Strategies for an efficient official publicity campaign

We consider the process of opinion formation, in a society where there is a set of rules, $B$. These rules change over time due to the drift of public opinion, driven in part by publicity campaigns. Public opinion is formed by the integration of the voters' attitudes which can be either conservative (in agreement with $B$) or liberal (in agreement with peer voters). These attitudes are represented in the phase space of the system by stable fixed points. In the present letter we study the properties that an official publicity campaign must have in order to turn the public opinion in favor of $B.$


Introduction.
In the present letter we analyze what impact different official publicity strategies have on the opinion-formation process, on a population of interacting agents. We assume that the agents, or voters, live in a working society, i.e. a society in which there exists a set of rules B that determine the acceptibility of a social issue. B can be thought as laws, social conventions, or otherwise that fix a reference for what is considered normal social behavior [1][2][3]. B also represents what we call society's official position.
Opinions are highly dynamic mental representations of individuals' beliefs, resulting from processes of inference frequently done with insufficient information. They play a fundamental role in individuals' reaction to social situations that can trigger collective responses [4][5][6]. To model this process of opinion formation in a community of interacting voters, we start by modeling the voters by adaptive agents, each one of them provided with a simple neural network that endows them with the capacity to learn from the social reference B and from each other. Agents interact with neighbors, with whom they are connected according to a directed graph [7][8][9]. The combination of these two sources of disorder, introduced through the set of examples for the learning process, and through the graph that fixes the set of connections, produce a very exiting model with predictive capabilities.
By modeling a publicity campaign using a periodic perturbation, we can analyze the strategy (represented by the amplitude and frequency of the perturbation) is most adequate to change the public opinion in favor of the official position. The relevance of the present studies can be easily exemplified. The campaign for the 2016 UK referendum was based on incomplete or unreachable information: internal polls showed that 85% of the British population wanted more information from the Government. It also consumed vast amounts of resources (Vote Leave, the official leave campaign, obtained the right to spend up to £7,000,000, a free mailshot, TV broadcasts and £600,000 in public funds, whereas the official position of Government was backed by a £9,300,000 campaign [10]), and produced immediate effects. Understanding a process that consumes this quantity of resources is paramount.
We start with the description of our model by assuming that agents {a} M a=1 form opinions {σ a } M a=1 on social issues S that are presented to them. We also assume that the social issues trigger a strong response from the agents, thus the opinions can be modeled by a binary variable, i.e. σ a ∈ {±1} [11]. Social issues can be codified as binary vectors S ∈ {±1} N with N sufficiently large. The way the reference B and the agents {a} produce an opinion on a given issue S is by processing such an issue through the neural network they have been provided with. In order to balance the level of sophistication of the model with its analytical tractability, we provided the agents and the reference with a perceptron [12]. Each perceptron is characterized by an internal representation vector (B ∈ R N for the reference J a ∈ R N for the agents) such that the opinions become σ B (S) = sgn(B · S) and σ a (S) = sgn( Both reference B and agents {a} evolve over time according to a learning algorithm. Assuming that the population of interacting agents receives information drawn from S ≡ {(σ B,n , S n ), n = 1, . . . , T }, we implement the following Hebbian algorithm [13] for the agents: where N −1/2 |J a,n | ∼ O(1) is a factor that has been only considered for technical purposes [14], the factor in parenthesis represents the learning rate of the algorithm which balances the importance f given by a to the opinion of B, with the importance g a,c given by agent a to its neighbors, placed in the neighborhood N a = {c : 1 ≤ c ≤ M, and g a,c > 0}, and where the last factor is a unit length vector pointing in the direction of S n if S n is socially acceptable (σ B,n = 1), and in the opposite direction otherwise. The construction of the learning rate is such that if agent a agrees with B on issue S n (i.e. σ a,n = σ B,n ) then the internal representation of a grows in the direction of B, whereas if σ a,n = σ B,n and the integrated contribution from the agreeing neighbors (i.e. Θ(σ a,n σ c,n ) = 1, where Θ(x) = 1 if x > 0 and 0 otherwise) is larger than f then the internal representation of a grows opposite to B. Observe that in algorithm (1) there is an implicit interaction between the disorder introduced through the training set S and the graph The internal representation of the reference B evolves according to the algorithm [3]: where λ n is the factor that controls the speed of change in the social position, and the population average is over the components of the vectors J c,n perpendicular to B n . Such a modification to the internal representation of B is such that the new internal representation B n+1 is on a direction closer to the average of the population with a length that remains unchanged (i.e. |B n+1 | − |B n | ∼ O(f 2 N −1 )). This algorithm mimics the process of a social reference moving towards the direction of the public opinion.
By defining the overlap R a ≡ (|J a ||B|) −1 J a · B, which is a self averaging quantity [15], it is possible to proof (see the full details of the derivation in Reference [3]) that for sufficiently large systems (i.e. N → ∞) the evolution of the overlap R a is given by the equation: where η a,c ≡ lim f →0 f −1 g a,c are the social strengths, and θ a ≡ arccos(R a ). The quantity R a represents the average agreement of agent a with the reference B, and the phase θ a is the angle between the internal representations J a and B. Observed that if all agents have, in average, the same number of connections ν ≡ M −1 M a=1 |N a |, and the social strengths {η a,c } are drawn from a narrow distribution with mean η, the (mean field) evolution of the overlap R a becomes: where we have assumed that . These model considers that variations in the evolution of the social rule B are mostly constant and proportional to' λ o , perturbed with a periodic wave of amplitude λ o A and frequency ω. This perturbation represents a bounded publicity campaign in favor of B's position (i.e. 1 ≥ v(ωt) ≥ 0 for all t), thus pushing the average agreement of a with B towards 1 [16]. With this model we can express the right-hand-side of Equation (4) as the sum of three terms: a) an autonomous term that can bee expressed as minus the gradient of a potential −∂ R V (R), b) an interaction with the neighborhood N a , and c) a periodic perturbation. It has been observed in [3] that there are four roots to and R = −R r and R = 1 are the (liberal and conservative) stable points. There is a particular value of the average social strength η o such that both stable points become energetically equivalent, i.e. V (1) = V (−R r ). By numerical calculations we found out that the bi-stability condition is satisfied when (1), and thus R r = 0.454754(1).
The objective of our investigation is to study the effects of a periodic perturbation to change the opinion of the voters in favor of the reference B. In such a case we can suppose that the agents have their initial conditions set into the basin of attraction of the liberal stable point, i.e. R a (0) ∈ (−1, R r ). For such a case we can transform the Equation (4) into: where θ a (0) ∈ (θ r , π), θ r ≡ arccos(R r ) = 1.0987 (1). By re-scaling the time where The first term of the right-hand-side of (6) is the average interaction over the neighborhood of the agent, the second term is a perturbation mainly proportional to the rate of change of the social rule B.
The perturbation term has two contributions, one autonomous and one time dependent, proportional to the constant A. A can be seen as the amount of resources needed to change a liberal agent into a conservative one. Observe that for every neighborhood, there must be an agent m such its phase is the Such an agent has a phase equation of the form: The associated homogeneous equation to (7) has a solution of the form: where π − θ r and θ r are the (stable and unstable) fixed points corresponding to −R r and R r respectively.
Observe that for all initial condition θ m,h (0) ∈ (θ r , π) the solution to the homogeneous equation asymptotically approaches the stable point π − θ r . Observe also that the interaction term in (6) is zero only if θ a = θ m . If the interaction is not zero, and thus negative, the derivative of θ a becomes negative and θ a is pulled closer to the value of θ m . In consequence, if the perturbation Av(ωt) is sufficiently strong to pull m into a conservative attitude [i.e. 0 < θ m < θ r ], the other phases are attracted into the conservative basin (0, θ r ) too. The hypothesis we will work with is that the agent with the smallest initial phase will keep this quality during the time evolution of the process, and in this form to know whether the perturbation is strong enough to pull the agents into the conservative basin we only need to know if the perturbation is strong enough to change the attitude of the agent with the smallest phase.
If the M agents in the population have been given initial conditions drawn randomly from a uniform distribution in (θ r , π), it can be proven that the expected initial value for the minimum phase is Given that the expected initial condition for the agent with the smallest phase is close to the lower bound of the liberal basin θ m (0) = θ r + cM −1 , and according to equation (8) the phase should not exceed π − θ r , we can approximate (7) by: where we have re-scaled the time and frequency such that Λt → t and ωΛ −1 → ω, and where Ω c ≡ (1) is the characteristic frequency of the system and Φ r ≡ π − 2θ r = 0.9442(1).
For equation (12) the perturbative solution is such that at short times the phase (10) becomes θ(t ≪ ω −1 ) = π − θ r − Lω 2 where 0 < L ∼ O(1). This indicates that in the low frequency regime the phase becomes very close to the liberal stable point π − θ r in a short time. Changes in the agents' attitude are seen only at later times, when the perturbation (publicity) becomes sufficiently strong. At those times we have that the equation that rules the dynamics of the system is (13), where the perturbation behaves linearly in ωt. Thus by considering a perturbative expansion as a solution of (13) with the initial condition θ(τ = 0) = π − θ r , we have that the minimum amplitude A c needed to take the phase θ from the stable point θ(τ = 0) = π − θ r to the unstable point θ(τ > 0) = θ r is: At high frequencies Ω c ≪ ω, the number of cycles cover by the perturbation during a characteristic time of the system is ωΩ −1 c ≫ 1, thus we can substitute v(ωt) by its average over a period, i.e. v ≡ (2π) −1´2 π 0 dz v(z), in equation (11), thus the Schrödinger equation at high frequencies becomes: with the initial condition: The minimal value of the perturbation's amplitude A c that ensures that the phase reaches the unstable point θ r at t > 0 for high values of the frequency ω is: which depends on the value of the phase at t = 0 but it is independent of the frequency.
Observe that the behavior of the critical amplitude at low and high frequencies, equation (14) and (17) respectively, are such that no interpolation to intermediate values of the frequency are meaningful. To illustrate the case we will explore the particular case of a perturbation v(ωt) = sin 2 ωt 2 that can give us the Schrödinger equation (11) that can be transformed into Mathieu's Equation ψ ′′ (x)+[a−2q cos(2x)]ψ(x) = 0 [20,21], with variable x = ωt/2 and parameters a ≡ 4Ω The general solution to the Mathieu's Equation can be expressed as a linear combination of even M c (a, q; x) and odd M s (a, q; x) Mathieu's functions [21], such that ψ(x) = C c M c (a, q; x) + C s M s (a, q; x). Given that the equation of the phase (9) is of the first order, we expect the solution to present only one free constant (that can be adjusted through the particular initial conditions). Thus The critical amplitude as a function of the frequency ω is A c (ω) = min A ω . We observed that for values of A < A c the Schrödinger's wave function is different from zero for all 0 < t, whereas for A ≥ A c , there By analyzing the eigenvalue of the Schrödinger's equation (11) at the critical amplitude ε c (ω) ≡ we observe that for sufficiently low frequencies the critical amplitude (14) is such that ε c (ω ≪ Ω c ) > 0 and for sufficiently high frequencies and sufficiently large systems, i.e. O(1) ∼ 4c(Φ r v) −1 < M which is a very mild assumption, the critical amplitude (17) is such that ε c (ω ≫ Ω c ) < 0. Therefore we define the critical frequency of the system ω c the frequency at which the eigenvalue of the Schrödinger equation becomes zero, i.e. A c (ω c ) = κ o − 1.
We have observed that in the high frequencies (ω ≫ Ω c ) regime the critical amplitude depends on the size of the system through the initial conditions. Thus, we computed the curve ε c (ω) for systems sizes To obtain the perturbation's critical amplitude we have reduced the system represented by the set of equations (6) to the study of the single equation correspondent to the smallest phase (7) by assuming that the agent with the smallest phase (the most conservative of all agents) remains the same through all the dynamical process. To test this assumption we performed numerical integration of systems of differential equations, with sizes M = 5, 10, 15, 20, 25, 30, 35, 40, and a sinusoidal perturbation. By applying a second order Runge-Kutta method we integrated the trajectories in the intervals t ∈ (0, 10π/ω), where ω is the frequency of the perturbation. The agents were assigned initial phases θ a (0) drawn from a flat distribution θ a (0) ∈ (θ r , π), and the critical amplitude was found by applying a bisection method. The results are presented in figure 3. The first feature we observe from these curves is that all collide to the same curve for small values of the frequency ω ≪ Ω c . The linear, least-square fit of the data A c.Low (ω) = A 0 + A 1 ω produces an intersect A 0 = 0.123(1) indistinguishable from (κ o − 1) and a slope A 1 = 2.379(1) that, by applying equation (14) corresponds to a time t ′ = 1.074(1)ω −1 . Both results are consistent with equation (14) and with assumption t ′ ≈ ω −1 leading to equation (13). Observe that the range of frequencies covered in figure 3 is bellow Ω c . We did not managed to obtain meaningful results for the high frequency regime, due to the technical difficulty associated to find zeros of highly oscillating functions. Even so, the numerical analysis of the results presented a tendency A c,High ∼ O(M −0.7 (3) ) which is consistent with equation (17).
Observe that the error bars for the low and high-frequency regime behave very differently. Error bars were computed by integrating 100 realizations of each system of differential equations (6). For highfrequencies the estimated error associated to each data point becomes one order of magnitude less than the amplitude A c itself [O(10 −1 A c )], whereas for the low-frequency regime, the error associated to each data point is negligible. The difference in behavior is due to the fact that for higher frequencies the perturbation effectively acts at very short times, t ≪ Ω −1 c , thus the noise introduce through the initial conditions has an impact in the results. At low values of the perturbation frequency all phases have time to relax towards the stable point π − θ r , thus for the time when the perturbation is strong enough to produce changes in the system (Ω c ≪ t ′ ), all phases are almost identical θ a (t ′ ) = π − θ r − ε a , with 0 < ε a ∼ O(10 −6 ). Thus the estimate of the variance computed from different realizations of the system is almost negligible.  By imposing mild conditions on the perturbation v(z), i.e. v is twice differentiable and bounded, we managed to reduce the the analysis of the system of differential equations (6) to the analysis of the equation (7) that rules the evolution of the smallest phase θ m = min{θ a ≡ arccos(R a )}. By applying a quadratic approximation to (7) we obtained the Riccati equation (9), which admits an exact solution (10). Such a solution is linked to the solution of the Schrödinger equation (11) that describes the behavior of an electron in a periodic potential. By exploring the behavior of the solution of the Schrödinger equation (11) for values of the perturbation's frequency ω much larger (smaller) than the characteristic frequency of the system Ω c = 0.260(1), we estimated the value of the minimum perturbation's amplitude A c needed to move agents with liberal attitude [i.e. with phases θ in the basin (θ r , π)] into the conservative basin (0, θ r ), as a function of ω. We observed that for very low frequencies, the critical amplitude A c,Low (ω) is a linear function of ω, equation (14), whereas for high values of ω the critical amplitude strongly depends on the initial conditions θ m (0). Given that the initial conditions of the system with M agents are drawn from a uniform distribution in the interval (θ r , π), the expected value of the minimum phase is θ r + cM −1 with c ∼ O(1). Thus, we have obtained that A c,High ∝ M −1 .
To validate our results we performed a number of numerical integration of the set of equations (6), for system sizes M = 5, 10,15,20,25,30,35,40, and for a periodic perturbation of the form v(z) = sin 2 (z).
For this particular case, the Schrödinger equation (11)  In summary, our model indicates that if the government desires to regularly perturb the population of voters with publicity campaigns, it is more profitable (for the government) to do so with a frequency higher than the characteristic frequency of the system Ω c . In doing so, the amplitude of the oscillation decays with the size of the population A c,High ∝ M −1 ,whereas for low frequencies ω ≪ Ω c the amplitude is always larger than a minimum value A c,Low > κ o − 1.

AKNOWLEDGMENTS
The author would like to acknowledge the constructive discussions with Dr. R. C Alamino and Dr I.
Yurkevich. The advise of Dr. C. M Juarez is kindly appreciated.