaneous ordering against an external field in non-equilibrium systems. New J Phys 12: 013010. [Google Scholar]
- 41. Candia J, Mazzitello KI (2008) Mass media influence spreading in social networks with community structure. J Stat Mech Theor Exp P07007. [Google Scholar]
- 42. Rodríguez AH, del Castillo-Mussot M, Vázquez GJ (2009) Induced monoculture in Axelrod model with clever mass media. Int J of Modern Physics C 20: 1233–1245. [Google Scholar]
- 43. Rodríguez AH, Moreno Y (2010) Effects of mass media action on the Axelrod model with social influence. Phys Rev E 82: 016111. [DOI] [PubMed] [Google Scholar]
- 44. Gargiulo F, Lottini S, Mazzoni A (2008) The saturation threshold of public opinion: are aggressive media campaigns always effective?. arXiv 0807.3937. [Google Scholar]
- 45. Kuramoto Y, Battogtokh D (2002) Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom Complex Syst 5: 380–385. [Google Scholar]
- 46. Abrams DM, Strogatz SH (2004) Chimera states for coupled oscillators. Phys Rev Lett 93: 174102. [DOI] [PubMed] [Google Scholar]
- 47. Abrams DM, Mirollo R, Strogatz SH, Wiley DA (2008) Solvable model for chimera states of coupled oscillators. Phys Rev Lett 101: 084103. [DOI] [PubMed] [Google Scholar]
- 48. Laing C R (2010) Chimeras in networks of planar oscillators. Phys Rev E 81: 066221. [DOI] [PubMed] [Google Scholar]
- 49. Omelchenko I, Maistrenko Y, Hövel P, Schöll E (2011) Loss of coherence in dynamical networks: spatial chaos and chimera states. Phys Rev Lett 106: 234102. [DOI] [PubMed] [Google Scholar]
- 50. Tinsley MR, Nkomo S, Showalter K (2012) Chimera and phase-cluster states in populations of coupled chemical oscillators. Nature Phys 8: 662–665. [DOI] [PubMed] [Google Scholar]
- 51. Hagerstrom AM, Murphy TE, Roy R, Hövel P, Omelchenko I, Schöll E (2012) Experimental observation of chimeras in coupled-map lattices. Nature Phys 8: 658–661. [Google Scholar]
- 52. Vazquez F, Redner S (2007) Non-monotonicity and divergent time scale in Axelrod model dynamics. EPL 78: 18002. [Google Scholar]
- 53. Mathews CG, Lesku JA, Lima SL, Amlaner CJ (2006) Asynchronous eye closure as an anti-predator behavior in the western fence lizard (Sceloporus Occidentalis). Ethology 112: 286–292. [Google Scholar]
- 54. Huth A, Wissel Ch (1992) The simulation of the movement of fish shoals. J Theor Biol 156: 365–385. [Google Scholar]
- 55. Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Schochet O (1995) Novel type of phase transition in a system of self-driven particles. Phys Rev Lett 75: 1226–1229. [DOI] [PubMed] [Google Scholar]
- 56. Grégoire G, Chaté H, Tu Y (2003) Moving and staying together without a leader. Physica D 181: 157–170. [Google Scholar]
- 57. Shibata T, Kaneko K (2003) Coupled map gas: structure formation and dynamics of interacting motile elements with internal dynamics. Physica D 181: 197–214. [Google Scholar]
- 58. Fuentes MA, Kuperman MN, Kenkre VM (2003) Nonlocal interaction effects on pattern formation in population dynamics. Phys Rev Lett 91: 158104. [DOI] [PubMed] [Google Scholar]