In search for the social hysteresis -- the symmetrical threshold model with independence on Watts-Strogatz graphs
IIn search for the social hysteresis – the symmetrical threshold model withindependence on Watts–Strogatz graphs
Bartłomiej Nowak ∗ and Katarzyna Sznajd-Weron † Department of Theoretical Physics, Wroclaw University of Science and Technology, Wrocław, Poland. (Dated: March 3, 2020)We study the homogeneous symmetrical threshold model with independence (noise) by pair ap-proximation and Monte Carlo simulations on Watts–Strogatz graphs. The model is a modifiedversion of the famous Granovetter’s threshold model: with probability p a voter acts independently,i.e. takes randomly one of two states ± ; with complementary probability − p , a voter takes a givenstate, if sufficiently large fraction (above a given threshold r ) of individuals in its neighborhood isin this state. We show that the character of the phase transition, induced by the noise parameter p , depends on the threshold r , as well as graph’s parameters. For r = 0 . only continuous phasetransitions are observed, whereas for r > . also discontinuous phase transitions are possible. Thehysteresis increases with the average degree (cid:104) k (cid:105) and the rewriting parameter β . On the other hand,the dependence between the width of the hysteresis and the threshold r is non-monotonic. The valueof r , for which the maximum hysteresis is observed, overlaps pretty well the size of the majorityused for the descriptive norms in order to manipulate people within social experiments. We putresults obtained within this paper in a broader picture and discuss them in the context of two othermodels of binary opinions, namely the majority-vote and the q -voter model. I. INTRODUCTION
It is not surprising that binary opinion models are ex-tremely popular among sociophysicists, given that the / -spin Ising model is not only one of the most popularmodels of theoretical physics, but also absolutely fun-damental for the theory of phase transitions. However,what is probably more surprising, the binary-choice mod-els have received considerably more theoretical attentionthan other choice models among social psychologists, so-ciologists and economists [1, 2]. One of the most impor-tant class of such models are the threshold models [3, 4]taking roots in the pioneering paper by Granovetter [5].The idea behind these models is extremely simple – anagent takes state (which can be interpreted as agree,adopt the innovation, join the riot, etc.) if sufficientlylarge fraction (above a given threshold) of people in hisneighborhood is in state . Originally model has beeninvestigated under the assumption of perfect mixing (all-to-all interactions). However, in 2002 Watts has adaptedGranovetter’s threshold model to a network framework[3]. We will use the same approach here and thereforeindividuals will be influenced only by the nearest neigh-bors, i.e. interactions will take place only between agentsthat are directly linked.There are two important differences between the Wattsthreshold model and other models of binary opinions,such as the Galam model [6–8], the majority-vote (MV)[9–20], the q -voter (qV) [21–28] or the threshold q -voter(TqV) model [29–32]. The first difference, often consid-ered as the most important, is the heterogeneity – eachagent is described by an individual threshold and there-fore some agents adopt a new state very easily, whereas ∗ [email protected] † [email protected] others don’t [3]. The second difference, that should beparticularly important for physicists, is the lack of theup-down symmetry. Once an agent adopts a state itcannot go back to the previous one. To make the thresh-old model comparable with other binary opinion models,we have introduced recently the homogeneous symmet-rical threshold model [33]. Here we will call this modelsimply symmetrical threshold (ST) model for brevity.Previously, we have studied two versions of the STmodel, each with a different type of nonconformity (an-ticonformity or independence) on the complete graph[33]. Therefore we were able to obtain exact analyti-cal results within the mean-field approach. Analogouslyas in other models of binary opinions, the introductionof nonconformity, whether in the form of anticonformityor in the form of independence, resulted in the appear-ance of the agreement–disagreement phase transitions.We have shown, that for the threshold r = 0 . , whichcorresponds to the majority-vote model, the phase tran-sition is continuous, whereas for r > . a discontinuousphase transitions appear within the model with indepen-dence. For the model with anticonformity phase transi-tions are continuous for an arbitrary value of r . Similarphenomenon has been observed previously for the q -votermodel – within the model with anticonformity only con-tinuous phase transitions are observed, whereas withinthe model with independence (known also as the nonlin-ear noisy voter model) a discontinuous phase transitionsappear for q > [28, 34, 35].In this paper we focus on the ST model with indepen-dence, because it occurs that the hysteresis and tippingpoints, two signatures of a discontinuous phase transi-tions, are common features of complex social systems[36–38]. We study the model on Watts-Strogatz (WS)graph [39] because it allows to tune the structure from(1) the complete graph, for which the mean-field approx-imation gives exact result, through (2) random graphs a r X i v : . [ phy s i c s . s o c - ph ] F e b for which the pair approximation should work properly,to (3) small-world networks which resembles the basicfeatures of the real social networks. Because it has beenshown recently that the size of the hysteresis may dependon the graph’s properties, we focus on this issue and checkto what extend results found within the MV model andthe qV model are universal [14, 15, 19, 20, 28, 40]. II. MODEL
We consider a system of N individuals placed in thenodes of an arbitrary graph. Each node represents ex-actly one individual (interchangeably called an agent , aspin , or a voter ). We consider a model of binary opin-ions/believes/decisions and thus each voter at time t isdescribed by a binary dynamical variable S i ( t ) = ± ↑ / ↓ ) . At each elementary update ∆ t :1. a site i is randomly chosen from the entire graph,2. an agent at site i acts independently with proba-bility p , i.e. changes its opinion to the opposite one S i ( t + ∆ t ) = − S i ( t ) with probability ,3. with complementary probability − p it conformsto its k i neighbors if the fraction of its neighbors inthe same state is larger than r :(a) S i ( t + ∆ t ) = 1 if more than rk i neighbors arein the state or(b) S i ( t + ∆ t ) = − if more than rk i neighborsare in the state − .As usual, a single Monte Carlo step consists of N up-dates, i.e. ∆ t = 1 /N , which means that one time unitcorresponds to the mean update time of a single individ-ual. Under the above algorithm the following changes arepossible in the system: ↑↑ . . . ↑ (cid:124) (cid:123)(cid:122) (cid:125) > (cid:98) rk i (cid:99) ⇓ − p −→ ↑↑ . . . ↑ (cid:124) (cid:123)(cid:122) (cid:125) > (cid:98) rk i (cid:99) ⇑ , ↓↓ . . . ↓ (cid:124) (cid:123)(cid:122) (cid:125) > (cid:98) rk i (cid:99) ⇑ − p −→ ↓↓ . . . ↓ (cid:124) (cid:123)(cid:122) (cid:125) > (cid:98) rk i (cid:99) ⇓ ,. . . . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) anyconfiguration ⇑ p/ −→ . . . . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) anyconfiguration ⇓ ,. . . . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) anyconfiguration ⇓ p/ −→ . . . . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) anyconfiguration ⇑ , (1)where ⇓ and ⇑ denotes states of a target agent, and (cid:98) rk i (cid:99) is the floor function of rk i . In any other situation, thestate of the system does not change.In the Watts threshold model flipping from ↑ to ↓ , wasforbidden [3]. Therefore, the model was asymmetrical oncontrary to the majority–vote or the q -voter. In the original threshold model an arbitrary value of r ∈ [0 , is possible, which is a reasonable assumptionfor the asymmetrical model describing the adoption tothe new state. In the symmetrical case, the situationfor r < . is less obvious. It can be easily seen withinthe following example: let the threshold r < . and theneighborhood of a target voter consists of positiveand negative agents. It means that both opinions(positive and negative) could be adopted by the voter.Which one should be chosen in such a situation?There are several possibilities to solve the above am-biguity, e.g. we can assume that: (1) a voter prefersto change opinion and therefore will always change it tothe opposite one whenever possible [30, 32], (2) a voterprefers to keep an old opinion; this assumption overlaps r ≥ . [29, 33] (3) a voter makes a random decision toflip or keep an old state. Each of these scenarios canbe used. However, for modeling opinion/belief formationthe second one, i.e. r ≥ . , seems to be the most justi-fied from the social point of view [31]. III. ANALYTICAL APPROACH WITHIN PAIRAPPROXIMATION
Our analytical approach is based on the pair approxi-mation (PA), an improved version of the standard mean-field approximation (MFA), which has been already ap-plied to various binary–state dynamics on complex net-works [28, 41, 42].Because at each elementary update only one voter canchange his opinion thus the number of agents with pos-itive opinion N ↑ increases or decreases by or remainsconstant. As in [34] we denote by c = N ↑ /N the concen-tration of the positive opinion, which in an elementarytime step increases or decreases by N or remains con-stant. We also denote transition probabilities as in [22]: γ + = P rob (cid:18) c ( t + ∆ t ) = c ( t ) + 1 N (cid:19) ,γ − = P rob (cid:18) c ( t + ∆ t ) = c ( t ) − N (cid:19) ,γ = P rob ( c ( t + ∆ t ) = c ( t )) = 1 − γ + − γ − . (2)For N → ∞ we can safely assume that random variable c localize to the expectation value and we get the followingcontinuous time dynamical system: dcdt = γ + − γ − , (3)in the rescaled time units t . The simplest and the mostpopular approach under which formulas for transitionprobabilities γ ± can be derived analytically is the sim-ple mean-field approach [21–23, 29–31, 33]. It gives verygood agreement for the complete graph, but rarely formore complicated structures, because it neglects all fluc-tuations in the system by assuming that the local con-centration of spins up is equal to the global one.Another method, which works particularly well forrandom graphs with low clustering coefficient, is thepair approximation. Within PA we describe the sys-tem by two differential equations – one for the time evo-lution of the concentration c of spins up and the sec-ond one for the time evolution of the concentration b ofactive bonds/links (bonds between two opposite spins)[2, 28, 41]: dcdt = − (cid:88) j ∈{ , − } c j (cid:88) k P ( k ) k (cid:88) i =0 (cid:18) ki (cid:19) θ ij (1 − θ j ) k − i × f ( i, r, k ) j, (4) dbdt = 2 (cid:104) k (cid:105) (cid:88) j ∈{ , − } c j (cid:88) k P ( k ) k (cid:88) i =0 (cid:18) ki (cid:19) θ ij (1 − θ j ) k − i × f ( i, r, k )( k − i ) , (5)where: • c j is the concentration of spins in state j = ± andthus c = c , c − = 1 − c , • P ( k ) is the degree distribution of a graph and (cid:104) k (cid:105) is the average node degree, • θ j is the conditional probability of selecting a nodethat is in the opposite state to its neighbor in astate j , which is equivalent to the probability ofchoosing an active link from all links of a node instate j and can be approximated by [2, 28]: θ j = b (2 c j ) , (6) • f ( i, r, k ) is the flipping probability, i.e. the proba-bility that a node in state j changes its state underthe condition that exactly i from its k links areactive.Within our version of the threshold model, a voterflips with probability / due to the independence, whichtakes place with probability p or due to the conformity,which takes place with probability − p if more than (cid:98) rk (cid:99) of its nearest neighbors are in the opposite state and thus: f ( i, r, k ) = p − p ) { i> (cid:98) rk (cid:99)} , (7)where { i> (cid:98) rk (cid:99)} is the indicator function, i.e. gives for i > (cid:98) rk (cid:99) and otherwise.We consider the model on the WS graph and thus the degree probability P ( k ) equals [43]: P ( k ) = f ( k,K ) (cid:88) n =0 (cid:18) K/ n (cid:19) (1 − β ) n β K/ − n × ( βK/ k − K/ − n ( k − K/ − n )! e − βK/ . (8)PA works properly for small clustering coefficients whichcorrespond to large values of β . Moreover, under theassumption β → , calculations simplify substantially,since Eq. (8) reduces to: P ( k ) = ( K/ k − K/ ( k − K/ e − K/ . (9)Therefore, we take in further calculations P ( k ) given byEq. (9).After inserting f ( i, r, k ) , given by Eq. (7), into Eqs.(4) – (5) we obtain: dcdt = − (cid:88) j ∈{ , − } c j (cid:88) k P ( k ) (cid:34) jp j (1 − p ) k (cid:88) i = (cid:98) rk (cid:99) +1 (cid:18) ki (cid:19) θ ij (1 − θ j ) k − i (cid:35) , (10) dbdt = 2 (cid:104) k (cid:105) (cid:88) j ∈{ , − } c j (cid:88) k P ( k ) (cid:34) pk (cid:18) − θ j (cid:19) ++ (1 − p ) k (cid:88) i = (cid:98) rk (cid:99) +1 (cid:18) ki (cid:19) θ ij (1 − θ j ) k − i ( k − i ) (cid:35) . (11)The steady states can be obtained by solving equations: dcdt = 0 , (12) dbdt = 0 . (13)Analogously as for the q -voter model with independence,we are not able to solve above equations explicitly but wecan obtain inverse relation p = p ( c ) , instead of c = c ( p ) [34]. For the concentration of active bonds we can presentonly implicit solution.One solution of Eq. (12), namely c = 1 / , is straight-forward because it is seen that for this value the rightside of Eq. (10) equals to zero, i.e. point c = 1 / is thefixed point for all values of p . On the other hand, theright side of Eq. (11) is nonzero at c = 1 / , thus fromEq. (13) for c = 1 / we can derive the relation p ( b ) : p = (cid:80) k P ( k ) k (cid:80) i = (cid:98) rk (cid:99) +1 (cid:0) ki (cid:1) b i (1 − b ) k − i ( k − i ) −(cid:104) k (cid:105) (cid:0) − b (cid:1) + (cid:80) k P ( k ) k (cid:80) i = (cid:98) rk (cid:99) +1 (cid:0) ki (cid:1) b i (1 − b ) k − i ( k − i ) (14)We see that b → gives p = 0 and b → / gives p = 1 .To show the behavior of the system for c (cid:54) = 1 / we insertEq. (10) to Eq. (12), which allows to derive the relation: p = (cid:80) k P ( k ) k (cid:80) i = (cid:98) rk (cid:99) +1 (cid:0) ki (cid:1) (cid:16) cθ i ↑ (1 − θ ↑ ) k − i − (1 − c ) θ i ↓ (1 − θ ↓ ) k − i (cid:17) − c + (cid:80) k P ( k ) k (cid:80) i = (cid:98) rk (cid:99) +1 (cid:0) ki (cid:1) (cid:16) cθ i ↑ (1 − θ ↑ ) k − i − (1 − c ) θ i ↓ (1 − θ ↓ ) k − i (cid:17) , (15)where we denoted θ / − by θ ↑ / ↓ for clarity. Note that theabove equation is in fact the relation p = p ( c, b ) , becauseboth b and c are implicitly included in θ ↑ and θ ↓ according to Eq. (6). Thus, to solve the above equation we needthe relation b = b ( c ) , which can be obtained by insertingthe above equation into Eq. (13): (cid:88) k P ( k ) k (cid:88) i = (cid:98) rk (cid:99) +1 (cid:18) ki (cid:19)(cid:34) cθ i ↑ (1 − θ ↑ ) k − i (cid:16) (cid:104) k (cid:105) (1 − b ) + (1 − c )( k − i ) (cid:17) ++ (1 − c ) θ i ↓ (1 − θ ↓ ) k − i (cid:16) (1 − c )( k − i ) − (cid:104) k (cid:105) (1 − b ) (cid:17)(cid:35) . (16)As we have noticed above, Eq. (15) gives the relation p = p ( c, b ) , which can be plotted in 3 different planes, asshown in Fig. 1. There are two critical points, seen in thisplot: (1) p = p ∗ , in which solution c = 1 / losses stability(so called lower spinodal), (2) p = p ∗ , in which solution c = c ( p ) (cid:54) = 1 / , given by Eq. (15), loses stability. Thereare several possibilities to calculate p = p ∗ [22, 28, 31].Here we use method based on the observation that p = p ∗ corresponds to the point c = 1 / in the relation b = b ( c ) (right bottom panel Fig. 1). Therefore, first we take alimit c → / in Eq. (16), which gives: (cid:88) k P ( k ) k (cid:88) i = (cid:98) rk (cid:99) +1 (cid:18) ki (cid:19) b i (1 − b ) k − i (cid:34) k − (cid:104) k (cid:105) (1 − b ) (cid:18) kb − b (cid:19) + − i + (cid:104) k (cid:105) (1 − b ) (cid:18) b − b (cid:19) i (cid:35) . (17)and then derive b from the above equation. Finally weinsert this value of b to Eq.(14), which gives p = p ∗ . Theupper spinodal, i.e. point p = p ∗ , where p = p ( c ) has twomaxima (see Fig. 1), can be calculated numerically fromEq.(15) by taking a maximum value of p . IV. DISCUSSION OF THE PAIRAPPROXIMATION RESULTS
It was shown that for the majority-vote model withinertia there are two ingredients responsible for the dis-continuous phase transitions: (1) the level of inertia and(2) the average node degree (cid:104) k (cid:105) [15, 19]. Similarly, forthe q -voter model (1) the size of the influence group q and (2) (cid:104) k (cid:105) and are key factors influencing the type ofthe phase transition [28, 34, 40]. The question is if thesame can be seen within the ST model.The first ingredient influencing the phase transition was studies already in the previous paper within themean-field approach [33]. We have observed continu-ous phase transitions for r = 0 . and discontinuous for r > . . We have obtained similar result within PA, asshown in Fig.2: for small values of r we observe a contin-uous, whereas for larger r a discontinuous phase transi-tion. This result is similar to results obtained within theMV model with inertia and the qV model. In both mod-els discontinuous phase transitions were observed onlyfor the sufficiently large value of inertia θ [15, 19] or thelarge size of the influence group q [28, 34]. It should benoticed that both the large size of the influence group q and the high threshold r corresponds to the high valueof inertia: qV model: it is unlikely to find a unanimous group ofsize q if q is large, ST model: it is unlikely to find a fraction of agents inthe same state larger than r if r is large. . . . p ∗ p ∗ c . . . . . . p ∗ b p . . . p ∗ c FIG. 1. Dependencies between the stationary value of theconcentration of spins up c , and active bonds b and the noise p obtained within PA for sample values of parameters (cid:104) k (cid:105) =80 and r = 0 . . Results are presented in three phase-spaceprojections: ( c, p ) , ( b, p ) and ( b, c ) . For p < p ∗ the only stablesolution is the ordered phase, in which the symmetry between ↑ and ↓ states is broken, whereas for p > p ∗ the only stablesolution is the disordered phase. . . . . . . . . . . . . p r . . . . . . . . . . . . o r d e r e d c o e x i s t e n ce disordered FIG. 2. Phase diagrams for the average degree (cid:104) k (cid:105) = 50 and r = 0 . . Lines with represents spinodals obtained within PAfrom Eqs. (15) - (17) i.e., limits of the region with metasta-bility, in which the final state depends on the initial one. . . . . . . C o m p l e t e G r a ph (a) c p . . . C o m p l e t e G r a ph (b) p FIG. 3. Dependence between the stationary concentration ofspins up c and the noise p for several values of the averagenode degree (cid:104) k (cid:105) and two values of the threshold: (a) r = 0 . and (b) r = 0 . . Thin (red and blue colors online) lines referto different values of (cid:104) k (cid:105) ∈ { , , , } from left to right,whereas thick black lines represent the mean-field solution.Arrows indicate the direction in which (cid:104) k (cid:105) increases. Therefore, in both cases a voter is unlikely influenced byneighbors, i.e. its inertia is larger.Now it is time to investigate the second ingredient,namely to check whether (cid:104) k (cid:105) influences phase transitionswithin ST model. In Fig. 3 we present the dependencebetween the stationary concentration of spins up c andthe noise p for several values of the average node degreeof the network (cid:104) k (cid:105) and two values of the threshold r .Again we see that for r = 0 . only continuous phasetransitions are observed independently on (cid:104) k (cid:105) . However,for r = 0 . the character of the phase transition changeswith (cid:104) k (cid:105) . Similarly as for the MV model with inertia andthe qV model, the width of the hysteresis increases with (cid:104) k (cid:105) [15, 19, 40].Due to our knowledge, the dependence between the sizeof the hysteresis and (cid:104) k (cid:105) was not investigated preciselyfor the MV model with inertia. However, for the q -votermodel it has been shown that (cid:104) k (cid:105) influences substantiallythe width of the hysteresis and has almost no influenceto the jump of the order parameter, defined as [40]: m = N ↑ − N ↓ N = 2 N ↑ N − c − . (18)In this paper we did not introduce order parameter m ,because we made all calculations in terms of c . Of coursewe could easily reformulate all results using the simplerelation between m and c , given by Eq. (18).In [40] the jump of m has been measured at upperspinodal. Therefore we also measure a jump of c at thispoint, i.e. c ( p ∗ ) − . . As we see in Fig. 4 both hysteresis,as well as the jump of c depend on (cid:104) k (cid:105) . However, thesedependencies are very different. There is only one com-mon feature seen in both relations – below certain valueof (cid:104) k (cid:105) both p ∗ − p ∗ , as well as c ( p ∗ ) − . are equal zero,which indicates continuous phase transition. Above thisvalue the width of hysteresis increases with (cid:104) k (cid:105) almostlinearly. On the other hand the jump of concentration ofspins up increases only slightly but this growth is very . . . . . .
510 45 80 115 150 p ∗ − p ∗ h k i . . . . . c ( p ∗ ) − . h k i FIG. 4. The width of hysteresis p ∗ − p ∗ (left panel) and thejump of the public opinion c (right panel) as a function of theaverage node degree (cid:104) k (cid:105) for threshold r = 0 . obtained withinPA. rapid and takes place in a relatively small range of (cid:104) k (cid:105) .For larger values of (cid:104) k (cid:105) the jump of c does not change,similarly as for the q -voter model [40].Until now we have analyzed the influence of (cid:104) k (cid:105) on thephase transition only for r = 0 . . Of course the same canbe done for an arbitrary value of r , as shown in Fig. 5.We see that the width of the hysteresis indeed increasesmonotonically with (cid:104) k (cid:105) . However, the dependence onthe threshold r is much more interesting. There is anoptimal value of r , which decreases with (cid:104) k (cid:105) , for whichthe hysteresis has the maximum size. Because empiricalstudies suggest that the mean number of friends variestypically from to , depending on the rated emo-tional closeness between them, [44], optimal value of r ,for which the maximum size of hysteresis appear lies in (0 . , . . We find this result particularly interestingfrom the social point of view, which will be commentedin the Conclusions. V. MONTE CARLO SIMULATIONS
We validate our analytical PA results by Monte Carlo(MC) simulations on WS graphs [39]. As we have writ-ten in the introduction, WS algorithm allows to tune thestructure of the graph from a regular ( β = 0 ) to a randomone ( β = 1 ). It also reduces to the complete graph for (cid:104) k (cid:105) = N − . Moreover, in the whole spectrum of param-eter β the average node degree is conserved. This makesthe WS graph particularly interesting for our studies.We start with β = 1 , for which PA should be themost accurate. Indeed, as seen in Fig. 6, Monte Carlooverlap PA results, even for small values of (cid:104) k (cid:105) . More-over, this agreement is seen in all dependencies, namely c = c ( p ) , b = b ( p ) , b = b ( c ) . The question is if and howparameter β will influence results.In Fig. 7 parameter β vary from . to . As seen,the width of the hysteresis p ∗ − p ∗ is increasing with β .Such a tendency is seen for all values of r . As usually, ingeneral PA gives consistent results with MC simulationsonly for sufficiently large values of the rewiring probabil-ity β . However, as seen in Fig. 7, the value of the upperspinodal is less sensitive to β than the lower spinodal and is predicted correctly even for β = 0 . . VI. CONCLUSIONS
The notion of the tipping point, similarly as the notionof the hysteresis, two signatures of discontinuous phasetransitions, has been present in social sciences for manyyears [36]. Although it may seem that the social hys-teresis and the tipping point are just fancy buzzwords,empirical social studies have confirmed that they are notjust abstract ideas [36, 38, 45].These findings, among others, inspired researchers tolook for the hysteresis in models of opinion dynamics . . . . . . . . . . . . . (a) r h k i p ∗ − p ∗ . . . . . . . . . . . . (b) r h k i c ( p ∗ ) − . FIG. 5. The size of the hysteresis (a) and the jump of theconcentration c at upper spinodal p ∗ (b) as a function of thethreshold r and the average degree of a graph (cid:104) k (cid:105) obtainedwithin PA. . . . . . . (a) c h k i = 10 h k i = 20 h k i = 40 h k i = 80 . . . . . . (b) b . . . . . . (c) b . . . . . . . . . . . . (d) c p h k i = 10 h k i = 20 h k i = 40 h k i = 80 . . . . . . . . . . . . (e) b p . . . . . . . . . . . . (f) b c FIG. 6. Comparison between results obtained within PA (denoted by lines) and Monte Carlo simulations (denoted by symbols)for r = 0 . (upper panels) and r = 0 . (bottom panels). Solid lines correspond to stable, whereas dashed lines to unstablesolutions of Eqs. (12)–(13). For all diagrams the size of the system N = 10 , the thermalization time t = 10 and the initialconcentration of spins up c (0) = 1 . Results are averaged only over samples, but for this size of the system it is sufficient, asseen above. . . . . . . . . . . . . (a) c p c (0) = 1 . c (0) = 0 . . . . . . . . . . . . . (b) c p . . . . . . . . . . . . (c) c p FIG. 7. Dependence between the stationary concentration of spins up c and the noise p for r = 0 . , (cid:104) k (cid:105) = 150 , and severalvalues of rewriting parameter: (a) β = 0 . , (b) β = 0 . , (c) β = 1 . Monte Carlo results for two types of initial conditions and N = 10 are denoted by symbols, whereas lines correspond to PA results. As in Fig.6, thermalization time t = 10 and resultsare averaged over samples. [15, 19, 40]. For example, an additional noise has beenintroduce to the MV model, but is was shown that itdoes not affect the type of the phase transition and it re-mains continuous irrespective of the network degree andits distribution [14, 20]. On the other hand it was shownthat discontinuous phase transitions may appear in theMV model with inertia, when the inertia is above anappropriate level [15]. Later the question about the fun-damental ingredients for discontinuous phase transitions in the inertial majority vote model has been asked [19].It was shown that low (cid:104) k (cid:105) leads to the suppression of thephase coexistence. Similar result has been also reportedfor the q -voter model [40].This motivated us to check if the same behavior will beobserved within the symmetrical threshold model intro-duced in [33]. We have shown, using PA and MC simula-tions, that indeed the type of the phase transition withinST model depends on threshold r , as well as the prop-erties of the network (cid:104) k (cid:105) and β , i.e. hysteresis increaseswith (cid:104) k (cid:105) and β . On the other hand, the dependence on r is non-monotonic, which will be commented below.We discuss ST in the context of MV and qV mod-els, because they have a lot in common, which has beenalready discussed in [33]. In particular, ST model withanticonformity is the generalization of the basic majority-vote model, which corresponds to r = 0 . . Moreover, STmodel with r = 1 reduces to the q -voter model on therandom regular graph with degree q , i.e. if ∀ ik i = k = q .Finally, ST model with an arbitrary value of r corre-sponds to the threshold q -voter model on the randomregular graph with ∀ ik i = k = q [29–32].Moreover, as we have noticed in Sec. IV, the param-eters that are mainly responsible for the discontinuousphase transitions, namely: the level of inertia θ in theMV model with inertia, the size of the influence group q in the qV model and the threshold r needed for the socialinfluence in the ST model, play in a sense a similar role.The larger q or r is, the harder it is to influence a voter,which in result increases inertia on the microscopic level.Because the hysteresis can be viewed as an inertia ofthe system on the macroscopic level, it would not be sur-prising that the inertia on the microscopic level supportsthe hysteresis. However, as shown in Fig. 5, the rela-tion between the size of the hysteresis and parameter r is not that trivial, i.e. it is non-monotonic, having themaximum value for a given value of r , which depends on (cid:104) k (cid:105) . This is particularly interesting result from the socialpoint view and worth to be discussed here.It is known that social influence increases with the sizeof the influence group as well as the unanimity of thegroup. However, this dependence is far from being trivial.First of all, it occurs that it increases only up to a certainlevel. The social influence is stronger if the group ofinfluence consists of , instead of people. However,above a certain threshold it remains on the same level.Moreover, above this threshold, around − people, thesocial influence decreases [46].Therefore, in social experiments, in which descriptivenorms are used to influence people, social psychologistsneither use unanimity nor simple majority. Instead theyuse certain super-majority, often around . For exam- ple they manipulate people to reuse towels in hotels withthe fake descriptive norm saying something like: of our guests are reusing towels". There is no strongevidence that is the magic number and in someother experiments larger majorities were used as brieflyreviewed in [31]. The main message we want to passhere is that the larger majority does not always resultin stronger social influence. It seems that some optimalvalues exist and these values probably depend on thesize of the influence group: for small groups unanimity isneeded but for large groups some threshold value is moreappropriate, significantly larger than , but smallerthan . How this is related with the results obtainedhere?As we have already written in Section IV, it was foundempirically that in real social networks (cid:104) k (cid:105) ∈ (5 , .For these values the optimal threshold of r , for whichthe largest social hysteresis is observed, lies in the range (0 . , . , depending on the average size of the influ-ence group (cid:104) k (cid:105) . We admit that what we measure is notthe power of social influence, but the size of the hys-teresis. However, having in mind that the hysteresis isusually observed in social systems, we can speculate thatthere are some optimal values in the level of social influ-ence and these values influence the hysteresis, which isusually observed in social systems.We are aware that it maybe merely intriguing but themeaningless coincident. However, we believe that thisfinding deserves more attention and studies within othermodels of opinion dynamics. ACKNOWLEDGMENTS
This work is supported by funds from the Na-tional Science Centre (NCN, Poland) through grant no.2016/21/B/HS6/01256 and by PLGrid Infrastructure.
DATA AVAILABILITY STATEMENT
Data sharing is not applicable to this article as no newdata were created or analyzed in this study. [1] D. Watts and P. Dodds,
Threshold models of social influ-ence (2017) pp. 475–497.[2] A. Jędrzejewski and K. Sznajd-Weron, Comptes RendusPhysique (2019), 10.1016/j.crhy.2019.05.002.[3] D. J. Watts, Proceedings of the National Academy ofSciences of the United States of America , 5766 (2002).[4] M. Grabisch and F. Li, Dynamic Games and Applications(2019), 10.1007/s13235-019-00332-0.[5] M. Granovetter, Am. J. Sociol. (1978),10.1086/226707.[6] S. Galam, J. Stat. Phys. , 943 (1990). [7] S. Galam, International Journal of Modern Physics C ,409 (2008).[8] S. Galam, Sociophysics: A Physicist’s Modeling ofPsycho-Political Phenomena (Springer-Verlag, 2012).[9] T. M. Liggett,
Interacting Particle Systems (Springer,1985).[10] T. Tome, M. De Oliveira, and M. Santos, Journal ofPhysics A: General Physics , 3677 (1991).[11] M. de Oliveira, Journal of Statistical Physics , 273(1992).[12] F. Lima and K. Malarz, International Journal of ModernPhysics C , 1273 (2006). [13] J. Santos, F. Lima, and K. Malarz, Physica A: StatisticalMechanics and its Applications , 359 (2010).[14] A. Vieira and N. Crokidakis, Physica A: Statistical Me-chanics and its Applications , 30 (2016).[15] H. Chen, C. Shen, H. Zhang, G. Li, Z. Hou, andJ. Kurths, Physical Review E (2017), 10.1103/Phys-RevE.95.042304.[16] A. Fronczak and P. Fronczak, Physical Review E (2017), 10.1103/PhysRevE.96.012304.[17] A. Krawiecki, European Physical Journal B (2018),10.1140/epjb/e2018-80551-9.[18] A. Krawiecki and T. Gradowski, Acta Physica PolonicaB, Proceedings Supplement , 91 (2019).[19] J. Encinas, P. Harunari, M. De Oliveira, and C. Fiore,Scientific Reports (2018), 10.1038/s41598-018-27240-4.[20] J. Encinas, H. Chen, M. de Oliveira, and C. Fiore, Phys-ica A: Statistical Mechanics and its Applications , 563(2019).[21] C. Castellano, M. A. Muñoz, and R. Pastor-Satorras,Physical Review E , 041129 (2009).[22] P. Nyczka, K. Sznajd-Weron, and J. Cisło, Phys. Rev.E , 011105 (2012).[23] P. Moretti, S. Liu, C. Castellano, and R. Pastor-Satorras, Journal of Statistical Physics , 113 (2013).[24] M. Mobilia, Physical Review E , 012803 (2015).[25] M. A. Javarone and T. Squartini, Journal of Statisti-cal Mechanics: Theory and Experiment , P10002(2015).[26] A. Mellor, M. Mobilia, and R. Zia, EPL (EurophysicsLetters) , 48001 (2016).[27] A. Mellor, M. Mobilia, and R. Zia, Physical Review E , 012104 (2017).[28] A. Jędrzejewski, Phys Rev. E , 012307 (2017).[29] P. Nyczka and K. Sznajd-Weron, J. Stat. Phys. , 174(2013). [30] A. R. Vieira and C. Anteneodo, Physical Review E ,052106 (2018).[31] P. Nyczka, K. Byrka, P. R. Nail, and K. Sznajd-Weron,PLoS ONE (2018), 10.1371/journal.pone.0209620.[32] A. Vieira, A. F. Peralta, R. Toral, M. San Miguel, andC. Anteneodo, arXiv:2002.04715v1 (2020).[33] B. Nowak and K. Sznajd-Weron, Complexity (2019), 10.1155/2019/5150825.[34] P. Nyczka, J. Cisło, and K. Sznajd-Weron, PhysicaA: Statistical Mechanics and its Applications , 317(2012).[35] A. Peralta, A. Carro, M. San Miguel, and T. R, Chaos , 075516 (2018).[36] M. Scheffer, F. Westley, and W. Brock, Ecosystems ,493 (2003).[37] R. Vallacher, A. Nowak, and S. Read, Computationalsocial psychology (2017) pp. 1–381.[38] D. Centola, J. Becker, D. Brackbill, and A. Baronchelli,Science , 1116 (2018).[39] D. J. Watts and S. H. Strogatz, Nature , 440 (1998).[40] A. Abramiuk and K. Sznajd-Weron, Entropy (2020),10.3390/e22010120.[41] J. P. Gleeson, Phys. Rev. X , 021004 (2013).[42] A. Peralta, A. Carro, M. San Miguel, and R. Toral, NewJournal of Physics , 103045 (2018).[43] A. Baronchelli and R. Pastor-Satorras, Physical ReviewE , 011111 (2010).[44] R. Dunbar, V. Arnaboldi, M. Conti, and A. Passarella,Soc. Netw. , 39 (2015).[45] G. Doering, I. Scharf, H. Moeller, and J. Pruitt, NatureEcology and Evolution , 1298 (2018).[46] S. E. Asch, Sci. Am.193