week ending PRL103,018701(2009) PHYSICAL REVIEW LETTERS 3 JULY 2009 Dynamic Opinion Model and Invasion Percolation Jia Shao, Shlomo Havlin,-and H. Eugene stanle Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA Minerva Center and Department of Physics, Bar-Man University, 52900 Ramat-Gan, Israel (Received 25 March 2009: published I July 2009) We propose a"nonconsensus"opinion model that allows for stable coexistence of two opinions by forming clusters of agents holding the same opinion. We study this nonconsensus model on lattices, several model complex networks, and a real-life social network. We find that the model displays a phas transition behavior characterized by a large spanning cluster of nodes holding the same opinion appearing when the concentration of nodes holding the same opinion(even minority)is above a certain threshold Because of the clustering(community support) of agents holding the same opinion, these clusters cannot be invaded by the other opinion( similar to incompressible fluids ). Our extensive simulations show that the nonconsensus opinion model appears to belong to the same universality class as invasion percolation. DOI: 10. 1103/PhysRevLett 103.018701 PACS numbers: 89.75 Hc, 64.60 ah 87.23. Ge N Social dynamics has been studied extensively in recent formation can be mapped to a known physics percolation ears using concepts and methods based on ideas from problem statistical physics. An important approach is complex net The structure of clusters formed by agents holding the works, where the nodes represent agents and the links same opinion is relevant in many real-life scenarios, such represent the interactions between them. There is consid- as the propagation of ideas in human populations and erable current interest in the problem of how two compet- communications among people holding the same opinion ing opinions evolve in populations [1]. Various versions of Thus, it is of interest to identify and understand the topo- the opinion model have been proposed [1], among which logical properties of the formed clusters, as well as the are the Sznajd model [2], the voter model [3], the majority distribution of the sizes of clusters and the average distance rule model [4], and the social impact model [5]. Models between agents belonging to the same cluster. incorporating the evolution of two competing states can be The basic assumption of the NCO model is that opinion mapped into spin models and can be applied in a much formation is a process where an agent's opinion is influ- broader range of disciplines from chemistry, physics, and enced both by his own current opinion and that of his biology to social science [1] friends represented as nearest neighbors in the network The main models which are based on spin systems with This assumption implies that a person is influenced by the short range interactions lead to a steady state with either majority opinion of the group which includes his friends consensus of a single opinion or equal concentrations of and himself. The idea of incorporating the role played by the two opinions [3, 4]. In real life, however, a stable the current state of an agent on its future state has been coexistence with unequal concentrations of two opinions considered elsewhere[8]. The two opinions are denoted by is commonly seen. a-and o+. At time t=0, opinions are randomly assigned In this Letter, we propose a spin-type nonconsensus to all nodes: With probability f a node will be assigned pinion(NCO)model, which demonstrates novel nontri- opinion o- and with probability 1-f opinion o+ For a vial stable states in which stable coexistence of minority randomly chosen node i, the node and its nearest neighbors and majority opinions occurs. This stable state is reached form a set of nodes A At each time step, node i will from a random initial configuration after a dynamical convert to its opposite opinion, if it is in the local minority process in a relatively short time. Further, we find that, opinion. If the two opinions are equally represented in A when the population of one opinion is above a certain will have probability p to convert to its opposite opin critical threshold, even still minority, a large spanning and probability l- p to remain unchanged. For simplicity, cluster of a size proportional to the total population forms. we present here results only for P=0[9]. At each simu- Using extensive simulations, we find that the phase tran- lation step, every node is tested to see whether its opinion sition in the NCO model belongs to the same universality needs to be changed. All of the updates are made simulta lass as invasion percolation with trapping (TIP)[6, 7]. neously at each simulation step. The system is considered Once agents holding the same opinion form a cluster, to reach a stable state if no more changes occur. where each member of the cluster gets enough commu- The NCO model shares features with the majority-voter nity support to hold its opinion, the cluster becomes stable model [3]. The difference is that the majority-voter model, and cannot be penetrated by the other opinion such as which converges to consensus in its stable state, takes into incompressible fluids in TIP. This is the first time that an account only the opinion of neighbors of a selected node i opinion model as well as clustering behavior in opinion Here we include the opinion of node i itself into consid 0031-9007/09/103(1)/018701(4) 018701-1 c 2009 The American Physical Society
Dynamic Opinion Model and Invasion Percolation Jia Shao,1 Shlomo Havlin,2 and H. Eugene Stanley1 1 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA 2 Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel (Received 25 March 2009; published 1 July 2009) We propose a ‘‘nonconsensus’’ opinion model that allows for stable coexistence of two opinions by forming clusters of agents holding the same opinion. We study this nonconsensus model on lattices, several model complex networks, and a real-life social network. We find that the model displays a phase transition behavior characterized by a large spanning cluster of nodes holding the same opinion appearing when the concentration of nodes holding the same opinion (even minority) is above a certain threshold. Because of the clustering (community support) of agents holding the same opinion, these clusters cannot be invaded by the other opinion (similar to incompressible fluids). Our extensive simulations show that the nonconsensus opinion model appears to belong to the same universality class as invasion percolation. DOI: 10.1103/PhysRevLett.103.018701 PACS numbers: 89.75.Hc, 64.60.ah, 87.23.Ge Social dynamics has been studied extensively in recent years using concepts and methods based on ideas from statistical physics. An important approach is complex networks, where the nodes represent agents and the links represent the interactions between them. There is considerable current interest in the problem of how two competing opinions evolve in populations [1]. Various versions of the opinion model have been proposed [1], among which are the Sznajd model [2], the voter model [3], the majority rule model [4], and the social impact model [5]. Models incorporating the evolution of two competing states can be mapped into spin models and can be applied in a much broader range of disciplines from chemistry, physics, and biology to social science [1]. The main models which are based on spin systems with short range interactions lead to a steady state with either consensus of a single opinion or equal concentrations of the two opinions [3,4]. In real life, however, a stable coexistence with unequal concentrations of two opinions is commonly seen. In this Letter, we propose a spin-type nonconsensus opinion (NCO) model, which demonstrates novel nontrivial stable states in which stable coexistence of minority and majority opinions occurs. This stable state is reached from a random initial configuration after a dynamical process in a relatively short time. Further, we find that, when the population of one opinion is above a certain critical threshold, even still minority, a large spanning cluster of a size proportional to the total population forms. Using extensive simulations, we find that the phase transition in the NCO model belongs to the same universality class as invasion percolation with trapping (TIP) [6,7]. Once agents holding the same opinion form a cluster, where each member of the cluster gets enough community support to hold its opinion, the cluster becomes stable and cannot be penetrated by the other opinion such as incompressible fluids in TIP. This is the first time that an opinion model as well as clustering behavior in opinion formation can be mapped to a known physics percolation problem. The structure of clusters formed by agents holding the same opinion is relevant in many real-life scenarios, such as the propagation of ideas in human populations and communications among people holding the same opinion. Thus, it is of interest to identify and understand the topological properties of the formed clusters, as well as the distribution of the sizes of clusters and the average distance between agents belonging to the same cluster. The basic assumption of the NCO model is that opinion formation is a process where an agent’s opinion is influenced both by his own current opinion and that of his friends represented as nearest neighbors in the network. This assumption implies that a person is influenced by the majority opinion of the group which includes his friends and himself. The idea of incorporating the role played by the current state of an agent on its future state has been considered elsewhere [8]. The two opinions are denoted by and þ. At time t ¼ 0, opinions are randomly assigned to all nodes: With probability f a node will be assigned opinion and with probability 1 f opinion þ. For a randomly chosen node i, the node and its nearest neighbors form a set of nodes Ai. At each time step, node i will convert to its opposite opinion, if it is in the local minority opinion. If the two opinions are equally represented in Ai, i will have probability p to convert to its opposite opinion and probability 1 p to remain unchanged. For simplicity, we present here results only for p ¼ 0 [9]. At each simulation step, every node is tested to see whether its opinion needs to be changed. All of the updates are made simultaneously at each simulation step. The system is considered to reach a stable state if no more changes occur. The NCO model shares features with the majority-voter model [3]. The difference is that the majority-voter model, which converges to consensus in its stable state, takes into account only the opinion of neighbors of a selected node i. Here we include the opinion of node i itself into considPRL 103, 018701 (2009) PHYSICAL REVIEW LETTERS week ending 3 JULY 2009 0031-9007=09=103(1)=018701(4) 018701-1 2009 The American Physical Society
week ending PRL103,018701(2009) PHYSICAL REVIEW LETTERS 3 JULY 2009 (a)=0 (b)t (c)t2 FIG. I. Dynamics of the NCO model showing the approach to (b)ER a stable state on a network with N=9 nodes. For simplicity, we assume p=0.(a) At t=0, five nodes are randomly assigned to 1.D be o+(filled circle). The remaining four nodes are assigned o (open circle). In the set comprising of node A and its 4 neighbor ing nodes(dashed box), node A is in a local minority opinion(2 o+ nodes and 3 o-nodes), while the remaining nodes are not. azb At the end of simulation step I=0, node A is converted into o (b)At t=l, in the set of nodes comprising node B and 6 neighboring nodes(dotted box), node B becomes in a local minority opinion (3 0+ nodes and 4 o-nodes), while th remaining nodes are not node b is converted into o at th FIG. 2(color online). Plot of the normalized size of the largest cluster si(dotted line), the second largest cluster s,(full line). end of simulation step t= 1. (c)At t=2, the nine-nodes system and the fraction of o- nodes F(dashed line)in the stable state as reaches a stable state a function of f for(a)a SF network with A= 2.5 and N= 10 (b)an ER network with(k)=4 and N= 10, (c)a HEP network eration,which makes the formation of stable clusters pos- with A-2.9, and (d)a hX lattice of size 1000 X 1000 [20]. Each sible even for nonzero p. curve represents an average over 100 realizations. The sharp A demonstration of the dynamics in the NCO model is ncrease of si and the peak of s2 atf indicate a second-order hown in Fig. 1. At time t=0, nodes C, D and E form a phase transition. The insets in(a)and (d)show g(S N)[Eq(1)I stable cluster. No matter what the opinion is of the nodes and f-A(D), where A=0.01 for SF and A=0.005 for HX outside the cluster, the community support inside the clus- Atf, &(S, N) approaches a constant, indicating the size of the ter is enough for nodes C, D, and E to keep their opinions largest custer Si proportional to Ns at f"(Eq(1),which o and not to be invaded by the opposite opinion, which is like another characteristic of a second-order phase transition a trapped liquid (pore)in TIP. In the stable state, all clusters formed by both opinions are stable clusters. Unlike TlP, networks. We find that F is a monotonically increasing one opinion in the NCO model can behave at the same time function of f with symmetry around ( F)=(0.5, 0.5)- both as an invading liquid and as an invaded liquid, expected for opinion dynamics, so the NcO model can as expected, since the two opinion states are symmetrical At a certain critical value f=f, s, shows a sharp increase demonstrate several unique properties, as we discuss later. from a very small value to a finite fraction of the entire We perform simulations of the NCO model in network models and in a real-life network. The network models system, while s2 displays a sharp peak, a characteristic of a include Erdos-Renyi(ER)networks, scale-free(SF)net second-order phase transition [15] works[10], and also two-dimensional (2D)regular lattices the The values of f and F(f)depend only on the type of the network and are almost independent of N. As can be including hexagonal (HX), square (SQ), and triangular seen in Fig. 2. in the stable state. if the final concentration (TR) lattices). As an example of real-life networks, we of o opinion node is above the threshold FO"),the o- analyze the high energy physics(HEP)citations netw nodes will be able to form a large spanning cluster of [l1]. ER networks are characterized by a Poisson degree order of the system size N. Below this threshold only distribution with average degree(k). SF networks are isolated small clusters can form. Note that" for ER, SF, characterized by a power-law degree distribution P(k)- and HEP networks are all less than 0.5, implying that a k- A, with kmin =k<kmax. To ensure network compact- phase transition occurs for nodes holding the minority ness, we choose kmin-2 We use the known natural cutoff opinion. Only for the HX lattice, f*=0.567[16] for kmax [ 12]. Simulations on the SF network with A-2.5 Note that for the Nco model--in contrast to the social [13] are reported here impact model [5]stable clusters of nodes holding the Next, we show the emergence of a phase transition in the minority opinion can persistently survive without asst stable state as the minority opinion o becomes more and ing influential or strong-willed nodes residing nside m- more infiuential (increasing concentration), even well be- clusters fore becoming the majority. We denote by F the fraction of Next, we present results indicating that the NCO model -nodes and S the sizes of the clusters formed by the o- is in the same universality class as TIP For regular(site and nodes in the stable state. The size of the largest and second bond) percolation at criticality, the probability density largest clusters are denoted by SI and S2, respectively, and function of the cluster size S follows a power law P(S) ve define S,= S,/N and S,= S2/N. The system reaches S-, where T=2.055 for 2D lattices and T=2.5 for a stable state after a few simulation steps[14]. In Fig. 2, we higher dimensional networks such as ER and SF[I7]. In show F, Si, and s, as a function of f for four different contrast, the TIP model shows a power-law distribution of 018701-2
eration, which makes the formation of stable clusters possible even for nonzero p. A demonstration of the dynamics in the NCO model is shown in Fig. 1. At time t ¼ 0, nodes C, D, and E form a stable cluster. No matter what the opinion is of the nodes outside the cluster, the community support inside the cluster is enough for nodes C, D, and E to keep their opinions and not to be invaded by the opposite opinion, which is like a trapped liquid (pore) in TIP. In the stable state, all clusters formed by both opinions are stable clusters. Unlike TIP, one opinion in the NCO model can behave at the same time both as an invading liquid and as an invaded liquid, as expected for opinion dynamics, so the NCO model can demonstrate several unique properties, as we discuss later. We perform simulations of the NCO model in network models and in a real-life network. The network models include Erdo˝s-Re´nyi (ER) networks, scale-free (SF) networks [10], and also two-dimensional (2D) regular lattices [including hexagonal (HX), square (SQ), and triangular (TR) lattices]. As an example of real-life networks, we analyze the high energy physics (HEP) citations network [11]. ER networks are characterized by a Poisson degree distribution with average degree hki. SF networks are characterized by a power-law degree distribution PðkÞ k, with kmin k<kmax. To ensure network compactness, we choose kmin ¼ 2. We use the known natural cutoff for kmax [12]. Simulations on the SF network with ¼ 2:5 [13] are reported here. Next, we show the emergence of a phase transition in the stable state as the minority opinion becomes more and more influential (increasing concentration), even well before becoming the majority. We denote by F the fraction of nodes and S the sizes of the clusters formed by the nodes in the stable state. The size of the largest and second largest clusters are denoted by S1 and S2, respectively, and we define s1 S1=N and s2 S2=N. The system reaches a stable state after a few simulation steps [14]. In Fig. 2, we show F, s1, and s2 as a function of f for four different networks. We find that F is a monotonically increasing function of f with symmetry around ðf; FÞ¼ð0:5; 0:5Þ— as expected, since the two opinion states are symmetrical. At a certain critical value f ¼ f, s1 shows a sharp increase from a very small value to a finite fraction of the entire system, while s2 displays a sharp peak, a characteristic of a second-order phase transition [15]. The values of f and FðfÞ depend only on the type of the network and are almost independent of N. As can be seen in Fig. 2, in the stable state, if the final concentration of opinion node is above the threshold FðfÞ, the nodes will be able to form a large spanning cluster of the order of the system size N. Below this threshold only isolated small clusters can form. Note that f for ER, SF, and HEP networks are all less than 0.5, implying that a phase transition occurs for nodes holding the minority opinion. Only for the HX lattice, f 0:567 [16]. Note that for the NCO model—in contrast to the social impact model [5]—stable clusters of nodes holding the minority opinion can persistently survive without assuming influential or strong-willed nodes residing inside the clusters. Next, we present results indicating that the NCO model is in the same universality class as TIP. For regular (site and bond) percolation at criticality, the probability density function of the cluster size S follows a power law PðSÞ S, where ¼ 2:055 for 2D lattices and ¼ 2:5 for higher dimensional networks such as ER and SF [17]. In contrast, the TIP model shows a power-law distribution of FIG. 1. Dynamics of the NCO model showing the approach to a stable state on a network with N ¼ 9 nodes. For simplicity, we assume p ¼ 0. (a) At t ¼ 0, five nodes are randomly assigned to be þ (filled circle). The remaining four nodes are assigned (open circle). In the set comprising of node A and its 4 neighboring nodes (dashed box), node A is in a local minority opinion (2 þ nodes and 3 nodes), while the remaining nodes are not. At the end of simulation step t ¼ 0, node A is converted into . (b) At t ¼ 1, in the set of nodes comprising node B and its 6 neighboring nodes (dotted box), node B becomes in a local minority opinion (3 þ nodes and 4 nodes), while the remaining nodes are not. Node B is converted into at the end of simulation step t ¼ 1. (c) At t ¼ 2, the nine-nodes system reaches a stable state. 0.000 0.002 0.004 0.006 0.008 S 104 105 106 N 0 1 2 3 g(S ,N) 0.0 0.2 0.4 0.6 0.8 f 0.0 0.2 0.4 0.6 0.8 1.0 S s1 s2 F (a)SF 1 1 2 0.000 0.002 0.004 0.006 S 0.2 0.4 0.6 0.8 f 0.0 0.2 0.4 0.6 0.8 1.0 S (b) ER 1 2 0.0 0.2 0.4 0.6 0.8 1.0 S 0.2 0.4 0.6 0.8 f 0.000 0.002 0.004 0.006 0.008 S (c)HEP 1 2 0 0.2 0.4 0.6 0.8 f 0.0 0.2 0.4 0.6 0.8 1.0 S 104 105 106 N 0.3 0.6 0.9 1.2 g(S ,N) 0.0 0.1 0.2 S (d)HX 1 1 2 FIG. 2 (color online). Plot of the normalized size of the largest cluster s1 (dotted line), the second largest cluster s2 (full line), and the fraction of nodes F (dashed line) in the stable state as a function of f for (a) a SF network with ¼ 2:5 and N ¼ 105, (b) an ER network with hki ¼ 4 and N ¼ 105, (c) a HEP network with 2:9, and (d) a HX lattice of size 10001000 [20]. Each curve represents an average over 100 realizations. The sharp increase of s1 and the peak of s2 at f indicate a second-order phase transition. The insets in (a) and (d) show gðS1; NÞ [Eq. (1)] as a function of N at the critical threshold f (d), f þ (e), and f (h), where ¼ 0:01 for SF and ¼ 0:005 for HX. At f, gðS1; NÞ approaches a constant, indicating the size of the largest cluster S1 proportional to N at f [Eq. (1)], which is another characteristic of a second-order phase transition. PRL 103, 018701 (2009) PHYSICAL REVIEW LETTERS week ending 3 JULY 2009 018701-2
week ending PRL103,018701(2009) PHYSICAL REVIEW LETTERS 3 JULY 2009 the sizes of pores [see Fig 3(a)] with a cumulative distri- 1.84 +0.01 and de a 1.54+0.02, in close agreement bution function having the form P(S>S)-S-, with with the fractal dimensions of invading fluid in TIP. T 1.90+0.01 for 2D lattices, different from regular These results provide further supports for the NCO ercolation model belonging to the same universality class as the TIP We find that for the nco model in 2d lattices. the model cumulative distribution function of cluster sizes at critical It is known that for 3D or higher dimensional systems, ity [see Fig 3(a)] is P(S>S)-S-, with T= 1.89+ which include ER and SE, trapping becomes not effective 0.01, which is close to the T from the distribution of pore and Tip falls into the same universality class of regular sizes in the TIP model. This fact leads us to hypothesize percolation [6]. Our simulations of the NCO model on ER that for a 2D lattice the nco model belongs to the same(not shown) and SF networks indeed show the same P(S> universality class as TIP. S)Isee Fig. 3(a)] and the same fractal dimension(not To further test this hypothesis, we study the fractal shown) as for regular percolation. For the NCO model, dimensions of the stable clusters formed by the NCO at a-nodes serve as both invading liquid and replaced liquid criticality For regular percolation in 2D lattices, the fractal (similarly for the o+ nodes), which is the reason why at dimensions of the clusters at criticality can be calculated criticality a-nodes have both the d and de of the invad from the power-law relation between S and the cluster ing fluid and the T of the pores diameter, which is represented by either the radius of To further support the existence of a second-order phase gyration Rg or the average hopping distances between all transition, in the insets in Figs. 2(a) and 2(d), we plot pairs of nodes e: SRg and S g(S1,N) as a function of N at f and f±△, where 91/48=1.896 and de≈1.678±0.003[17]. For TIP in g(S1,N)≡S1/N 2D lattices, the invading liquid has fractal dimensions of dr=1.83±0.0 I and d e=1.51±0.0l6 Here 5=0.667 for high-dimensional networks like SF and Our simulations for 2D lattices show that the stable ER, and 4=de/2 for 2D lattices [17]. It is another char lusters of the NCO model at criticality are also fractals. acteristic of a second-order phase transition that at criti To test whether they have the same fractal dimensions as cality g(SI, N)approaches a constant. Indeed, for SF with TIP(as our hypothesis ), we plot Rg/sl/dy and e/s/de as aA=2.5 (=0.450), when s=0.667, g(S,, N)ap- function of S in Figs. 3(b)and 3(c). We test different trial proaches a constant. For HX (=0.567), when s= values of df and de to find the best power-law fits for the 0.92, 8(S1 N) also approaches a constant. The fact that simulation results. We find that for d =1.84, for both sQ the theoretically predicted values of s fit well the simula and TR,Re/sl/df approaches asymptotically a constant.In tion results provides further evidence for our findings To understand why, in contrast to regular percolation contrast,when we choose d/=1.896(regular percolation [13], the NCO model in the SF network with Ak)of the a-nodes in the stable state. Fig function of S. We conclude that, for the NCO model, df= ure 4(a)shows(k()) for SF networks with A-2.5 and sQd=1.896 074sQd=1.678 △ NCO sQ FIG. 3(color online). (a) The cumulative distribution function of cluster sizes P(s> S)atf for a NCO model on a sQ lattice with N=9 10(3000 x 3000)and a SF network with A= 2.5 and N=10. P(S>S)-S- for both SQ and SF, where T* 1.89 (SQ)and T= 2.5(SF). P(S> S)of the sizes of pores of TIP on SQ with N=9 x 10 is also shown, which also takes the form P(S>S)-S-T with T 1.90. Averages over 100 realizations are shown for all curves (b) For the NCO model, at criticality, Ra/s/ dy as functions of S for SQ and TR with N=9 X 106. For the trial value d,=1.84, Rg/slay approaches a constant.For e/s/dt as a function of S for so and TR with N =9x 106. For the trial value de = 1. 54. e/5/de approaches a constant. For d, a' 1.678(regular percolation fractal dimension), e/s d is an increasing function of S In(b)and(c), each curve is averaged over 1000 realization 018701-3
the sizes of pores [see Fig. 3(a)] with a cumulative distribution function having the form PðS0 > SÞ S1, with 1:90 0:01 for 2D lattices, different from regular percolation. We find that, for the NCO model in 2D lattices, the cumulative distribution function of cluster sizes at criticality [see Fig. 3(a)] is PðS0 > SÞ S1, with 1:89 0:01, which is close to the from the distribution of pore sizes in the TIP model. This fact leads us to hypothesize that for a 2D lattice the NCO model belongs to the same universality class as TIP. To further test this hypothesis, we study the fractal dimensions of the stable clusters formed by the NCO at criticality. For regular percolation in 2D lattices, the fractal dimensions of the clusters at criticality can be calculated from the power-law relation between S and the cluster diameter, which is represented by either the radius of gyration Rg or the average hopping distances between all pairs of nodes ‘: S Rdf g and S ‘d‘ , where df ¼ 91=48 ¼ 1:896 and d‘ 1:678 0:003 [17]. For TIP in 2D lattices, the invading liquid has fractal dimensions of df 1:83 0:01 and d‘ 1:51 0:01 [6]. Our simulations for 2D lattices show that the stable clusters of the NCO model at criticality are also fractals. To test whether they have the same fractal dimensions as TIP (as our hypothesis), we plot Rg=S1=df and ‘=S1=d‘ as a function of S in Figs. 3(b) and 3(c). We test different trial values of df and d‘ to find the best power-law fits for the simulation results. We find that for df ¼ 1:84, for both SQ and TR, Rg=S1=df approaches asymptotically a constant. In contrast, when we choose df ¼ 1:896 (regular percolation fractal dimension), for Rg=S1=df we observe an increasing function with S. On the other hand, when d‘ ¼ 1:54, ‘=S1=d‘ approaches a constant. In contrast, when d‘ ¼ 1:678 (as regular percolation), ‘=S1=d‘ is an increasing function of S. We conclude that, for the NCO model, df 1:84 0:01 and d‘ 1:54 0:02, in close agreement with the fractal dimensions of invading fluid in TIP. These results provide further supports for the NCO model belonging to the same universality class as the TIP model. It is known that for 3D or higher dimensional systems, which include ER and SF, trapping becomes not effective and TIP falls into the same universality class of regular percolation [6]. Our simulations of the NCO model on ER (not shown) and SF networks indeed show the same PðS0 > SÞ [see Fig. 3(a)] and the same fractal dimension (not shown) as for regular percolation. For the NCO model, nodes serve as both invading liquid and replaced liquid (similarly for the þ nodes), which is the reason why at criticality nodes have both the df and d‘ of the invading fluid and the of the pores. To further support the existence of a second-order phase transition, in the insets in Figs. 2(a) and 2(d), we plot gðS1; NÞ as a function of N at f and f , where gðS1; NÞ S1=N : (1) Here ¼ 0:667 for high-dimensional networks like SF and ER, and ¼ df=2 for 2D lattices [17]. It is another characteristic of a second-order phase transition that at criticality gðS1; NÞ approaches a constant. Indeed, for SF with ¼ 2:5 (f 0:450), when ¼ 0:667, gðS1; NÞ approaches a constant. For HX (f 0:567), when ¼ 0:92, gðS1; NÞ also approaches a constant. The fact that the theoretically predicted values of fit well the simulation results provides further evidence for our findings. To understand why, in contrast to regular percolation [13], the NCO model in the SF network with kÞ of the nodes in the stable state. Figure 4(a) shows hkðfÞi for SF networks with ¼ 2:5 and 100 102 104 106 S 10-8 10-6 10-4 10-2 100 P(S >S) NCO SQ NCO SF Pores TIP 0.89 0.90 (a) 1.5 101 102 103 104 105 S 0.3 0.4 0.5 0.6 0.7 R /S(1/d ) SQ d =1.84 SQ d =1.896 TR d =1.84 TR d =1.896 f f g f (b) f f 101 102 103 104 105 S 0.2 0.3 0.4 0.5 0.6 0.7 0.8 l/S(1/d ) SQ d =1.54 SQ d =1.678 TR d =1.54 TR d =1.678 l l (c) l l l FIG. 3 (color online). (a) The cumulative distribution function of cluster sizes PðS0 > SÞ at f for a NCO model on a SQ lattice with N ¼ 9 106 (3000 3000) and a SF network with ¼ 2:5 and N ¼ 105. PðS0 > SÞ S1 for both SQ and SF, where 1:89 (SQ) and 2:5 (SF). PðS0 > SÞ of the sizes of pores of TIP on SQ with N ¼ 9 106 is also shown, which also takes the form PðS0 > SÞ S1 with 1:90. Averages over 100 realizations are shown for all curves. (b) For the NCO model, at criticality, Rg=S1=df as functions of S for SQ and TR with N ¼ 9 106. For the trial value df ¼ 1:84, Rg=S1=df approaches a constant. For df ¼ 1:896 (regular percolation fractal dimension), Rg=S1=df is an increasing function of S. (c) For the NCO model, at criticality, ‘=S1=d‘ as a function of S for SQ and TR with N ¼ 9 106. For the trial value d‘ ¼ 1:54, ‘=S1=d‘ approaches a constant. For d‘ ¼ 1:678 (regular percolation fractal dimension), ‘=S1=d‘ is an increasing function of S. In (b) and (c), each curve is averaged over 1000 realizations. PRL 103, 018701 (2009) PHYSICAL REVIEW LETTERS week ending 3 JULY 2009 018701-3
PHYSICAL REVIEW LETTERS week ending PRL103,018701(2009) 3 JULY 2009 [2 K. Sznajd-Weron and J Sznajd, Int J Mod. Phys. C l1 l157(2000. 3 T M. Liggett, Stochastic Interacting Systems: Contact Voter, and Exclusion Processes(Springer, Berlin, 1999) R. Lambiotte and S. Redner, Europhys. Lett. 82, 18 00 (2008) 0.1 02 03 Q4 03 [4]S Galam, Eur. Phys. J B 25, 403(2002): P L. Krapiy and S. Redner, Phys. Rev. Lett. 90, 238701(2003) [5] B Latane, Am. Psychol. 36, 343(1981): A Nowak et FIG. 4(color online).(a) The average degree (k())of o Psychol. Rev. 97, 362(1990) nodes in the stable state as a function of f for a SF network with [6] s. Schwarzer et al., Phys. Rev. E 59, 3262 (1999) d=25 and N= 105 and the hEP network. It is seen that. for [7 Invasion percolation describes the evolution of the front f0.5. tween two immiscible liquids in a random medium since the high degree nodes join the majority opinion [18] when one liquid is displaced by injection of the other. If (b)The cumulative degree distribution P(k'>k)of a-nodes the invasion into the trapped liquid is forbidden because of in the stable state for a sf network with d=2 5 and N= 10 an incompressibility constraint, the model is regarded Pr(>k) for f =0.450 and 0.5 are plotted. Notice that f* s invasion percolation with trapping 0.450. The absence of high degree o- nodes in a stable state fo [8] P S. Dodds and D J. Watts, Phys. Rev. Lett. 92, 218701 fk) for different values of f in nonzero p Fig. 4(b), which provides further evidence for lower de- [10J A.L. Barabasi and R. Albert, Science 286, 509(1999) greesandtheappearanceofsignificantlyfewerhighde-v.BatageljandA.Mrvar,Pajekdatasetshttp://vlado oub/networks/data/ gree nodes in the minority opinion [18]. This explains why, [12]R. Cohen et al., Phys. Rev. Lett. 85. 4626(2000) for sf networks with d= 2.5 and for hEP we observe a ase transition in the NCO model. The minority opinion [13] We use the sF network with A=2.5. There is no random nodes do not include high degree nodes which are respon percolation threshold (Pc =0) for the SF networks with A<3[12]. In contrast, for the NCO model, we obtain a sible for the formation of large spanning cluster for Pc-0 clear phase transition in regular percolation. This process is analogous to remov- [14] The number of agents who change their opinions de- ing the hubs from a SF network, for which P becomes creases exponentially with time. The average number of finite [19 hanges for each node is about 1.5 with standard deviation In summary, we propose a nonconsensus opinion model 0.5 for the sf network with a= 2.5 which allows the stable coexistence of minority and ma- [15] The critical behavior of the NCO model disappears when jority opinions. In the stable state, nodes holding the same applied to networks with a high average degree, because opinion demonstrate a phase transition from small clusters when the average node degree is large, an individual to large spanning clusters when the concentration of that opinion becomes irrelevant and the majority-voter model opinion increases Our simulations suggest that the phase 3] becomes valid. Thus, the minority opinion will not be transition belongs to the same universality class as invasion able to survive, and the system will eventually reach percolation, which is physically reasonable because, due to consensus, unless one introduces a large weight on the the clustering ("community support")of agents holding nodes own opinion the same opinion, stable clusters cannot be invaded by [16] Other 2D lattices also show critical beha for TR and f s 0.506 for SQ other opinion(similar to incompressible fluids). Thus, an [17] Fractals and Disordered Systems, edited by A. Bunde and opinion model can be mapped to a known physics perco S Havlin(Springer, New York, 1996) lation problem [18] High degree nodes join the majority opinion, since their We thank the ONR, EU project Epiwork, and the Israel wn opinion becomes negligible and they have mor Science Foundation for financial support. We thank L A neighbors holding the majority opinion than those holding Braunstein, S V. Buldyrev, and X.C. Yang for helpful [19] D.S. Callaway et al., Phys. Rev. Lett. 85, 5468(2000) R. Cohen et aL., ibid. 86, 3682(2001) [20] Note that for NCO in a 2D regular lattice, some nodes changing their states with period 2. Since they [I C. Castellano et al., Rev. Mod. Phys. 81, 591 (2009): resent a very small fraction of the system, we neglect S Galam, Europhys. Lett. 70, 705(2005) 018701-4
for the HEP network ( 2:9). Note that hkðfÞi shows a significant increase at f ¼ 0:5, which demonstrates that the minority opinion nodes have a significantly lower degree compared to those of the majority opinion. We also show Pfðk0 > kÞ for different values of f in Fig. 4(b), which provides further evidence for lower degrees and the appearance of significantly fewer high degree nodes in the minority opinion [18]. This explains why, for SF networks with ¼ 2:5 and for HEP, we observe a phase transition in the NCO model. The minority opinion nodes do not include high degree nodes which are responsible for the formation of large spanning cluster for pc ! 0 in regular percolation. This process is analogous to removing the hubs from a SF network, for which pc becomes finite [19]. In summary, we propose a nonconsensus opinion model, which allows the stable coexistence of minority and majority opinions. In the stable state, nodes holding the same opinion demonstrate a phase transition from small clusters to large spanning clusters when the concentration of that opinion increases. Our simulations suggest that the phase transition belongs to the same universality class as invasion percolation, which is physically reasonable because, due to the clustering (‘‘community support’’) of agents holding the same opinion, stable clusters cannot be invaded by the other opinion (similar to incompressible fluids). Thus, an opinion model can be mapped to a known physics percolation problem. We thank the ONR, EU project Epiwork, and the Israel Science Foundation for financial support. We thank L. A. Braunstein, S. V. Buldyrev, and X. C. Yang for helpful discussions. [1] C. Castellano et al., Rev. Mod. Phys. 81, 591 (2009); S. Galam, Europhys. Lett. 70, 705 (2005). [2] K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C 11, 1157 (2000). [3] T. M. Liggett, Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes (Springer, Berlin, 1999); R. Lambiotte and S. Redner, Europhys. Lett. 82, 18 007 (2008). [4] S. Galam, Eur. Phys. J. B 25, 403 (2002); P. L. Krapivsky and S. Redner, Phys. Rev. Lett. 90, 238701 (2003). [5] B. Latane´, Am. Psychol. 36, 343 (1981); A. Nowak et al., Psychol. Rev. 97, 362 (1990). [6] S. Schwarzer et al., Phys. Rev. E 59, 3262 (1999). [7] Invasion percolation describes the evolution of the front between two immiscible liquids in a random medium when one liquid is displaced by injection of the other. If the invasion into the trapped liquid is forbidden because of an incompressibility constraint, the model is regarded as invasion percolation with trapping. [8] P. S. Dodds and D. J. Watts, Phys. Rev. Lett. 92, 218701 (2004). [9] We find that the value of p does not hinder the appearance of critical behavior; only f changes with p. When a tie of two opinions occurs in Ai, nonzero p will make node i finally take its opposite state, which will break the tie and lead to a local frozen configuration. Thus, the formation of stable clusters and phase transition can be observed for nonzero p. [10] A.-L. Baraba´si and R. Albert, Science 286, 509 (1999). [11] V. Batagelj and A. Mrvar, Pajek data sets, http://vlado. fmf.uni-lj.si/pub/networks/data/. [12] R. Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). [13] We use the SF network with ¼ 2:5. There is no random percolation threshold (pc ¼ 0) for the SF networks with k) f=0.50 f=0.450 SF f 1.5 (b) FIG. 4 (color online). (a) The average degree hkðfÞi of nodes in the stable state as a function of f for a SF network with ¼ 2:5 and N ¼ 105 and the HEP network. It is seen that, for f 0:5, since the high degree nodes join the majority opinion [18]. (b) The cumulative degree distribution Pfðk0 > kÞ of nodes in the stable state for a SF network with ¼ 2:5 and N ¼ 105. Pfðk0 > kÞ for f ¼ 0:450 and 0.5 are plotted. Notice that f 0:450. The absence of high degree nodes in a stable state for f < 0:5 is again confirmed. PRL 103, 018701 (2009) PHYSICAL REVIEW LETTERS week ending 3 JULY 2009 018701-4