The critical behavior of Hegselmann-Krause opinion model with smart agents
aa r X i v : . [ phy s i c s . s o c - ph ] F e b The critical behavior of Hegselmann-Krause opinion model withsmart agents
Yueying Zhu , ∗ , Jian Jiang , and Wei Li , Research Center of Nonlinear Science, College of Mathematics and Computer Science,Wuhan Textile University, 430200 Wuhan, China Complexity Science Center & Institute of Particle Physics,Central China Normal University, 430079 Wuhan, China Max-Planck Institute for Mathematics in the Sciences,Inselst. 22, 04103 Leipzig, Germany ∗ Correspondence author: [email protected]
Abstract
The Hegselmann-Krause (HK) model allows one to characterize the continuous change of agents’ opinionswith the bounded confidence threshold ε . To consider the heterogeneity of agents in characteristics, we study theHK model on homogeneous and heterogeneous networks by introducing a kind of smart agent. Different fromthe averaging rule in opinion update of HK model, smart agents will consider, in updating their opinions, theenvironmental influence following the fact that an agent’s behavior is often coupled with environmental changes.The environment is characterized by a parameter that represents the biased resource allocation between differentcliques. We focus on the critical behavior of the underlying system. A phase transition point separating a com-plete consensus from the coexistence of different opinions is identified, which occurs at a critical value ε c for thebounded confidence threshold. We state analytically that ε c can take only one of two possible values, depend-ing on the behavior of the average degree k a of a social graph, when agents are homogeneous in characteristics.Results also suggest that the phase transition point weakly depends on the network structure but is strongly corre-lated with the fraction of smart agents and the environmental parameter. We finally establish the finite size scalinglaw that stresses the role that the system size has in the underlying opinion dynamics. Meanwhile, introducingsmart agents does not change the functional dependence between the time to reach a complete consensus and thesystem size. However, it can drive a complete consensus to be reached faster, for homogeneous networks that arefar from the mean field limit. Keywords : opinion dynamics, convergence threshold, critical behavior, finite size effect
The dynamics regarding human social behavior has been widely studied in many disciplines, as well as in their in-tersections [1–3]. In this field, one hot topic is the opinion dynamics that has been largely studied in the literature,both analytically and by means of large-scale numerical simulations [4–6]. In the field of opinion research, peo-ple have been trying to find out how a common opinion or decision is formed on top of a given population [7, 8].Historically, the classical modeling of opinion evolution assumes homogeneously mixed populations, which im-plies that any two individuals in the population are connected at the same probability [9, 10]. However, such an pproach leads to fluctuations in fitting to real systems, due to the complexity in individual’s characteristics andthe stochastic nature in communications among people [11]. With the advent of modern network science, wehave witnessed significant influence of heterogeneously mixed patterns in real systems on the critical behavior ofopinion dynamics [12, 13].A set of agents are considered with each one assigned a special opinion from a real space. The opiniondynamics is then formed by the fact that an arbitrary agent may update its opinion when it is aware of theopinions of others. Discrete opinion space is first focused by researchers especially those in physics because ofits simplicity to analytic work [14–17]. It represents a reasonable description in several instances but fails toexplain how an individual varies smoothly its attitude from one extreme of the range of possible choices to theother. The political orientation of an individual, as an example, is not limited to the binary choices of left or rightbut may take any one possible position in between. Hence, several mathematical models of continuous opiniondynamics are then designed to reveal the underlying mechanism that determines the smooth change in humanpsychology [18, 19].As a typical example of continuous opinion dynamics, HK model was first proposed by Hegselmann andKrause in 2002 [20]. It introduces a realistic aspect of human communications that a discussion between twopeople usually happens only if their opinions differ from each other no larger than a certain parameter ε . Thisparameter is named the bounded confidence threshold. In HK model, an agent i takes a real value between 0 and1 as its initial opinion. The update rule for its opinion at time t is defined mathematically by o i ( t + ) = | N i ( t ) | ∑ j ∈ N i ( t ) o j ( t ) , (1)where | · | for a finite set denotes the number of elements. N i ( t ) collects compatible neighbors of agent i , saying N i ( t ) = { j , j ∈ N || o i ( t ) − o j ( t ) | ≤ ε and a ( i , j ) = } , (2)where a ( i , j ) is the element of the adjacency matrix A that mathematically formulates the network of agents andtheir interpersonal relations. a ( i , j ) = i to agent j and a ( i , j ) = endencies to change or keep their own opinions. In general, human beings are recognised as "higher animal"because of self-awareness and the freedom from nature’s determinism that allows us to choose, whether forgood or ill. Some people, however, may behave irrationally in facing some activities. This maybe becausethe underlying activity does not interest them or they do not care about their personal gains. Based on thisphenomenon, we consider in our system two kinds of agents: smart ones and general ones. In the opinionevolution, smart agents will instinctively learn from their friends who have earned the most scores from thegambling activity, because of their primary intend of wining money and/or material goods. General agents willfollow the averaging rule of HK model in updating their opinions, as illustrated in Eq. (1). With the model, weattempt to describe the formation of a common opinion in a real-world system of agent’s heterogeneity and thecoupling between agents’ behaviors and environmental changes, from both numerical and theoretical viewpoints.Furthermore, finite size effect analysis is also conducted to help understand the critical behavior of the underlyingsystem.The remainder of the paper is organized as follows. In the second section we introduce smart agents toHK model and define the opinion update rule for heterogeneous agents. In section 3, we state the consensusthreshold ε c of the underlying opinion dynamics, from both analytic and numerical points of view. Section 4focuses on the critical behavior of the underlying system. We establish the scaling law of order parameter, andexplain analytically the influence of environmental and heterogeneous parameters. Some conclusions are drawnin section 5. Opinions of agents are assigned initially by real values between -1 and 1. Agents in our system are then dividedinto two cliques based on the sign of their opinions: G + (for positive opinions) and G − (for negative opinions).This setting, similar to the El Farol game of agents having two possible choices, i.e. whether to go or not to go toa bar [40], has certain practical significance.The coupling mechanism between the evolution of agents’ opinions and environmental changes is designedby a virtual gambling game. The environment of the underlying system is quantified by the resource allocationamong agents. Considering the biased resource allocation in a real-world system, we introduce a parameter γ toquantify the resource discrepancy between two cliques: γ = R + / R − , where R + labels the resource allocated to G + and R − to G − . The value of γ is restricted to the closed region [0, 1] as the system’s behavior is symmetricwith respect to γ =
1. In our system, we can regard the absolute value of an agent’s opinion as the cost or thedegree of attention that it pays for the activity of gambling. Moreover, an agent who pays more cost or attentionwill share more resource (or information) according to the general rule of survival: the more you contribute,the more you get [41–43]. On this basis, the resource r i ( t ) that agent i can share, at time t , from its clique isproportional to the absolute value of its current opinion o i ( t ) : r i ( t ) = o i ( t ) Ω + ( t ) · R + if o i ( t ) > o i ( t ) Ω − ( t ) · R − if o i ( t ) ≤ , (3)where Ω + ( t ) = ∑ j ∈ G + o j ( t ) , Ω − ( t ) = ∑ j ∈ G − o j ( t ) . (4)The total resource R (= R + + R − ) in the system is supposed to be conserved (unchanged with time evolution andsystem size). For simplicity, R is set to be 1000 in numerical simulations, but its value does not influence thepresented results.Generally speaking, the primary intend of people joining in a gambling game is to gain money and/or materialgoods [44]. In our model, smart agents have the ability to gain more scores by learning from their friends.Parameter p is used to control the fraction of general agents who follow the averaging rule of HK model inupdating their opinions. The fraction of smart agents in the system then is ( − p ) . p can take any one real valuebetween 0 and 1. In the system, opinions determine the resource distribution among agents and further control thescores that agents gain from the gambling activity. Following the primary intend of smart agents to gain moneyand/or material goods, a smart agent will update its opinion by the average one of compatible friends who have ained the most scores. If its scores are the highest compared to its compatible friends, it will keep the currentopinion. A rewards and punishment mechanism is then designed based on the resource that an agent shares fromthe system. An agent is regarded as a winner and to be rewarded with fixed scores f if its resource is no less thanthe global average A r ( = R / N ). Otherwise, it is labeled as a loser and to be punished with losing f scores. Theupdate rule for the number of cumulative scores of agent i at time t reads f i ( t + ) = f i ( t ) + f if r i ( t ) ≥ A r f i ( t ) − f if r i ( t ) < A r . (5) f is constant for each agent and set to be 5 in our numerical simulations, but its value does not influence theresults that we presented in this paper.The update rule for the opinion of smart agent i is given mathematically by o i ( t + ) = δ f i ( t ) m i ( t ) o i ( t ) + ( − δ f i ( t ) m i ( t ) ) | M i ( t ) | ∑ j ∈ M i ( t ) o j ( t ) , (6)where δ xy is the Kronecker delta function with δ xy = x = y and δ xy = m i ( t ) = max ( f i ( t ) , { f j ( t ) , j ∈ N i ( t ) } ) , (7) M i ( t ) = { j , j ∈ N i ( t ) | f j ( t ) = m i ( t ) } , (8)where max () is a function by returning the maximum value in a set of real numbers. N i ( t ) is defined in Eq. (2).For a general agent i , specifically, its opinion will be updated by the averaging rule of HK model, as illustrated inEq. (1).It is of interest to identify the critical behavior of the extended HK model, and to explore how smart agents,biased resource allocation, as well as graph topology influence the behavior of the underlying system. Basically,we carry out simulations of the model on different network topologies and determine in each case the value ofthe consensus threshold and the critical behavior. We analyze three different types of networks: (1) a completenetwork where any two agents can talk to each other (the mean field limit); (2) a regular network of the nearest-connections (an example of homogeneous networks); (3) a Barabási-Albert (BA) scale-free network (an exampleof heterogeneous networks). In Monte Carlo simulations, we chose to update the opinions of agents in orderedsweeps over the population. The system is considered to reach a stable state if any opinion changes by less than10 − . Specifically, a complete consensus is defined whenever the opinion distance between any two agents is lessthan 0.01. Simulated results illustrated hereafter are all obtained by averaging over 1000 independent realizations. A bifurcation diagram has been presented in previous work for HK model with uniform initial density in the opin-ion space [18]. It states that the dynamics of HK model leads to the pattern of stationary states with the number offinal opinion clusters decreasing with the increase of ε . Thus, it is of interest to identify the consensus threshold ε c of the bounded confidence parameter ε , that specifies a phase transition from disordered state (coexistence ofdifferent opinions) to an ordered one (a complete consensus). In particular, for ε above consensus threshold ε c ,a group opinion is certainly formed. Most investigations regarding the opinion formation of HK model often setthe opinion space to be [ , ] . Fortunato, on this basis, has claimed based on numerical results that the threshold ε c for a complete consensus can only take one of two possible values, depending on the behavior of the averagedegree k a of the underlying social graph, when the number of agents N approaches infinity [45]. If k a stays finitein the limit of large N , ε c = .
5. Instead, if k a → ∞ when N → ∞ , ε c = .
2. It is natural to extend the statementto HK model of initial opinion space being [-1, 1] with ε c = k a and ε c = . k a diverges. This isbecause the consensus threshold ε c linearly depends on the maximal opinion distance g o at initial time, see Fig.1 ( a ) . An analytic explanation for the underlying results is presented in Appendix A.We first discuss how smart agents impact the consensus threshold ε c . Take the case of p = t = k a =N-1 c g o Simulated result Analytic result k a =4 c g o N=100 (regular) N=1000 (regular) N=100 (BA) N=1000 (BA) ( a ) ε c changes with g o for p = k a =N-1 c g o Simulated result Analytic result k a =4 c g o N=100 (regular) N=1000 (regular) N=100 (BA) N=1000 (BA) ( b ) ε c changes with g o for p = Figure 1: (color online) Analytic values are compared to numerical ones for the linear correlation between consensus threshold ε c and theinitial opinion change distance g o for HK model ( a ) and the case of all agents being smart ( b ) with γ =
0. In the left panels, N = he opinions of others. Following this statement, the opinion at initial time is natural to be obtained, with whichan agent can gain scores from the beginning of gambling, | o i | ≥ ( + γ ) O γ o i > , ( + γ ) | O | o i ≤ , (9)where agents’ opinions are supposed to be uniformly distributed, at t =
0, in the real range [ O , O ] with O = − O . Considering a particular case where the resource is concentrated to the clique of agents holding negativeopinions ( γ = R [ O , O ] and R ( O , O ] . Agents with opinions being interval R R O and O should be no larger than the previously specified confidencethreshold ε to form a group opinion, that is O − O ≤ ε . (10)Recalling the condition O = − O , we then get the functional dependence between the initial opinion gap g o (= O − O ) and the consensus threshold ε c : ε c = g o , (11)which agrees completely with numerical simulations of the agent-based model, see Fig. 1 ( b ) . For sparse net-works, this linear correlation slightly depends upon the underlying system size, but is independent of the networkstructure. This differs from the behavior of HK model where the correlation between ε c and g o depends neitheron the system size nor on the network structure, even when the network is very sparse. Furthermore, compared toHK model, the introduction of smart agents can strengthen, in the mean field limit, the linear correlation between ε c and g o .Interestingly, the opinion dynamics of smart agents leads to the pattern of stationary states with the number offinal opinion clusters decreasing if ε increases, similar to that of HK model. The consensus threshold ε c can alsoonly take two possible values when the system size N tends to be infinity. ε c = k a is finitein the limit of large N . If instead k a → ∞ when N → ∞ , ε c = .
25. This is verified by the functional relationshipbetween ε c and k a (Ref. [46]): ε c = + k a , if k a ≥ ;2 , if k a < . (12)However, for a social system of finite number of agents, the consensus threshold ε c will decrease with theincrease of k a in both cases of p = p =
0, as presented in Fig. 2. Homogeneous and heterogeneousnetwork structures are both considered here. In HK model, the reliance of ε c on k a is almost independent of thenetwork structure, as stated previously. When all agents are smart, however, ε c of BA scale-free (heterogeneous)networks is larger than that of regular (homogeneous) ones, for each specified k a . This suggests that a lowerconfidence threshold could guarantee the formation of a group opinion in the homogeneous network, comparedto the heterogeneous one. Simulation results will gradually agree with analytic ones (Eq. (12)) that are obtainedby the mean field approximation. Compared to the bifurcation diagram, the structure of clusters formed by agents holding the same opinion providesmore detailed information in uncovering the behavior of opinion fusion [47]. In our model, a cluster is formed byagents whose opinions differ from each other less than 0.01. Denote the number of clusters at stationary state by M . At t =
0, opinions of agents are randomly assigned by real values between -1 and 1, making the maximum of M to be M (=2/0.01=200).Analytically, the underlying system is more disordered when M / M is closer to 1 and more ordered if M / M is closer to 0. Numerical results for the special case of mean field limit ( k a = N −
1) are exhibited in Fig. 3( a ).
20 40 60 800.40.60.81.0 p=1.0 c k a BA network Regular network p=0.0 c k a BA network Regular network Analytic result
Figure 2: (color online) The consensus threshold ε c as a function of the average degree k a for BA scale-free and regular networks with γ = N = k a is quite small, ε c = p = ε c = p =
0. If instead k a is large enough, ε c = . p = ε c = . p =
0. Each data point is averaged over 1000 realizations and the curve without markers in the right panel is obtained by the mean-fieldapproximation.
In the presence of smart agents, a continuous phase transition point is identified that separates an ordered statefrom the disordered one. The value of confidence threshold at the transition point is independent of the systemsize, but weakly relies on the fraction of smart agents. In HK model ( p = ε decrease with the increase of system size. Whenthe system size N approaches infinity, the phase transition point tends to the analytic value 0.4. This is verified bythe fluctuation δ M of M , see Fig. 3( b ), where a peak happens at the transition point. More results are displayed inAppendix B where the influence of resource allocation parameter γ and network structure is considered. Opinion clustering analysis allows one to explain the transition from a coexistent state of many different opinionsto that of a few of ones. Actually, it still has some difficulties in determining with precision the transition pointseparating a complete consensus from the coexistence of different opinions. We label the normalized size ofthe largest opinion cluster as m L . An order parameter, denoted by P m L = , is then natural to be introduced forunderstanding further the critical behavior of the underlying system [48]. It is defined as the probability to havecomplete consensus in different configurations. We first study P m L = as a function of ε in the special case of allagents being smart ( p = P m L = is calculated as the fraction of samples with all agentssharing the same opinion over 1000 independent configurations. The underlying results are illustrated in Fig. 4.For spare networks, the complete consensus comes to be generated only when the confidence threshold ε is closeto 2, as already discussed in section 3. The curve corresponding to larger average degree k a is definitely abovethe curve relative to smaller k a , which suggests that when k a diverges, P m L = will probably attain the value 1 at afixed position. A discontinuous change happening to the curve for complete network ( k a = N −
1) indicates thisposition is ε = .
25. This states again the consensus threshold ε c = k a , and ε c = . k a , when all agents in the system are smart.Furthermore, homogeneous and heterogeneous networks of agents and their interpersonal relations drive somekinds of difference in the distribution of P m L = . For quite spare networks: k a = P m L = is almost the same forboth regular and BA scale-free networks, which is less than that on star network where complete consensus beginsto be generated at ε = .
25. The deviation of results on regular and BA scale-free networks gradually appearswith the increase of average degree. This implies a homogeneous network is more benefit than a heterogeneousone for a complete consensus when k a is large. Interestingly, the normalized size m L of the largest cluster linearlydepends upon the confidence threshold ε until the consensus state is reached, in the mean field limit. For networksaway from the mean field limit, m L follows a nonlinear function of ε . .0 0.5 1.0 1.5 2.00.00.20.40.60.81.0 p=0 N=100 N=1000 N=10000 M / M M p=0.5 M / M N=100 N=1000 N=10000 M p=1.0 M / M N=100 N=1000 N=10000 M p=0 M / M N=100 N=1000 N=10000 p=0.5 M / M N=100 N=1000 N=10000 p=1.0 M / M N=100 N=1000 N=10000
Figure 3: (color online) Distribution of normalized number of opinion clusters M / M and its standard variance δ M / M along the axis ofconfidence threshold ε for the pattern of stationary states. Results are obtained by averaging over 1000 realizations with γ = k a = N − k=2 k=4 k=8 k=16 k=N-1 P m L = k=2 k=4 k=8 k=16 k=N-1 m L ( a ) P m L = and m L vary with ε for BA scare-free networks. P m L = k a =2 Regular BA Star P m L = k a =16 Regular BA ( b ) The dependence of P m L = with ε for homogeneous and heterogeneous networks. Figure 4: (color online) The order parameter P m L = as a function of the confidence threshold ε for homogeneous and heterogeneous networksof different values of k a with γ = p = N = .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 k a =2 k a =4 k a =8 k a =16 k a =N-1 P m L = p k a =2 k a =4 k a =8 k a =16 k a =N-1 m L p ( a ) P m L = and m L distribute with p on BA scale-free networks. k a =2 P m L = p Regular BA Star P m L = k a =16 p Regular BA ( b ) P m L = as a function of p for homogeneous and heterogeneous networks. Figure 5: (color online) P m L = as a function of p for homogeneous and heterogeneous networks of different values of k a with γ = ε = N = .2.1 The influence of parameter p As stated previously, the presence of smart agents will affect the position to reach a complete consensus. Partic-ulary, this effect is discussed further by setting γ = ε =
2. Results in Fig. 5 ( a ) show that, for quite sparsenetworks, a complete consensus happens only at two boundary conditions: p = p =
1. For a completenetwork, however, a consensus appears in almost the whole space of parameter p . The underlying theoreticalanalysis is attached in the following. Furthermore, there is a valley in the distribution of m L at p ≈ .
7, corre-sponding to the most disordered state of opinion dynamics. This phenomenon, however, will gradually disappearwith the increase of k a .Analogously, the influence of network topology is also discussed on the dependence of P m L = with parameter p , see Fig. 5 ( b ) . The results on regular and BA scale-free networks are of negligible deviations for each specifiedaverage degree, which differ from the distribution behavior on a star network. The star network is known for itstopology of special structure with a hub node connecting to the rest N − P m L = acts as a linear function of parameter p on a star network: P m L = = − p . (13)On a complete graph with γ = ε =
2, general agents update gradually their opinions by global averagevalues in the opinion space, and smart ones adapt to the average opinions of their compatible neighbors whogained scores at t =
0. Specifically, an absorbing state of all agents’ opinions being 0 is certain to be reachedat t = p = < p < t ( ≥
4) can be denoted by [ − , ϕ ( t )] with ϕ ( t ) following the formula (as shown inAppendix C): ϕ ( t ) = − ( − p ) t + " t − ∑ m = t − m ( + p ) m + ( + p )( + p ) t − , (14)which can be simplified as ϕ ( t ) = t + − − F ( p ) − t + . (15) F ( p ) is a polynomial function of p : F ( p ) = t − ∑ m = c m p m , (16)with coefficients { c m } following c t − = , and t − ∑ m = c m = t + − . (17)Recalling the previous definition, a complete consensus is reached at time t if and only if ϕ ( t ) + < . , (18)which yields ∆ ( t , p ) = + F ( p ) t + < . . (19)In Fig. 6 it is presented that the real range of p to satisfy Eq. (19) will be enlarged with the increase of t . Thissuggests that a complete consensus is finally to be reached for all possible values of parameter p , when the socialgraph is completely connected and ε = , γ =
0. The time T c to reach a complete consensus is well confirmed bysimulation results, especially when the system size is large enough. γ Specifically, resource allocation parameter γ is set to be 0 in above discussion. However, this is hardly to bereached in a real-world system as it lacks some kind of fairness. To support for the fairness in resource allocationin practice, we now consider the fact that the resource held by the closed system is shared initially by both two .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 ( p ,t ) p t=6 t=10 t=20 t=50 (p,t)=0.01 T c p N=100 N=1000 N=10000 Analytic value
Figure 6: (color online) Analytic values for the influence of parameter p on the formation of a complete consensus in the mean field limit( k a = N −
1) with γ = ε =
2. Left panel: ∆ ( t , p ) acts as a function of p for different values of t . Right panel: T c (time to reach a completeconsensus) acts as a function of p , in good agreement with numerical results especially when the system size is large enough. cliques. Parameter γ controls the biased resource allocation between two cliques in the system, as defined insection 2.Next, we discuss the role of parameter γ acted in the generation of a complete consensus. For simplicity,we set k a = N − ε =
2, and suppose that all agents are smart. Following the gambling game, agents willupdate their opinions by learning from their compatible neighbors who have gained the most scores. Recallingthe definition of r i (Eq. (3)) and the condition for agent i to win: r i ≥ R / N , we can specify the opinion of winnersin both cliques at t = o i ( ) ≥ + γ γ , i ∈ G + ≤ − + γ , i ∈ G − . (20)If winners exist in both cliques at t =
0, the consensus state will be absent. To ensure its appearance, we shouldhave 1 + γ γ > , (21)yielding γ < / t = [ − , − + γ ] . At next timestep, agents with initial opinions in [ − + γ , ] will then update their opinions by the average opinion in [ − , − + γ ] as opinions are uniformly distributed at t =
0. The range of opinion for wining the gambling will be changed asmore agents attracted to share the resource of clique G − . With time evolution, the maximal opinion, denoted by φ , will be gradually close to -1. φ is a function of both γ and time t , expressed mathematically by φ ( t ) = − [ ∑ tm = m ( + γ ) t − m + +( + γ )( + γ ) t − ] t + , t ≥ φ ( t ) = − ( + γ )( + γ ) , t = φ ( t ) = − + γ , t = . (22)At any special time t , agents’ opinions are located in the real range [ − , φ ( t )] . A complete consensus will bereached only when the distance of opinions satisfies the inequality:1 + φ ( t ) < . . (23)Fixing t and recalling two general constraints: γ < and φ ( t ) > −
1, we then get the range of γ to have a complete -4 -3 -2 -1 P m L = N=100 N=1000 N=10000 Analytic value -4 -3 -2 -1 t max T c N=100 N=1000 N=10000 Analytic value
Figure 7: (color online) P m L = and T c (time to reach a complete consensus) as functions of the resource allocation parameter γ for a completenetwork ( k a = N −
1) with p = ε =
2. Red lines are obtained analytically by the mean filed theory. T c gets t max (the maximal time set topropose Monte Carlo simulations), representing the absence of a complete consensus. Numerical results are averaged over 1000 realizations. consensus: P m L = = ⇐ . ≤ γ ≤ . , T c = . ≤ γ ≤ . , T c = . ≤ γ ≤ . , T c = . ≤ γ ≤ . , T c = . ≤ γ ≤ . , T c = ≤ γ < . , T c =
8; (24)For the rest values of γ , a complete consensus is absent. This suggests that the order parameter P m L = does notchange continuously in the axis of γ but acts as a rectangular pulse. The theoretical analysis is valid only whenthe system size is large enough, as illustrated in Fig. 7. In the study of the critical behavior of a physical system, it often needs to predict the thermodynamical propertiesnear a critical point. The well developed techniques of finite size scaling allow one to extrapolate the results to N → ∞ [48]. In practice, however, agents considered can never be that large. And the results in thermodynamiclimit may vary with respect to those of finite-size systems. In the following, we will consider the finite size effectof the distribution behavior of order parameter P m L = .We first set γ to 0. Results of HK model are compared to that of the case with all agents being smart. Forboth cases, it is natural to propose the prediction that a complete consensus is certain to be reached only if theconfidence threshold is above a critical value ε c : P m L = = ε > ε c ε < ε c . (25)This analysis, however, is valid only in the thermodynamic limit. In fact, for a population of small size, thecondition ε > ε c does not automatically ensure a complete consensus. There are finite size fluctuations such thatthe actual result of the repeated discussion process is not well established if the difference between the fixed valueof confidence parameter ε and the consensus threshold ε c is of order | ε − ε c | ∼ N − α . This rounding off of thesharp transition can be summarized in the following finite-size scaling law: P m L = ( ε , N ) = P m L = (( ε − ε c ) N α ) , (26) P m L = ( - c )N p=1k a =4 P m L = N=100 N=300 N=500 N=1000 N=10000 -10 -8 -6 -4 -2 00.00.20.40.60.81.0 P m L = ( - c )N k a =4p=0 P m L = N=100 N=300 N=500 N=1000 N=10000 ( a ) Simulated results on a BA scare-free network with k a = -4 -2 0 2 40.00.20.40.60.81.0 P m L = ( - c )N k a =N-1p=1 P m L = N=100 N=300 N=500 N=1000 N=10000 -4 -2 0 2 40.00.20.40.60.81.0 P m L = ( c )N k a =N-1p=0 P m L = N=100 N=300 N=500 N=1000 N=10000 ( b ) Simulated results on a complete network ( k a = N − Figure 8:
Inset panels show P m L = as a function of ε and N for HK model ( p =
1) and the case of all agents being smart ( p =
0) on a BAscale-free network with k a = k a = N − N = N = (the transition sharpens with increasing system size). The main plots show the scaling law P m L = ( ε , N ) = P m L = (( ε − ε c ) N α ) with α = . k a =
4, in which ε c = .
02 for p = ε c = p =
0, and α = . k a = N −
1, in which ε c = . p = ε c = . p =
0. Each data point is obtained by averaging over 1000 realizations with γ = a result well confirmed by numerical simulations with α = . k a = α = . k a = N −
1, as presentedin Fig. 8. The statement is satisfied for both of the two cases: p = p =
0. This implies that the introductionof smart agents does not change the scaling law of HK model, but has statistically significant impact on theconsensus threshold ε c . Numerical results also indicate that ε c = p = ε c = p = k a is finite in the limit of large N . If instead k a → ∞ when N → ∞ , ε c = . p = ε c = .
25 for p =
0. These are in good agreement with analytic results illustrated in section 3.More dramatic finite size effects appear if we go beyond the boundary cases ( p = p =
1) and considerthe coexistence of smart and general agents (0 < p < ε = γ =
0, the first feature that appearsin the numerical simulations is the smooth transition between two different extreme states as a function of theproportion p of general agents. This is evident in the inset of the left panel in Fig. 9, where we plot the orderparameter P m L = as a function of p . The transition point clearly decreases with population size N , when theaverage degree k a is small. A finite size scaling analysis shows that the data can be well fitted by the formula: P m L = ( p , N ) = P m L = (cid:16) pN β (cid:17) , (27)with β = . k a = β = .
15 for k a =
16, and β ≈ k a = N −
1. An strict statistical mechanics analysisbased on the thermodynamic limit N → ∞ would conclude that the actual critical point is p ≈ =
1) forfinite k a . However, for divergent k a , the system size has no statistically significant impact on the phase transitionbehavior in the axis of parameter p , as shown in the right panel of Fig. 9. It is implied that a complete consensusis certain to be reached at a finite time for almost all possible values of parameter p when the social graph iscompletely connected. This is in good agreement with our analytic discussion in section 4.2. -2 -1 pN P m L = k a =8 P m L = N=100 N=300 N=500 N=1000 N=10000 p -2 -1 k a =16 P m L = pN P m L = N=100 N=300 N=500 N=1000 N=10000 p -2 -1 k a =N-1 P m L = pN -0.005 P m L = N=100 N=300 N=500 N=1000 N=10000 p Figure 9: P m L = as a function of p for different values of N on BA scale-free networks with k a = k a =
16, and on a complete network( k a = N − N =
100 to N = . The main plots show the scaling law P m L = ( p , N ) = P m L = ( pN β ) with β = . k a = β = .
15 for k a =
16, and β = − .
005 for k a = N −
1. The insets show the unscaled results, indicating the transition point shiftingtoward p = N increases. It is possible to compute the time it takes the population to reach the complete consensus. Fixing the networkstructure to be scale-free, we first focus on the distribution of system convergence time T c in the axis of populationsize N . In numerical simulations, the program stops if no agent changes the opinion after an iteration. Ourcriterion is to check whether any opinion varies by less than 10 − after a sweep. It is shown in Fig. 10 that T c experiences a logarithmic growth in the presence of smart agents but keeps almost unchanged in HK model, withthe increase of population size: T c ∝ log N for p = T c ≈ C for p = , (28)where C is a constant. In the case of p =
0, the dependence of T c with system size N experiments a decrease withthe increase of average degree k a . Furthermore, smart agents can drive system converge faster than general onesin HK model when the population is finite and sparsely connected. If smart and general agents are coexisted inthe system, however, it is of more difficulty to get the system to a stationary state.According to the previous analysis of HK model, a generally accepted phenomenon is that the time to reacha complete consensus is bounded by N O ( N ) and conjectured to be polynomial [49]. This general phenomenon,however, is demonstrated only for regular lattices of different dimensions. To certificate again the universalityof this scaling behavior, numerical simulations are proposed on general regular graphs of different degrees. It isexhibited in Fig. 11 that T c scales as a power law function of N , which is valid not only for HK model ( p = p = T c ∝ N λ , (29)with λ ∼ . p = λ ∼ . p =
0. This says that a regularly connected population of smart agentswill take less time to get a complete consensus, compared to that of general agents in HK model.Different from regular networks, BA scale-free graphs will take longer time for the population to reach acomplete consensus for p = p =
1, as shown in Fig. 12. Actually, for BA scale-free networks, T c weakly depends on the average degree k a , especially for the case of p =
0. For regular networks, however, T c actsas a power law function of k a with exponent being about -1.9 for p = p =
0, respectively. We maydraw the conclusion that the dependence of T c on the average degree k a experiences a strong change for differentnetwork structures. In summary, we propose a continuous opinion model by introducing smart agents to HK model and consideringalso the coupling effect between human behaviors and environmental changes. It allows one to understand thepropagation of human ideas or attitudes in real-world systems. Social environment is represented by a biased p=0 T c N p=1 T c N ( a ) T c as a function of N for BA scale-free networks of k a = ε = γ =
0. Down trianglesrefer to k a = N − p=0 =1 =2p=1 =1 =2 T c N p=0 p=0.5 p=1 T c N ( b ) T c as a function of N for a BA scale-free network with k a = γ = ε = Figure 10: (color online) Plot of the system convergence time T c as a function of network size N in considering different values of k a , p and ε .Each data point is obtained by simulations averaged over 1000 realizations and the straight lines are obtained by logarithmic fittings. p=0 T c N p=1 T c N Figure 11: (color online) The time T c to reach a complete consensus as a power law function of system size N on regular networks of differentdegrees with ε = γ =
0. Square scatters correspond to k a =
4, circle ones to k a =
8, and up triangles to k a =
16, which have been averagedover 1000 realizations. Straight lines are power law fittings with slope being about 0.7 for p=0 and 1.7 for p=1. BA network p=0 p=1 T c k a Regular network p=0 p=1 T c k a Figure 12: (color online) The time T c to reach a complete consensus as a function of the average degree k a for BA scale-free and regularnetworks with ε = γ = N = resource allocation among people. We identify the influence of smart agents, resource allocation parameter andnetwork structure on the critical behavior of the underlying system.(1) The consensus threshold ε c is first discussed for a boundary case of all agents being smart. Both theoreticaland numerical results claim that ε c can only take two possible values, depending on the behavior of the averagedegree k a of the social graph, when the system size N approaches infinity. ε c = k a is finite in the limit of large N , and ε c = .
25 if instead k a → ∞ when N → ∞ .(2) An order parameter P m L = is introduced, which is defined as the average, over realizations of the discussedprocess, of the complete consensus. There is a clear change of behavior in the sense that it is possible to identify atransition point separating a complete consensus from the coexistence of different opinions. The transition pointoccurs at a pseudo-critical value ε c for the bounded confidence threshold. We certify from an analytic viewpointthat the generation of a complete consensus strongly depends on the fraction of smart agents and the resourceallocation parameter, but it is weakly connected with the topological structure of the network.(3) Introducing smart agents does not change the finite size scaling law of HK model with the order parameterscaling as: P m L = ∼ ( ε − ε c ) N α , but has statistically significant impact on the consensus threshold ε c . The criticalbehavior depends on the system size in such a way that the phase transition tends to be no-continuous when thesystem size N increases to infinity. While in the axis of parameter p (the fraction of general agents), a finitesize scaling analysis shows that the phase transition can be summarized by the scaling law: P m L = ∼ pN β . Theexponent β decreases with the average degree and gets close to 0 when the network is completely connected.This indicates the appearance of a complete consensus for almost all possible values of p . Meanwhile, smartagents do not change the power law functional relationship between system convergence time and the systemsize on regular networks. They can drive, however, the complete consensus to be reached faster than generalagents in HK model when the network structure is homogeneous and far from the mean field limit. In contrast, aheterogeneously structured network will make smart agents take more time to reach a complete consensus. Acknowledgments
We acknowledge financial support from the National Natural Science Foundation of China through the project11947063 and from the Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University,Wuhan 430079, China.
A The consensus threshold ε c of HK model Denote the gap between initial maximal opinion O and minimal one O by g o : g o = O − O . With the meanfield limit: k a = N −
1, a group opinion is formed at time t only when the difference between the maximal andminimal opinions at time ( t − ) is no larger than the bounded confidence threshold ε . or the simplicity in theoretical analysis, we first divide the initial opinion space into three adjacent intervals,as presented in Fig. A.1. With the limitation of bounded confidence, agents taking the minimal opinion O will beaffected by those ones taking opinions in interval R ( t = ) . Analogously, agents taking the maximal opinion O will be affected by those ones holding opinions in interval R ( t = ) . Stochastic distribution of initial opinionsdetermines the opinion space of the next step: O min ( t = ) = ( O + O + ε ) = O + ε . (A.1)and O max ( t = ) = ( O + O − ε ) = O − ε . (A.2)Agent i of initial opinion in R ( t = ) will keep its opinion as it will be affected by those ones of opinion in [ o i − ε , o i + ε ] , yielding o i ( t = ) = ( o i − ε + o i + ε ) = o i . (A.3)The average opinions in both intervals R ( t = ) and R ( t = ) are calculated as Ω R ( t = ) = ( O + ε + O + ε ) = O + ε , (A.4)and Ω R ( t = ) = ( O − ε + O − ε ) = O − ε . (A.5)An agent of opinion in R ( t = ) ( R ( t = ) ), however, will be persuaded by agents holding opinions in [ O , o i + ε ] ( [ o i − ε , O ] ). This guides, respectively, the average opinions in intervals R ( t = ) and R ( t = ) to be Ω R ( t = ) = ε Z O + ε O ( O + o i + ε ) d o i = O + ε , (A.6)and Ω R ( t = ) = ε Z O O − ε ( O + o i − ε ) d o i = O − ε . (A.7)The opinion space at t = R ( t = ) and R ( t = ) : O min ( t = ) = h ( O + ε ) ε + ( O + ε ) ε i ε ≈ O + . ε , (A.8) O max ( t = ) = h ( O − ε ) ε + ( O − ε ) ε i ε ≈ O − . ε . (A.9)They will be further influenced by opinions in intervals R ( t = ) and R ( t = ) .The average opinions in both R ( t = ) and R ( t = ) can be obtained as Ω R ( t = ) = ε Z O + ε O + ε ( O + ε ) ε + ( O + o i + ε )( o i − O ) ε + o i − O d o i = O + (
14 ln 53 + ) ε , (A.10) Ω R ( t = ) = ε Z O − ε O − ε ( O − ε ) ε + ( O + o i − ε )( O − o i ) O + ε − o i d o i = O − (
14 ln 53 + ) ε . (A.11)We then get respectively the average opinions in both R ( t = ) and R ( t = ) : Ω R ( t = ) = ε Z O + ε O + ε ( O + ε ) · O + ε − o i ε ε + ( O + ε + o i )( o i − O ) O + ε − o i ε ε + o i − O d o i = O − ( + ) ε , (A.12) Ω R ( t = ) = ε Z O − ε O − ε ( O − ε ) · o i + ε − O ε ε + ( O − ε + o i )( O − o i ) o i + ε − O ε ε + O − o i d o i = O + ( + ) ε ,. (A.13) igure A.1: (color online) Division of the opinion space at different time steps. Computing further the average opinions in intervals R ( t = ) and R ( t = ) , we have the opinion space at t = O min ( t = ) = h O + ( ln + ) ε i · ε + h O − ( + ln ) ε i · ε ε ≈ O + . ε , (A.14) O max ( t = ) = h O − ( ln + ) ε i · ε + h O + ( + ln ) ε i · ε ε ≈ O − . ε . (A.15)Following the above time sequence of the opinion space, it is natural to derive the general approximations: O min ( t + ) − O min ( t ) = . − . · t , (A.16) O max ( t + ) − O max ( t ) = − . + . · t . (A.17)To have a complete consensus, we should first make O min ( t + ) − O min ( t ) = O max ( t + ) − O max ( t ) = t = O min ( t = ) = O + . ε , O max ( t = ) = O − . ε . (A.18)These two boundary opinions are then updated for the coming time step as O min ( t = ) = ( O + . ε + O + O ) , (A.19) O max ( t = ) = ( O − . ε + O + O ) . (A.20)A complete consensus requires O max ( t = ) − O min ( t = ) ≤ ε , (A.21)which yields ε c = g , T c = . (A.22)a result well confirmed by the numerical simulations, as shown in Fig. A.2. B Distribution of the number of opinion clusters
It is of great interest to reveal how smart agents affect the opinion clustering in the dimension of confidencethreshold ε when the resource is concentrated to one clique: γ =
0. Three situations are discussed here: p = p = p = . k a is quite small, there will be no apparent opinion clustering if bothgeneral and smart agents are coexisted in the system. This is different from the case where only one kind ofagents presents. However, with the increase of k a , coexistence of both kinds of agents can also gradually driveapparent opinion clustering. And the fraction of smart agents has ignorable influence on the system’s clusteringbehavior in opinion for large k a only if there is smart agents. Above conclusions are applicable to the underlyingopinion dynamics on both homogeneous and heterogeneous network structures. This suggests again networkstructure has no statistically significant impact on the opinion clustering whether there is smart agent or not.More detailed information is presented in Fig. B.1. .0 0.2 0.4 0.6 0.8 1.005101520253035 =0.4, T c g o =2 T c N=100 N=300 N=1000 N=3000 =1, T c g o =5 T c N=100 N=300 N=1000 N=3000
Figure A.2: (color online) Simulation results for the distribution of the time T c to reach a complete consensus with k a = N − p = g o denotes the opinion distance at t = k a =4 M p=0.0 p=1.0 p=0.5 M k a =16 p=0.0 p=1.0 p=0.5 ( a ) Results for regular networks. M k a =4 p=0.0 p=1.0 p=0.5 k a =16 M p=0.0 p=1.0 p=0.5 ( b ) Results for BA scale-free networks.
Figure B.1: (color online) The number M of opinion clusters in the stable state as a function of the confidence threshold ε for regular and BAscale-free networks with γ = N = The influence of parameter p on the generation of a completeconsensus Particularly, we set γ = ε = k a = N − [ − , − ] . N p general agents are then update their opinions for the next time by the global averageopinion 0 as opinions are uniformly distributed at t =
0. Smart agents holding opinions outside the real range [ − , − ] will adapt to the average opinion of this range at next time. For those agents holding opinions insidethis range, they will keep their own opinions.The update rule for the opinion of agent i at time t can be expressed as o i ( t + ) = N ∑ Nj = o j ( t ) ; i ∈ N g , | N w ( t ) | ∑ j ∈ N w ( t ) o j ( t ) ; i ∈ N s , (C.1)where N g , N s , and N w ( t ) denote in sequence the set of general agents, the set of smart ones, and the set of winners: N g = { j , j ∈ N | j is a general agent } , N s = { j , j ∈ N | j is a smart agent } , N w ( t ) = { j , j ∈ N | r j ( t ) ≥ R / N } . (C.2) | ⋆ | returns the number of elements in set ⋆ . Uniform distribution of initial opinions yields N w ( ) = { j , j ∈ N | − ≤ o j ( ) ≤ − } , (C.3)and o i ( ) = i ∈ N g , − ; i ∈ N s and o i ( ) ∈ ( − , ] , o i ( ) ; i ∈ N s and o i ( ) ∈ [ − , − ] . (C.4)This specifies the opinion space: [ − , ] at t =
1. A winner’s opinion is limited to the real range [ − , − ( − p )] according to the condition o i ( )( − p ) N · − · RN ≥ RN (C.5)The opinion of an arbitrary agent i is further updated by o i ( ) = − ( − p ) , i ∈ N g ; − − p , i ∈ N s and o i ( ) ∈ ( − ( − p ) , − ] ; o i ( ) , i ∈ N s and o i ( ) ∈ [ − , − ( − p )] , (C.6)driving opinion space to be [ − , − ( − p ) at t =
2, and winner’s opinion space to be [ − , − ( + p )( − p ) ] . Forthe next time step, we have o i ( ) = − + p ( − p ) ; i ∈ N g , − +( + p )( − p ) ; i ∈ N s , o i ( ) ; i ∈ N s , (C.7) n which N s = { j , j ∈ N s | − ( + p )( − p ) < o j ( ) ≤ − ( − p ) } , (C.8) N s = { j , j ∈ N s | − ≤ o j ( ) ≤ − ( + p )( − p ) } . (C.9)This determines the opinion space to be [ − , − ( + p )( − p ) ] at t =
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