Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias

We study the Galam majority rule dynamics with contrarian behavior and an oscillating external propaganda in a population of agents that can adopt one of two possible opinions. In an iteration step, a random agent interacts with three other random agents and takes the majority opinion among the agents with probability p(t) (majority behavior) or the opposite opinion with probability 1−p(t) (contrarian behavior). The probability of following the majority rule p(t) varies with the temperature T and is coupled to a time-dependent oscillating field that mimics a mass media propaganda, in a way that agents are more likely to adopt the majority opinion when it is aligned with the sign of the field. We investigate the dynamics of this model on a complete graph and find various regimes as T is varied. A transition temperature Tc separates a bimodal oscillatory regime for TTc in which m oscillates around zero. These regimes are characterized by the distribution of residence times that exhibit a unique peak for a resonance temperature T*, where the response of the system is maximum. An insight into these results is given by a mean-field approach, which also shows that T* and Tc are closely related.


I. INTRODUCTION
In the last decades, statistical physics has expanded its scope to venture into the field of sociology, giving rise to a discipline called sociophysics [1][2][3][4][5][6][7][8][9][10].A commonly studied phenomenon is the dynamics of opinion formation, by means of simple mathematical models.In these models, individuals are called agents, and each of them is characterized by the value of a variable that represents its opinion on a particular topic -such as the intention to vote for a candidate in a ballot-which, for simplicity, can take one of two possible values (+1 or −1).The opinion of each agent can change after interacting with other agents following simple rules.One of the most implemented interaction rule is that introduced in a model by Galam [11] and extensively studied later on [6,[12][13][14], to which we refer as the Galam Majority Model (GMM), in which all agents of a group chosen at random adopt the opinion of the majority in that group.This local dynamics drives a steady increase of the initial global majority opinion (provided the system's symmetry is not broken at ties for even size groups) which eventually ends at a consensus, i.e., an absorbing state where all agents share the same opinion.Multiple extensions of the GMM have been studied in the literature, including the possibility of a contrarian behavior, that is, all members of a chosen group taking the minority opinion [7].This work studied the effects of introducing a fixed fraction a of contrarian agents on the original GMM, where it was found that, instead of a frozen consensus as in the model with no contrarians, the system reaches an ordered stationary state for a < a c and a disordered stationary state for a > a c .The transition value a c separates an ordered phase where a large majority of agents hold the same opinion, from a disordered phase in which both opinions are equally represented in the population.
Many other opinion formation models with contrarians were also studied in [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].In particular, the effects of contrarian behavior was also investigated in the voter model (VM) for opinion formation [24], where agents interact by pairs and one adopts the opinion of the other with probability 1−p (imitation) or the opposite opinion with probability p (contrarian).It was shown that the model displays a transition from order to disorder when the probability of having a contrarian behavior overcomes the threshold p c = (N + 1) −1 in a system of N agents.The contrarian voter model [24] was recently studied under the presence of a mass media propaganda that influences agents' decisions [34].The propaganda was implemented in the form of an external oscillating field that tends to align agents' opinions in the direction of the field.It was found a stochastic resonance (SR) phenomena within an oscillatory regime, that is, there is an optimal level of noise for which the population effectively responds to the modulation induced by the external field [35,36].
In order to expand our knowledge on the combined effects of contrarians and propaganda on opinion models, we study in this article the GMM with contrarian behavior under the presence of an external field.Each agent in the population can either follow a majority rule that increases similarity with its neighbors or behave as a contrarian by adopting the opposite opinion, with respective probabilities p(t) and 1 − p(t).The majority probability p(t) varies in time according to an external field, based on a mathematical form introduced in [21,22] for the Sznajd model and implemented in [34,37] for the VM, so that agents tend to follow the majority when it is aligned with the field.By exploring the dynamics of the GMM model under the influence of an oscillating external field and the presence of contrarians, we aim to gain deeper insights into the manifestation of the SR phenomenon in opinion dynamics models.We show that this model exhibits unimodal and bimodal oscillatory regimes, as well as a SR that is observed close to the transition between the two regimes.
It is worth mentioning that, while GMM belongs to the class of "non-linear" models whose mean-field dynamics is associated to a double-well Ginzburg-Landau potential, the VM with contrarians described above belongs to a completely different class characterized by an associated zero potential that leads to a dynamics driven purely by noise [38].A main consequence of this difference is that the average magnetization is conserved in the VM, while it is not in the GMM.Another consequence is that, in the version of these models with contrarians, the order-disorder transition in the thermodynamic limit (N → ∞) takes place at a finite fraction of contrarian agents a c > 0 in the GMM, while in the VM the transition happens at a vanishing contrarian probability (p c → 0).We also need to mention that the SR effect has also been observed in other opinion models.For instance in [21,22] the authors found SR in a variation of the Sznajd model with stochastic driving and a periodic signal.The work in [14] analyzed a majority rule dynamics under the action of noise and an external modulation, and found a SR that depends on the randomness of the small-world network.There are also other works [39][40][41][42][43] that explored the combined effects of a stochastic driving and an external signal on a majority rule dynamics.However, none of these works have incorporated a contrarian behavior in the dynamics.
The rest of the article is organized as follows.We introduce the model in section II.In section III we present numerical simulation results for the evolution of the system and the behavior of different magnitudes that characterize the SR phenomena.In section IV we develop a mean-field (MF) approach that gives an insight into the system's evolution and the relation between the SR and the transition between different regimes.Finally, in section V we summarize our findings and discuss the results.

II. THE MODEL
We consider a population of N interacting agents where a given agent i (i = 1, .., N ) can hold one of two possible opinion states s i = +1, −1.We denote by σ + (t) and σ − (t) the fraction of nodes with respective states +1 and −1 at time t, such that σ + (t) + σ − (t) = 1 for all t ≥ 0. In a time step ∆t = 1/N of the dynamics, we follow the basic GMM using groups of size three to update individual opinions.However, here for our purpose of investigating the effects of propaganda on individuals, we implement the rule in a different setting, which does not modify the outcome.Instead of selecting three agents randomly to update all of them at once, we pick one agent i with state s i and a group of three other different agents j, k, l (i = j = k = l), all randomly chosen.In the N ≫ 1 limit, their respective states are (s j , s k , s l ) with probability σ sj σ s k σ s l .A majority of + choices is thus obtained for the configurations (+, +, +), (+, +, −), (+, −, +) and (−, +, +), yielding an overall probability Similarly, a majority of − occurs for (−, −, −), (+, −, −), (−, +, −) and (−, −, +), with the overall probability Then, agent i updates its state in two steps.i) First, the update follows the basic GMM, where agent i simply adopts the majority state of the group of the three agents j, k, l.We thus have s i → s i = +1 with probability P + , or s i → s i = −1 with probability P − = 1 − P + .ii) Second, agent i can either preserve this majority state (s i → s i ) with probability p si , or change to the opposite (minority) state (s i → −s i ) with the complementary probability 1 − p si , where p si is defined below.The implication of this second step is that each agent can behave as a "contrarian" by adopting the state opposed to the majority (minority state) with probability 1 − p si , or as a "majority follower" with probability p si .Thus, there is no fixed fraction of contrarian agents as in [7] At this point, we introduce the effect of an external field H on agent i in state s i within a Boltzmann scheme, by assuming that the probability p si to preserve the majority state is larger when s i is aligned with where T ≥ 0 is a parameter that plays the role of a social temperature analogous to the contrarian feature of the GMM.The related probability to oppose the field is 1 − p si,H .We assume that H is an oscillating periodic field H(t) = H 0 sin(ωt) with amplitude H 0 (0 ≤ H 0 ≤ 1), frequency ω = 2π/τ and period τ , which represents an external propaganda.Thus, according to Eq. ( 3), agents are more likely to keep the opinion that is aligned with the propaganda.
In addition to the external field, we introduce an individual "inertia" parameter I, which provides an agent with a weight to preserve its current state against a field favoring the opposite state.It is a self-interaction −Is i s i that modifies Eq. ( 3) as which can be rewritten as where I, H, T have been rescaled as 1, H I , T I using s 2 i = 1.At this stage we combine the GMM with the inertia and field effects by taking for the probability of agent i to keep the majority state s i , and 1 − p si (t) for the probability to adopt the opposite (minority) state −s i , which can be interpreted as a noise.Finally, combining Eqs. ( 1), ( 2) and ( 6), the probability P + for a randomly selected agent i to adopt the state + in a single time step ∆t is given by where the first term comes from following a local majority + among the three selected agents, which happens with probability P + p + (t), while the second term corresponds to opposing the state − in case of a majority of − among the three selected agents, which happens with probability P − [1 − p − (t)].Analogously, the state − is selected with probability P − ≡ 1 − P + .
As noted above, only the "focal agent" i updates its state, unlike in the original GMM where all agents in the chosen group update their states.Equation (6) shows that individuals are more prone to adopt the opinion of the majority when it is aligned with the propaganda.In addition, p + and p − approach the value 1 as T → 0, which makes this case equivalent to the original GMM, with neither contrarians nor external field.In the opposite limit T → ∞, p + and p − approach the value 1/2, which corresponds to the pure noise case where agents take one of the two opinions at random, independent of the field.

A. Evolution of the magnetization
We start by studying the time evolution of the mean opinion of the population or magnetization defined as m(t) ≡ 1 N N i=1 s i (t), for the simplest case of zero field H = 0, which corresponds to the contrarian GMM with symmetric majority probabilities p + = p − = p = (1 + e −2/T ) −1 .We run several independent realizations of the dynamics where, initially, each agent adopts state +1 or −1 with respective probabilities σ + (0) and σ − (0), leading to an initial average magnetization m(0) = σ + (0) − σ − (0).Due to the symmetry of the system, the evolution of the average value of m over many realizations starting from m(0) = 0 gives m (t) ≃ 0 for all t ≥ 0, which does not describe the correct behavior of the system.Instead, we looked at the evolution of the absolute value of the  11), while the dashed lines are the numerical integration of Eq. ( 8).The averages were done over 10 3 independent realizations starting from a symmetric condition m0 = 0.
magnetization, |m|, as we show in Fig. 1(a), for various values of p.In Fig. 1(b) we show in circles the stationary value of |m| as a function of T for H = 0. We observe that, as T increases, the system displays a transition from an ordered state (|m| > 0) for T < T 0 c , to a disordered state (|m| ≃ 0) for T > T 0 c , where T 0 c is a transition temperature.This order-disorder transition, reminiscent of the GMM with a fixed fraction of contrarian agents [7], is induced by the presence of a contrarian behavior that acts as a source of external noise, preventing the system to reach full consensus.When the noise amplitude, controlled by T , overcomes a threshold value T 0 c the system reaches complete disorder.In section IV we develop a mean-field approach that allows to estimate the transition temperature as T 0 c ≃ 1.24.When the field is turned on, these results change completely.In the case that the field remains constant in time (constant propaganda H=const), the symmetry of the system is broken in direction of H, increasing the stationary value of |m| as compared to the H = 0 case.This effect can be seen in Fig. 1(b), where we see that |m| * increases monotonically with H. Besides, the order-disorder transition disappears for H > 0 (see H = 0.1 and H = 0.5 curves).
If we now let the field oscillate in time, a series of different regimes emerge.In Fig. 2 we show the evolution of m in a single realization under the effects of an oscillating field, for three different amplitudes H 0 , period τ = 256 and various temperatures.For the H 0 = 0.1 and H 0 = 0.5 cases [panels (a) and (b)], we can see that for low temperatures m oscillates around a positive value or negative value, and that oscillations vanish for small enough T , where m stays in a value close to 1.0 (consensus), as we can see for T = 0.2 and T = 0.1 in panels (a) and (b), respectively.The center of oscillations can jump from positive to negative values and vice-versa (bimodal regime), as we can see in panel (b) for T = 0.5.Above a given temperature threshold, T c ≃ 1.0 for H 0 = 0.1 [panel (a)] and T c ≃ 0.5 for H 0 = 0.5 [panel (b)], the magnetization oscillates around m = 0 (unimodal regime).This behavior is reminiscent of the ordered and disordered phases in the model without field [Fig.1(b)], although the transition temperature T 0 c ≃ 1.24 for H = 0 is quite different from that of the model with oscillating field.An insight into this behavior shall be given in section IV.For H 0 = 1.0 [panel (c)] oscillations are centered at m = 0 even for small T , and thus the bimodal regime is not observed.Finally, at very large temperatures the high levels of noise leads to a purely stochastic dynamics where agents adopt an opinion at random, and thus m fluctuates around zero.

B. Residence times
In order to characterize the different regimes described in the last section, we study here the residence time t r , defined as the time interval between two consecutive changes of the sign of m, i.e., when m crosses the center value m = 0.In a single realization, m can change sign multiple times depending on the parameter values, leading to a distribution of the residence time that is particular of each regime.Results are shown in Fig. 3  until it changes to negative (positive) oscillations, and back to positive (negative) oscillations again, as we observe in Fig. 2(b) for T = 0.5.These changes are more likely to happen when H changes sign, in the first attempt at time t τ /2, or in the second attempt one period later (at t = 3τ /2), or in the third attempt at t = 5τ /2 and so on, leading to the different peaks in the RT D. Finally, for very large T the RT D shows an exponential decay due to the stochastic fluctuations of m around zero (panels for T = 10).

C. Stochastic resonance
The patterns of the RTD shown in section III B can be employed to quantify the phenomena of stochastic resonance, as it was done in related systems [14,35].The sensitivity or response of the system to the external oscillating field can be measured by the area A under the first peak around τ /2 in the RTD histogram.It is expected that A reaches a maximum at the resonance temperature T * , when m resonates with the field H.This method to quantify the resonance is an alternative to the study of the signal-to-noise ratio [21,22,34].Figure 4(a) shows the response A vs T for a field of amplitude H 0 = 0.1.Each curve corresponds to a different period τ .We observe that A reaches a maximum value at a temperature T * that depends on τ .The RTD for the resonance temperatures T * = 1.04 and T * = 0.98 for periods τ = 256 and τ = 1024, respectively, are shown in the top-right panels of Figs.3(a) and 3(b), where we see the existence of a well defined peak centered at t r = τ /2.For larger temperatures (see T = 1.3) there is also a peak at τ /2, although lower than that for T * , and the RTD exhibits another pronounced peak near t r = 0, corresponding to the short crossings of m(t) that become more frequent as T increases (larger fluctuations in m).

IV. MEAN-FIELD APPROACH
In this section we analyze the behavior of the model within a MF approach, by deriving a rate equation for the evolution of m that corresponds to the dynamics introduced in section II.Let us write the fractions of + and − 0.
which becomes, in the N → ∞ limit, the rate equation after replacing the expressions for P + and P − and doing some algebra.Here are the probabilities of adopting the state +1 and −1 of a majority, respectively, as defined in Eq. ( 6).
For the zero field case (H 0 = 0) is p + = p − = p = (1 + e −2/T ) −1 , and thus Eq. ( 8) is reduced to the simple equation Equation ( 10) has three fixed points corresponding to the possible stationary states of the agent based model.The fixed point m * 0 = 0 is stable for p < 5/6 and corresponds to a disordered active state with equal fractions of + and − agents (σ + = σ − = 1/2), whereas the two fixed points are stable for p > 5/6, and they represent asymmetric active states of coexistence of + and − agents, with stationary fractions σ * The stable fixed points are plotted by a solid line in Fig. 1(b), where we observe a good agreement with MC simulation results (solid circles).Equation (11) shows the existence of a transition from order to disorder as T overcomes the value T 0 c = 2/ ln(5) ≃ 1.24 (p 0 c = 5/6), as we already mentioned in section III A. Notice that the probability of behaving as a contrarian 1 − p 0 c = 1/6 is identical to the critical proportion of contrarians a c = 1/6 obtained in the GMM for groups of size 3 [7].Given that Eq. ( 10) can be rewritten as a Ginzburg-Landau equation with an associated double-well potential with two minima at m * ± , we expect a bistable regime for T < T c , where in a single realization m jumps between m * + and m * − .
For a field that is constant in time (H = const = 0) the fixed points of Eq. ( 8) are given by the roots of a cubic polynomial, and m = 0 is not longer a root.Only one root is real, and corresponds to the stationary state of the agent-based model.As the analytical expression for the real root is large and not very useful, we integrated Eq. ( 8) numerically to find the stationary value m * , which we plot by a dashed line in Fig. 1(b) for H = 0.1 and 0.5.We observe a good agreement with MC simulations (symbols).A positive field H > 0 breaks the symmetry in favor of the + state, given that p + > p − , leading to a positive stationary value m * > 0 that increases monotonically with H.
For an oscillating field H(t), we have that p + (t) and p − (t) oscillate in time according to H(t), which in turn leads to oscillations in m(t).In order to explore, within the MF approach, the behavior of m in different regimes described in section III A, we plot in Fig. 5(a) the evolution of m obtained from the numerical integration of Eq. ( 8) for H 0 = 0.1, τ = 256, and various temperatures.For low temperatures we see that m oscillates around a positive value (it could also be a negative value for other initial conditions), but when the temperature is increased beyond a threshold value oscillations turn to be around m = 0.At first sight, this transition that happens in the oscillatory regime of m, already reported in section III A from MC simulations, appears to be quite sharp, where the center of oscillations seems to jump from a large value to zero after a small increment of T .To better characterize the transition we plot in Fig. 5(b) the temporal average of m from t = 0 to t = 1000τ , called m, as a function of T and for several periods τ .The value of m can be seen as an order parameter, which takes a positive or negative value in the bimodal regime and a value close to zero in the unimodal regime.We can see that m decreases continuously with T for low τ (see curve for τ = 64), and that the transition becomes more abrupt as τ increases (see curves for τ ≥ 256).The inset shows a more detailed view of the transition in the value of m.
In Fig. 2(a) we compare the evolution of m obtained from the MF approach (dashed lines) with that from MC simulations, for H 0 = 0.1, τ = 256, and various temperatures.We observe a good agreement with single realizations of the dynamics, except for the temperature T = 1.0 that is close to the transition value T c ≃ 0.981, estimated from Fig. 5(b) as the point where m becomes zero.This discrepancy is due to the fact that the MF approach cannot reproduce the random jumps of m from the value m ≃ 0.564 in the bimodal regime to m ≃ 0 in the unimodal regime.These jumps are induced by finite-size fluctuations, and are more frequent when the control parameter T is close to the transition point T c .
An insight into the behavior of the resonance temperature T * with the period τ can be obtained from the MF approach assuming that the response A reaches a maximum value at a temperature similar to the transition point T c , that is, we expect T * ≃ T c .This is because in the bimodal regime T < T c the magnetization m oscillates around a positive or a negative value and eventually crosses m = 0 around times t = τ /2, 3τ /2, etc., by finite-size fluctuations, leading to multiple peaks in the residence time distribution.Then, at T = T c , oscillations start to be centered at m = 0, and thus we expect that the RT D shows a single peak at τ /2.By increasing T beyond T c we expect that the height of the peak for T = T c is reduced by the presence of a higher noise that induces another maximum of the RT D at t = 0, as explained in section III B, leading to a smaller A. Therefore, we expect that A is maximum at T ≃ T c . Figure 4(b) shows in diamonds the value of T c obtained from Fig. 5(b) for various periods τ .We see that T c decreases with τ , as it happens with T * (circles), although discrepancies between T c and T * increase as τ decreases.

V. SUMMARY AND DISCUSSION
In this article we studied the dynamics of the binary-state majority rule model introduced by Galam for opinion formation, under the presence of an external propaganda and contrarian behavior.When an agent has to update its opinion, it can either follow the majority opinion among three random neighbors, similarly to the original GMM, or adopt the opposite (contrary) opinion, i.e., the minority opinion.The probability to adopt the majority opinion p ± (t) is coupled to an external field that oscillates periodically in time (propaganda), in a way that agents are more likely to adopt the majority opinion when it is align with the field.This rule tries to reproduce a reinforcing mechanism by which individuals have a tendency to follow the majority opinion when it is in line with mass media propaganda.Besides, the majority probability p ± depends on a parameter T (temperature) that acts as an external source of noise, in such a way that by increasing T from zero the system goes from following the majority opinion only (p ± = 1 for T = 0) to adopting a random opinion for large temperatures (p ± = 0.5 for T ≫ 1).
We explored the model in complete graph (all-to-all interactions) and found different phenomena associated to different regimes as T is varied.For T below a threshold value T c the system is in a bimodal regime, where the mean opinion m oscillates in time around a positive or negative value, m ± , and performs jumps between m + and m − due to finite-size fluctuations, similarly to what happens in a bistable system.As the temperature is increased beyond T c there is a transition to an unimodal regime in which m oscillates around zero, where the amplitude of oscillations decreases with T and eventually vanishes in the T ≫ 1 limit that corresponds to pure noise.The transition at T c becomes more abrupt as the period τ of the field increases.We also studied the response of the system to the external field, by means of the distribution of residence times, i.e., the time interval between two consecutive changes of the sign of m.We found that there is an optimal temperature T * for which the response is maximum, that is, a stochastic resonance phenomenon induced by the external noise controlled by T .Also, we developed a mean-field approach that lead to a non-linear rate equation for the time evolution of m in the thermodynamic limit, whose numerical solution agrees very well with MC simulations of the model.We used this equation to give a numerical estimate of T c , and found that the behavior of T c with the period τ is qualitatively similar to that of T * .Although the transition temperature T c is similar to the resonance temperature T * only for large τ , this analysis shows that they are related.
A possible interpretation of these results in a social context is following.Reacting with a contrarian attitude occasionally (small T /low noise) on a given issue, that is, adopting an opposite position to that of the majority of our acquaintances, leads to a state of collective agreement in a population, which can be reversed completely after some time by means of a collective decision, independently of the external propaganda.This alternating behavior between opposite opinions might be seen as more "socially healthy" than a frozen full consensus in one of the two alternatives, which happens in populations with a total absence of contrarian attitudes (T = 0).However, having a contrarian behavior more often could induce a collective state where the mean opinion oscillates in time following the external propaganda, which can be interpreted as a society whose opinions are manipulated optimally by the mass media, in opposition to collective freedom.Finally, in the extreme case of having a very frequent contrarian attitude (T ≫ 1) the population falls into a state of opinion bipolarization, where there are two groups of similar size with opposite opinions.
The results presented in this article correspond to a fully connected network.Although we expect that the conclusions remain valid qualitatively for other interaction topologies, it might be worthwhile to study the model in complex networks like scale-free or Erdös Renyi networks, which better represent social interactions.It might also be interesting to explore how the stochastic resonance effect depends on the topology of the network.

FIG. 1 :
FIG. 1: (a) Time evolution of the average value of the absolute magnetization |m| in a population of N = 10 3 agents, zero field H = 0 and various values of majority probability p = (1 + e −2/T ) −1 , as indicated in the legend.(b) Stationary value of |m| vs T for constant fields H = 0.0 (circles), H = 0.1 (squares) and H = 0.5 (diamonds).The solid line is the analytical expression from Eq. (11), while the dashed lines are the numerical integration of Eq. (8).The averages were done over 10 3 independent realizations starting from a symmetric condition m0 = 0.

FIG. 2 :FIG. 3 :
FIG.2: Evolution of m in a single realization for a population of N = 1024 agents under an oscillating field with period τ = 256 and amplitudes H0 = 0.1, 0.5 and 1.0, panels (a), (b) and (c), respectively, and the temperatures indicated in the legends.Solid lines correspond to MC simulations, while dashed lines in panel (a) represent the numerical integration of Eq. (8).

FIG. 4 :
FIG. 4: (a) Response A as a function of the temperature T for a field of amplitude H0 = 0.1 and periods τ indicated in the legend.(b) Resonance temperature T * [maximum of A vs T curves from (a)] and transition temperature Tc vs period τ .

FIG. 5 :
FIG. 5: (a) Time evolution of the magnetization m from Eq. (8) for a field of amplitude H0 = 0.1, period τ = 256, and the temperatures indicated in the legend.Horizontal dashed lines represent the time average value of m, m, in the interval t ∈ (0, 1000τ ).(b) Time average of the magnetization, m, vs temperature T for the field's periods indicated in the legend.The inset shows a closer look around the transition values Tc.