复旦大学：《网络科学导论 Introduction to Network Science》教学参考文献_Opinion dynamic model-Galam model

PHYSICAL REVIEW E 78. 016104(2008) Coevolution onary dynamics of opinions and networks: From diversity to uniformity Feng Fu- and Long wang Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts 02138, USA Center for Systems and Control, College of Engineering and Key Laboratory of Machine Perception(Ministry of Education), Peking University, Beijing 100871, China (Received 6 February 2008; published 11 July 2008) We investigate the coevolutionary dynamics of opinions and networks based upon majority-preference(MP) and minority-avoidance(MA)rules. Under MP, individuals adopt the majority opinion among their neighbors while in MA individuals can break the link to one holding a minority and different opinion, and rewire either to neighbors of their neighbors with the same opinion or to a random one from the whole population except heir nearest neighbors. We study opinion formation as a result of combination of these two competing rules. with a parameter tuning the balance between them. We find that the underlying network can be self-organized into connected communities with like-minded individuals belonging to the same group; thus a broad variety of opinions coexist. Diverse opinions disappear in a population in which all individuals share a uniform opinion when the model parameter exceeds a critical value. Furthermore, we show that an increasing tendency to redirect to neighbors of neighbors is more likely to result in a consensus of opinion DO:10.11o3/ PhysrevE78.016104 PACS number(s): 89.75 Fb, 87.23. Ge, 64.60 aq L INTRODUCTION tends to remove the link to one holding a different and mi- cs or formation is a traditional nority opinion. In order to keep the total number of edges constant. a connection is formed between the focal research area in social science and social psychology. Re- dividual and another one chosen from the network. Here, we cently, in the physics community, it has attracted much atten- view the creation of new links as a balance between two tion for study of collective behaviors in social and economic opposing forces: order and randomness. Order indicates the systems. This results in a significant overlap between social tendency of an individual to connect to neighbors of neigh science and statistical physics, i.e., sociophysics [1]. In anal bors: randomness means that a new link is formed between ogy with the Ising model [2 ], some models were developed the focal individual and one randomly chosen from the rest to address this issue [3-8] the voter model [3], the majority- of the population. We take into account this trade-off in the rule model [4], and the bounded-confidence model [81, to network rewiring process and reveal that an increasing ten name a few. These models assume opinions as discrete or dency of rewiring to neighbors of neighbors is more likely to continuous variables. Much previous work studied the opin- reduce the number of surviving opinions. Furthermore, we ion formation process on fixed networks and focused on how focus on the final number of existing opinions in the network network topology affects the dynamics. More recently, in- as a result of the regime where opinion update or network stead of considering opinion dynamics on static networks, rewiring dominates the dynamics. Interestingly, our investi- some researchers took into account the dynamics on continu- gation will show that the coevolving network exhibits char ously evolving networks [9-15], namely, network change in acteristics ranging from interconnected communities(where response to opinion and opinion change in response to the like-minded individuals belong to the same group)to a single network. Indeed,many networks have an adaptive dynamical community(where everyone holds the same opinion nature. their evolution occurs on a fast or slow time scale compared with the dynamics taking place on them. Such considerations of the entangled dynamics of opinions networks are thus plausible and relevant. Motivated by IL. THE MODEL existing works, in this paper, we consider a simple coevolu- rewiring process. We are interested in understanding the con- paper. Initially, N individuals are embedded in a random tion model that combines opinion dynamics and a network Let us briefly introduce the model used in the present regular graph in which the links denote the acquaintances ditions in which the opinion evolves from diverse to uni- between them. The total number of edges M is fixed during form Our study is based on the majority-rule model, which as- the coevolution of opinions and networks. This constraint implies that the evolution is conservative, in the sense that their opinion update (e.g, herding behavior in public tions and the possibilities of resulting network configurations choices). In addition, each individual is assumed to show are limited. Each individual holds one of G possible opinions minority-avoidance behavior, that is, the focal individual on some topic of interest, denoted by g At each time step. an individual is chosen at random to update either its opinion or neighborhood according to the following two rules u@fas. harvar 1)Majority preference(MP): with probability p, the fo- ang@pku. edu. cal individual accepts the specific opinion held by a majority 1539-3755/2008/78(1)/016104(4) 016104-1 @2008 The American Physical Society

Coevolutionary dynamics of opinions and networks: From diversity to uniformity Feng Fu1,2,* and Long Wang2,† 1 Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts 02138, USA 2 Center for Systems and Control, College of Engineering and Key Laboratory of Machine Perception (Ministry of Education), Peking University, Beijing 100871, China
Received 6 February 2008; published 11 July 2008 We investigate the coevolutionary dynamics of opinions and networks based upon majority-preference
MP and minority-avoidance
MA rules. Under MP, individuals adopt the majority opinion among their neighbors; while in MA individuals can break the link to one holding a minority and different opinion, and rewire either to neighbors of their neighbors with the same opinion or to a random one from the whole population except their nearest neighbors. We study opinion formation as a result of combination of these two competing rules, with a parameter tuning the balance between them. We find that the underlying network can be self-organized into connected communities with like-minded individuals belonging to the same group; thus a broad variety of opinions coexist. Diverse opinions disappear in a population in which all individuals share a uniform opinion, when the model parameter exceeds a critical value. Furthermore, we show that an increasing tendency to redirect to neighbors of neighbors is more likely to result in a consensus of opinion. DOI: 10.1103/PhysRevE.78.016104 PACS number
s: 89.75.Fb, 87.23.Ge, 64.60.aq I. INTRODUCTION Opinion dynamics or opinion formation is a traditional research area in social science and social psychology. Re￾cently, in the physics community, it has attracted much atten￾tion for study of collective behaviors in social and economic systems. This results in a significant overlap between social science and statistical physics, i.e., sociophysics 1. In anal￾ogy with the Ising model 2, some models were developed to address this issue 3–8: the voter model 3, the majority￾rule model 4, and the bounded-confidence model 8, to name a few. These models assume opinions as discrete or continuous variables. Much previous work studied the opin￾ion formation process on fixed networks and focused on how network topology affects the dynamics. More recently, in￾stead of considering opinion dynamics on static networks, some researchers took into account the dynamics on continu￾ously evolving networks 9–15, namely, network change in response to opinion and opinion change in response to the network. Indeed, many networks have an adaptive dynamical nature: their evolution occurs on a fast or slow time scale compared with the dynamics taking place on them. Such considerations of the entangled dynamics of opinions and networks are thus plausible and relevant. Motivated by these existing works, in this paper, we consider a simple coevolu￾tion model that combines opinion dynamics and a network rewiring process. We are interested in understanding the con￾ditions in which the opinion evolves from diverse to uni￾form. Our study is based on the majority-rule model, which as￾sumes that individuals preferentially follow the crowd in their opinion update
e.g., herding behavior in public choices. In addition, each individual is assumed to show minority-avoidance behavior, that is, the focal individual tends to remove the link to one holding a different and mi￾nority opinion. In order to keep the total number of edges constant, a new connection is formed between the focal in￾dividual and another one chosen from the network. Here, we view the creation of new links as a balance between two opposing forces: order and randomness. Order indicates the tendency of an individual to connect to neighbors of neigh￾bors; randomness means that a new link is formed between the focal individual and one randomly chosen from the rest of the population. We take into account this trade-off in the network rewiring process and reveal that an increasing ten￾dency of rewiring to neighbors of neighbors is more likely to reduce the number of surviving opinions. Furthermore, we focus on the final number of existing opinions in the network as a result of the regime where opinion update or network rewiring dominates the dynamics. Interestingly, our investi￾gation will show that the coevolving network exhibits char￾acteristics ranging from interconnected communities
where like-minded individuals belong to the same group to a single community
where everyone holds the same opinion. II. THE MODEL Let us briefly introduce the model used in the present paper. Initially, N individuals are embedded in a random regular graph in which the links denote the acquaintances between them. The total number of edges M is fixed during the coevolution of opinions and networks. This constraint implies that the evolution is conservative, in the sense that individuals can maintain only a limited number of connec￾tions and the possibilities of resulting network configurations are limited. Each individual holds one of G possible opinions on some topic of interest, denoted by gi . At each time step, an individual is chosen at random to update either its opinion or neighborhood according to the following two rules.
1 Majority preference
MP: with probability p, the fo￾cal individual accepts the specific opinion held by a majority *fengfu@fas.harvard.edu † longwang@pku.edu.cn PHYSICAL REVIEW E 78, 016104
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FENG FU AND LONG WANG PHYSICAL REVIEW E 78. 016104(2008) of its neighbors (i.e, the opinion the largest number of indi- viduals among its neighbors hold) 2)Minority avoidance (MA): otherwise, with likelihood 1-P, the focal individual removes the link to one of its neighbors who holds a minority and different opinion (i.e the different opinion held by the least number of individuals within its neighborhood), and then with probability o rewires to a random neighbor of its neighbors who holds the same opinion, or otherwise to a random one selected from the FIG. 1.( Color online) Illustration of the coevolution of opinions whole population except its nearest neighbors(with probabil- and networks. Left panel(p=0.6, b=0.6): the coexistence of differ- ty 1-o) ent opinions, in which the individuals are self-organized into com- munities or groups sharing the same opinion (node color denotes MP represents the influence of individual acquaintances the type of opinion). Right panel (p=0.8, d=0.6): uniformity of the and also the tendency to adopting the majority opinion ndividual opinions exists, where individuals are clustered in a among the node,s acquaintances. Such preference reflects whole tight group Parameters: population size N=100, average de- one's reluctance to be in a minority among friends as well one 's willingness to be in agreement with most friends. Ma gree ()=6, and initial number of opinions G=5 characterizes the situation in which individuals avoid becom- ing a minority among their friends and try to form new ac- into agreement with the majority of their neighbors. In pa quaintanceships with the neighbors of neighbors such that ticular, when the rewiring process occurs in a triadic manner they would have a great chance to become a majority in the (e, friends of friends), it leads to an increase of the total MP updating, if presently in the minority. These two rules are number of triangles in the network, i. e, the clustering coef- in line with the conformism found in human society. In what ficient. We will show that this tendency of redirection to follows, we will investigate the competing roles of MP and neighbors of neighbors has a recognizable impact on the evo- MA in the coevolution of opinions and networks, and cor lution of opinion. In combination these two competing centrate on how the model parameters (i. e, p and o )affect update moves, the system will eventually reach a state de- the evolution pendent on the p value: small p values result in the coexist- ence of different opinions; large p values lead to a uniform opinion among the population IIL. SIMULATION RESULTS AND DISCUSSIONS We present the rank-size distribution of opinions in a net work in a consensus state in Fig. 2. With small p values, In our simulations, we keep the population size N and the tal number of edges M constant during the evolution. Thus most of the opinions have roughly equal numbers of holders whereas, with increasing p values, we observe a progressive the average connectivity ()=2M/N is fixed In addition, iso- broadening of the distribution, which clearly presents a lated vertices and duplicate links are not allowed. Initially, heavy ta. As p becomes very large, most opinions go into individuals are located on a random regular graph( the who- extinction in the population during the coevolutionary cess, but certain opinions dominate until uniformity in opi opinions are uniformly assigned to vertices at random. A ion is attained. Here, we should point out that, for suitable p cording to our specified update rule, in finite time steps, the values, community structure also emerges along with the co- system will evidently approach a"consensus state"in which each individuals opinion agrees with the majority of neighbors. We stop the simulations when this consensus achieved. At this stage we compute the number of survival p=0.2 0.5 opinions and the consensus time as a function of the param- eters p and respectively In order to gain intuition into the results of our model, we slope =-1 plot the emergent networks in the final stage corresponding to two different p values(see Fig. 1). Clearly, the p value governs the two competing processes in the coevolution. With small p values, the total number of difterent opinions a- remains unchanged and individuals with the same opinion are grouped in the same communities, where they have dense linked to other communities with different oplnolceAsp connections to others with the same opinion, but rare edges 1(a)]. In contrast, for larger p values, all individuals share the same unique opinion and are clustered as a tight group [see ig. 1(b)]. Based on these results, one can find the expected Rank qualitative behavior of our model: the MP update procedure FIG. 2.( Color online) Log-log plot of rank-size distribution, increases the number of nearest-neighbor vertex pairs with i.e, the rank versus the proportion of individuals holding the same the same opinions, while the MA update consolidates the opinion with p=0.2()and 0.5(O). The inset shows the case of chance that, without changing opinions, individuals come large p=0.8(A) Parameters: N=1000, (k)=10, and =0.6 016104-2

of its neighbors
i.e., the opinion the largest number of indi￾viduals among its neighbors hold.
2 Minority avoidance
MA: otherwise, with likelihood 1−p, the focal individual removes the link to one of its neighbors who holds a minority and different opinion
i.e., the different opinion held by the least number of individuals within its neighborhood, and then with probability
rewires to a random neighbor of its neighbors who holds the same opinion, or otherwise to a random one selected from the whole population except its nearest neighbors
with probabil￾ity 1−
. MP represents the influence of individual acquaintances and also the tendency to adopting the majority opinion among the node’s acquaintances. Such preference reflects one’s reluctance to be in a minority among friends as well as one’s willingness to be in agreement with most friends. MA characterizes the situation in which individuals avoid becom￾ing a minority among their friends and try to form new ac￾quaintanceships with the neighbors of neighbors such that they would have a great chance to become a majority in the MP updating, if presently in the minority. These two rules are in line with the conformism found in human society. In what follows, we will investigate the competing roles of MP and MA in the coevolution of opinions and networks, and con￾centrate on how the model parameters
i.e., p and
affect the evolution. III. SIMULATION RESULTS AND DISCUSSIONS In our simulations, we keep the population size N and the total number of edges M constant during the evolution. Thus the average connectivity k=2M /N is fixed. In addition, iso￾lated vertices and duplicate links are not allowed. Initially, individuals are located on a random regular graph
the who￾meets-whom relationship is random and homogeneous and opinions are uniformly assigned to vertices at random. Ac￾cording to our specified update rule, in finite time steps, the system will evidently approach a “consensus state” in which each individual’s opinion agrees with the majority of its neighbors. We stop the simulations when this consensus is achieved. At this stage we compute the number of survival opinions and the consensus time as a function of the param￾eters p and
, respectively. In order to gain intuition into the results of our model, we plot the emergent networks in the final stage corresponding to two different p values
see Fig. 1. Clearly, the p value governs the two competing processes in the coevolution. With small p values, the total number of different opinions remains unchanged and individuals with the same opinion are grouped in the same communities, where they have dense connections to others with the same opinion, but rare edges linked to other communities with different opinions see Fig. 1
a. In contrast, for larger p values, all individuals share the same unique opinion and are clustered as a tight group see Fig. 1
b. Based on these results, one can find the expected qualitative behavior of our model: the MP update procedure increases the number of nearest-neighbor vertex pairs with the same opinions, while the MA update consolidates the chance that, without changing opinions, individuals come into agreement with the majority of their neighbors. In par￾ticular, when the rewiring process occurs in a triadic manner
i.e., friends of friends, it leads to an increase of the total number of triangles in the network, i.e., the clustering coef- ficient. We will show that this tendency of redirection to neighbors of neighbors has a recognizable impact on the evo￾lution of opinion. In combination with these two competing update moves, the system will eventually reach a state de￾pendent on the p value: small p values result in the coexist￾ence of different opinions; large p values lead to a uniform opinion among the population. We present the rank-size distribution of opinions in a net￾work in a consensus state in Fig. 2. With small p values, most of the opinions have roughly equal numbers of holders; whereas, with increasing p values, we observe a progressive broadening of the distribution, which clearly presents a heavy tail. As p becomes very large, most opinions go into extinction in the population during the coevolutionary pro￾cess, but certain opinions dominate until uniformity in opin￾ion is attained. Here, we should point out that, for suitable p values, community structure also emerges along with the co￾FIG. 1.
Color online Illustration of the coevolution of opinions and networks. Left panel
p=0.6,
=0.6: the coexistence of differ￾ent opinions, in which the individuals are self-organized into com￾munities or groups sharing the same opinion
node color denotes the type of opinion. Right panel
p=0.8,
=0.6: uniformity of the individual opinions exists, where individuals are clustered in a whole tight group. Parameters: population size N=100, average de￾gree k=6, and initial number of opinions G=5. FIG. 2.
Color online Log-log plot of rank-size distribution, i.e., the rank versus the proportion of individuals holding the same opinion with p=0.2

and 0.5
. The inset shows the case of large p=0.8
. Parameters: N=1000, k=10, and
=0.6. FENG FU AND LONG WANG PHYSICAL REVIEW E 78, 016104
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COEVOLUTIONARY DYNAMICS OF OPINIONS AND PHYSICAL REVIEW E 78. 016104(2008) 31 04 FIG. 4. Final number of opinions versus the parameter Pa- rameters: N=1000, (k)=10, G=50, and p=0.6. Each data point re- IG. 3.( Color online) Contour plot of the number of survival ults from an average of 500 independent runs opinions as a function of the parameter space (p, ) The right color bar indicates the number of opinions. Parameters: N=1000,(k) Another quantity of interest is the average consensus 10, and G=50. Each data point results from an average of 10 time, which is the number of updates needed to reach con- dependent runs. sensus. We plot the mean consensus time as a function of p with different values of in Fig. 5. Clearly, the consensus evolution. We refer to""community structure"according to time peaks at the critical value of p (p=0.55). For p the qualitative definition given by Newman--it is groups of 0.55, the consensus time links between groups, but between groups there are fewer drops very quickly as p increases. In accordance with the edges [16]. Interestingly, when p is very small (i.e, p-0), result shown in Fig. 3, the p value at which the consensus the community size simply follows the Poisson distribution time peaks moves toward the left when is very large(i.e with mean N/G in the limit of large N. As p increases, a fe b-1). This result also indicates that there exists a phase particular opinions successfully become prevalent in the transition around pe =0.55. Furthermore, the system gets ab- population, indicating that a few giant communities accom- sorbed very fast into a uniform state for larger p values. In panied by some relatively small ones appear. Accordingly, by previous studies of opinion models, finite-size scaling analy varying the value of p, the system goes from a state in which sis is adopted to give insight into the phase transition in such only small communities exist and almost all opinions exist. systems [9]. Moreover, a comprehensive theoretical analysis one in which there is a giant community of like-minded in- of these coevolutionary processes was provided recently (s dividuals plus some smaller ones, and most opinions vanish Ref [15]. We confirm that varying the system size does not in the population. We show the final number of opinions as a function of the qualitatively change the results presented here So far, we have presented the main results of our opinion parameter space (p, )in Fig 3. For fixed o, there exists a model. Compared with previous works of opinion formation critical value p()of p; below p(o), a broad variety of opinions are preserved in the population, namely, diversity of on fixed and evolving networks, our proposed model reflects individual views is achieved; above p(), each individual more realistic scenarios and focuses on a nontrivial network holds the same opinion, that is, a uniform view is attained in the population. Interestingly, one finds that pe() monotoni- 25x10 cally decreases with increasing o value, especially when arge. In addition, for those p values not close to p(), the effect of varying values on evolution of opinion vanishes When p is close to P(o), a change of plays an important role in the evolution of opinion(see Fig. 3): for fixed p, increasing o leads to a decrease in the number of surviving opinions. More explicitly, we plot the number of opinions as a function of in Fig 4. We see that the number of opinions 50×10 is quickly reduced with -1. In other words, an increasing endency of an individual to rewire to neighbors of neighbors quickly reduces the number of survival opinions, and thus is more likely to result in consensus of opinion. Hence, these results demonstrate that the preference toward ordered or FIG. 5.( Color online) Plot of consensus time as a function of random rewiring plays a nontrivial role in evolution of opin-the parameter P, with different values of =0.15(),0.30(O), ion although the parameter p primarily determines the dy- and 0.60(A). Parameters: N=1000, (k)=10, and G=50. Each data namIcs point results from an average of 100 independent runs. 016104-3

evolution. We refer to “community structure” according to the qualitative definition given by Newman—it is groups of network vertices; within these groups there are dense internal links between groups, but between groups there are fewer edges 16. Interestingly, when p is very small
i.e., p→0, the community size simply follows the Poisson distribution with mean N/G in the limit of large N. As p increases, a few particular opinions successfully become prevalent in the population, indicating that a few giant communities accom￾panied by some relatively small ones appear. Accordingly, by varying the value of p, the system goes from a state in which only small communities exist and almost all opinions exist to one in which there is a giant community of like-minded in￾dividuals plus some smaller ones, and most opinions vanish in the population. We show the final number of opinions as a function of the parameter space
p,
in Fig. 3. For fixed
, there exists a critical value pc

of p; below pc

, a broad variety of opinions are preserved in the population, namely, diversity of individual views is achieved; above pc

, each individual holds the same opinion, that is, a uniform view is attained in the population. Interestingly, one finds that pc

monotoni￾cally decreases with increasing
value, especially when
is large. In addition, for those p values not close to pc

, the effect of varying
values on evolution of opinion vanishes. When p is close to pc

, a change of
plays an important role in the evolution of opinion
see Fig. 3: for fixed p, increasing
leads to a decrease in the number of surviving opinions. More explicitly, we plot the number of opinions as a function of
in Fig. 4. We see that the number of opinions is quickly reduced with
→1. In other words, an increasing tendency of an individual to rewire to neighbors of neighbors quickly reduces the number of survival opinions, and thus is more likely to result in consensus of opinion. Hence, these results demonstrate that the preference toward ordered or random rewiring plays a nontrivial role in evolution of opin￾ion although the parameter p primarily determines the dy￾namics. Another quantity of interest is the average consensus time, which is the number of updates needed to reach con￾sensus. We plot the mean consensus time as a function of p with different values of
in Fig. 5. Clearly, the consensus time peaks at the critical value of p
p0.55. For p 0.55, the consensus time monotonically increases with in￾creasing p values; while for p0.55, the consensus time drops very quickly as p increases. In accordance with the result shown in Fig. 3, the p value at which the consensus time peaks moves toward the left when
is very large
i.e.,
→1. This result also indicates that there exists a phase transition around pc0.55. Furthermore, the system gets ab￾sorbed very fast into a uniform state for larger p values. In previous studies of opinion models, finite-size scaling analy￾sis is adopted to give insight into the phase transition in such systems 9. Moreover, a comprehensive theoretical analysis of these coevolutionary processes was provided recently
see Ref. 15. We confirm that varying the system size does not qualitatively change the results presented here. So far, we have presented the main results of our opinion model. Compared with previous works of opinion formation on fixed and evolving networks, our proposed model reflects more realistic scenarios and focuses on a nontrivial network FIG. 3.
Color online Contour plot of the number of survival opinions as a function of the parameter space
p,
. The right color bar indicates the number of opinions. Parameters: N=1000, k =10, and G=50. Each data point results from an average of 100 independent runs. FIG. 4. Final number of opinions versus the parameter
. Pa￾rameters: N=1000, k=10, G=50, and p=0.6. Each data point re￾sults from an average of 500 independent runs. FIG. 5.
Color online Plot of consensus time as a function of the parameter p, with different values of
=0.15

, 0.30
, and 0.60
. Parameters: N=1000, k=10, and G=50. Each data point results from an average of 100 independent runs. COEVOLUTIONARY DYNAMICS OF OPINIONS AND … PHYSICAL REVIEW E 78, 016104
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FENG FU AND LONG WANG PHYSICAL REVIEW E 78. 016104(2008) rewiring process. Most existing studies allow the network to imposes a negative effect on the diversity of opinion. This be disconnected during the coevolutionary process. How- effect becomes large especially near the critical value of the ever, we argue that it is more reasonable to keep the network model parameter onnected, since this constraint facilitates the investigation Recently, many promising research directions are gaining of the possible resulting network configurations. Further- ground, such as the naming game [17], language games [18]. more, we take into account the effect of the tendency to and consensus problems [19, which may be viewed as gen- ordered or random rewiring on opinion dynamics. Interest- eralized opinion dynamics and formation. The fascinating ingly, we found that an increasing tendency of an individual language dynamics of the struggle between regular and ir to rewire to neighbors of neighbors is more likely to lead to regular English verbs examined in Ref. [20] could exemplify consensus of opinion. This is because, if individuals tend to the significance of this kind of research, both theoretically be grouped with others holding the same opinion, this readily and empirically. Therefore, it is of much interest and impor- leads to the situation in which some opinions held by the tance to provide a unifying framework to understand these minority are more likely collective dynamics stemming from diverse fields IV CONCLUSIONS ACKNOWLEDGMENTS We investigated the coevolutionary dynamics of opinions We thank two anonymous reviewers for their constructive and networks based upon two distinct update moves: MP and comments. Valuable discussion with Dr C. Taylor is grate- MA. These two competing processes capture the character- fully acknowledged. This work was supported by NSFC istics of an individuals opinion change and an individuals (Grant Nos. 60674050 and 60528007), the National 973 Pro- neighborhood adjustment, respectively. By varying the gram( Grant No. 2002CB312200), the National 863 Program model parameter determining the two update moves, the sys-(Grant No. 2006AA04Z258), and the 11-5 Project( Grant No tem goes from a diverse world where a broad variety of A2120061303). F.F. also gratefully acknowledges the sup. opinions are present to a uniform one in which everyone port of the China Scholarship Council ( Grant No hares the same view. Importantly and interestingly, we 2007U01235). The Program for Evolutionary Dynamics at found that the tendency to rewire to neighbors of neighbors Harvard University is sponsored by Jeffrey Epstein [1]D Stauffer et al., Biology, Sociology, Geology by Computa- [13]B Kozma and A Barrat, Phys. Rev. E 77, 016102(2008) tional Physicists(Elsevier, Amsterdam, 2006) [14]C. Nardini, B. Kozma, and A Barrat, Phys. Rev. Lett. 100 [2]R. Glauber, J. Math. Phys. 4, 294(1963) 158701(2008 [3]V. Sood and S. Redner, Phys. Rev. Lett. 94, 178701(2005). [15]F. Vazquez, V. M. Eguiluz, and M. S. Miguel, Phys. Rev. Lett. 4P. Chen and S. Redner, J Phys. A 38. 7239(2005); Phys. Rev 100,108702(2008) E71,036101(2005) [5]SGalam, B Chopard, A Masselot, and M Droz, Eur. Phys. J. [16]M. E J. Newman, Eur. Phys. J. B 38, 321(2004) B4,529(1998) [17J A. Baronchelli. L. Dall'Asta, A. Barrat, and V. Loreto, Phys 6]R. Lambiotte, M. Ausloos, and J. A. Holyst, Phys. Rev. E 75. Rev. E 73, 015102(R)(2006): L. Dall'Asta, A. Baronchelli, A 030101(R)(2007):R. Lambiotte, S. Thurner, and R. Hanel Barrat, and V Loreto, Europhys. Lett. 73, 969(2006): Phys. ibid. 76, 046101(2007); R. Lambiotte and M. Ausloos, J Stat Rev. E 74, 036105(2006: A Baronchelli, M. Felici, E Ca- Mech. Theory Exp (2007), P08026 glioti, V. Loreto, and L. Steels, J. Stat. Mech. Theory Exp [7K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C 11. 1157 (2006),PO6014 2000 [18M. A Nowak, N. L Komarova, and P. Niyogi, Nature(Lon- []G. Duffuant, F. Amblard, G. Weisbuch, and T. Faure, J. Artif. don)417,611(2002) Soc. Soc. Simul. 5. 4(2002). [ 19] This direction was initiated by the seminal paper of T. Vicsek [9]P. Holme and M. E. J. Newman, Phys. Rev. E 74, 056108 et aL, Phys. Rev. Lett. 75, 1226(1995), and has attracted increasing interest from researchers in the control community [10]D. Stauffer, M. Hohnisch, and S. Pittnauer, Physica A 370, for a recent review. we refer to R. olfati-Saber. J. A. Fax. and 734(2006) R. M. Murray, Proc. IEEE 95, 215(2007) [11]S. Gil and D. H. Zanette, Phys. Lett. A 356, 89(2006) [20JE. Lieberman, J.-B. Michel, J. Jackson, T. Tang, and M. A [12]D. H. Zanette and S. Gil, Physica D 224, 156(2006). Nowak, Nature (London)449, 713(2007) 016104-4

rewiring process. Most existing studies allow the network to be disconnected during the coevolutionary process. How￾ever, we argue that it is more reasonable to keep the network connected, since this constraint facilitates the investigation of the possible resulting network configurations. Further￾more, we take into account the effect of the tendency to ordered or random rewiring on opinion dynamics. Interest￾ingly, we found that an increasing tendency of an individual to rewire to neighbors of neighbors is more likely to lead to consensus of opinion. This is because, if individuals tend to be grouped with others holding the same opinion, this readily leads to the situation in which some opinions held by the minority are more likely to die out ultimately. IV. CONCLUSIONS We investigated the coevolutionary dynamics of opinions and networks based upon two distinct update moves: MP and MA. These two competing processes capture the character￾istics of an individual’s opinion change and an individual’s neighborhood adjustment, respectively. By varying the model parameter determining the two update moves, the sys￾tem goes from a diverse world where a broad variety of opinions are present to a uniform one in which everyone shares the same view. Importantly and interestingly, we found that the tendency to rewire to neighbors of neighbors imposes a negative effect on the diversity of opinion. This effect becomes large especially near the critical value of the model parameter. Recently, many promising research directions are gaining ground, such as the naming game 17, language games 18, and consensus problems 19, which may be viewed as gen￾eralized opinion dynamics and formation. The fascinating language dynamics of the struggle between regular and ir￾regular English verbs examined in Ref. 20 could exemplify the significance of this kind of research, both theoretically and empirically. Therefore, it is of much interest and impor￾tance to provide a unifying framework to understand these collective dynamics stemming from diverse fields. ACKNOWLEDGMENTS We thank two anonymous reviewers for their constructive comments. Valuable discussion with Dr. C. Taylor is grate￾fully acknowledged. This work was supported by NSFC
Grant Nos. 60674050 and 60528007, the National 973 Pro￾gram
Grant No. 2002CB312200, the National 863 Program
Grant No. 2006AA04Z258, and the 11-5 Project
Grant No. A2120061303. F.F. also gratefully acknowledges the sup￾port of the China Scholarship Council
Grant No. 2007U01235. The Program for Evolutionary Dynamics at Harvard University is sponsored by Jeffrey Epstein. 1 D. Stauffer et al., Biology, Sociology, Geology by Computa￾tional Physicists
Elsevier, Amsterdam, 2006. 2 R. Glauber, J. Math. Phys. 4, 294
1963. 3 V. Sood and S. Redner, Phys. Rev. Lett. 94, 178701
2005. 4 P. Chen and S. Redner, J. Phys. A 38, 7239
2005; Phys. Rev. E 71, 036101
2005. 5 S. Galam, B. Chopard, A. Masselot, and M. Droz, Eur. Phys. J. B 4, 529
1998. 6 R. Lambiotte, M. Ausloos, and J. A. Hołyst, Phys. Rev. E 75, 030101
R
2007; R. Lambiotte, S. Thurner, and R. Hanel, ibid. 76, 046101
2007; R. Lambiotte and M. Ausloos, J. Stat. Mech.: Theory Exp.
2007, P08026. 7 K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C 11, 1157
2000. 8 G. Duffuant, F. Amblard, G. Weisbuch, and T. Faure, J. Artif. Soc. Soc. Simul. 5, 4
2002. 9 P. Holme and M. E. J. Newman, Phys. Rev. E 74, 056108
2006. 10 D. Stauffer, M. Hohnisch, and S. Pittnauer, Physica A 370, 734
2006. 11 S. Gil and D. H. Zanette, Phys. Lett. A 356, 89
2006. 12 D. H. Zanette and S. Gil, Physica D 224, 156
2006. 13 B. Kozma and A. Barrat, Phys. Rev. E 77, 016102
2008. 14 C. Nardini, B. Kozma, and A. Barrat, Phys. Rev. Lett. 100, 158701
2008. 15 F. Vazquez, V. M. Eguíluz, and M. S. Miguel, Phys. Rev. Lett. 100, 108702
2008. 16 M. E. J. Newman, Eur. Phys. J. B 38, 321
2004. 17 A. Baronchelli, L. Dall’Asta, A. Barrat, and V. Loreto, Phys. Rev. E 73, 015102
R
2006; L. Dall’Asta, A. Baronchelli, A. Barrat, and V. Loreto, Europhys. Lett. 73, 969
2006; Phys. Rev. E 74, 036105
2006; A. Baronchelli, M. Felici, E. Ca￾glioti, V. Loreto, and L. Steels, J. Stat. Mech.: Theory Exp.
2006, P06014. 18 M. A. Nowak, N. L. Komarova, and P. Niyogi, Nature
Lon￾don 417, 611
2002. 19 This direction was initiated by the seminal paper of T. Vicsek et al., Phys. Rev. Lett. 75, 1226
1995, and has attracted increasing interest from researchers in the control community- ;for a recent review, we refer to R. Olfati-Saber, J. A. Fax, and R. M. Murray, Proc. IEEE 95, 215
2007. 20 E. Lieberman, J.-B. Michel, J. Jackson, T. Tang, and M. A. Nowak, Nature
London 449, 713
2007. FENG FU AND LONG WANG PHYSICAL REVIEW E 78, 016104
2008 016104-4