Dynamics

We are interested in the equilibria reached by the system when agents meet and interact in successive periods. We model this as a process where agents may increase or decrease their persuasions after each interaction depending on whether the opponent has positive or negative opinion. These social interactions produce cumulative changes that can eventually lead to the change of opinion: a shift in opinion occurs when the persuasion exceeds a given threshold.

The interaction dynamics between agents is the following: whenever two agents i and j interact, the agents’ persuasion values C_i and C_j are modified by an amount bounded by kΔ, depending on the opinion of both individuals. This parameter, Δ, is a measure of how an agent changes the persuasion after a given interaction.

This social influence mechanism works by moving the respective persuasions of their existing positions to new positions depending on the opponent. This way the opinion shift is not based on the imitation of opinions of the neighbours (like the typical imitation behaviours), but is due to a cumulative persuasive dynamics.

The way in which the two social ingredients of the interacting dynamics (positive and negative social influence) is implemented, will produce an “anti-flocking effect”: Dissimilar individuals repel (caused by mechanism of negative influence) if they differ in opinions and approach each other if their opinions are similar (positive social influence). This last is in line with the fact observed by Wood [38] where individuals sharing a common attribute tend to get closer in opinions.

More formally, at each time step t, the states of the agents after interaction are updated according to the following rules, also illustrated in Fig 1(b):

• If agents share the same opinions or are both undecided, O_i(t) = O_j(t), and C_i(t) > C_j(t) then they approach (agents influence each other so that their persuasions become closer)
• If agents hold different opinions, O_i(t) = +1, and O_j(t) = −1 then they repel
  If one of the agents has an opinion and another one is undecided, they approach. In this case the dynamics is asymmetric simulating the fact that is not the same convincing someone who does not have any opinion yet that making someone change his opinion.
• If O_i = ∓1 and O_j = 0 then

We also implement that if two agent i and j interacts and ∣C_i − C_j∣ < Δ, then Δ = Δ_eff, where .

Summarising, the main parameters of the model are:

• Δ is a sensitivity parameter which measures how much the persuasion of an agent changes after each interaction. The smaller it is, the more interactions agent needs to get convinced.
• P₀ is the initial fraction of undecided agents.
• C_T is the threshold beyond which the agent is no longer undecided and adopts an opinion.
• k is a variable which simulates an asymmetric dynamics between an undecided agent and the one with a formed opinion. Values k > 1 imply that the persuasion of the undecided is modified by a factor k with respect to the other one.

Given the details of the interaction dynamics, we are interested in the following question: Do the interactions among the agents with different opinions bring the group to consensus or bi-polarization?

We present the results of simulations in the next sections. All the simulations are done for systems with N = 10000 agents. Results are averages over ensembles N_ev = 100 equivalent configurations, corresponding to different realisations of the random initial conditions.

Equilibrium States

We start with analysing the steady states T_i ≡ S_i(t_asint) as a function of P₀ and Δ. We vary 0.01 ≤ P₀ ≤ 0.99 and 0.01 ≤ Δ ≤ 1.0 with steps of 0.01. Equilibria P₀ = 0 and P₀ = 1 are already absorbing states, in both cases the interaction will not change the opinions of the agents. If initially all agents are undecided, they will remain undecided. On the contrary, if agents are equally distributed between the two extreme opinions, the distribution of opinions will remain unchanged.

Results are obtained with asynchronous updating where the procedure is iterated until convergence. The steady state is reached when T₀ = 1 or T₀ = 0, i.e. agents are either all undecided or all have an opinion.

As a result of simulations, the system converges to one of these three equilibrium states:

1. (a). T₀: All undecided (T₀ = 1).
2. (b). T₊ T₋: Consensus (either T₊ = 1 or T₋ = 1).
3. (c). T_bp: Bi-Polarization (T₀ = 0, 0 < T₊ < 1 and 0 < T₋ < 1).

Given the constraint S₀(t) + S₊(t) + S₋(t) = 1 (∀t), if T₀ = 0, the equilibria are either consensus of one of the extreme opinion (b) or bi-polarization (c).

Fig 2 shows the Fundamental Phase Diagram (FPD) that depicts existence of different regions of the system under equilibrium. In this Phase Diagram we analyse the prevalence of each equilibria for every pair of values of P₀ and Δ within the specific range.

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Fig 2. Fundamental Phase Diagram.

Dominant steady state solution as a function of P₀ and Δ. We identify three different regions: region T₀, convergence of undecided; region T₊ T₋, consensus of opinions +1/-1 and region T_bp, bi-polarization. In the simulations C_T = 1, k = 2, C_max = 3 were used.

https://doi.org/10.1371/journal.pone.0139572.g002

The Phase Diagram exhibits three different regions (see Fig 2):

1. T₀, delimited by large values of P₀ and small values of Δ, where the equilibrium state is characterised by convergence of undecided agents ( 〈T₀〉 = 1).
2. T₊ T₋, delimited by large values of P₀ and intermediate values of Δ, where the equilibrium state is characterised by consensus of either of the two opinions ( 〈T₊〉 = 1 or 〈T₋〉 = 1).
3. T_bp, delimited by low values of P₀ when Δ is small and by all values of P₀ when Δ is large, where the steady state is defined by a bi-polarization of opinion ( 〈T₀〉 = 0, 0 < 〈T₊〉 < 1 and 0 < 〈T₋〉 < 1).

The Fundamental Phase Diagram (FPD) can be understood in terms of the positive and negative social influence described before: interactions between agents sharing a common attribute (i.e., same opinion) force them get closer in persuasion C [38], bringing them to the center, meanwhile interactions between dissimilar agents, move them away from T₀. The result of these two competing opposing forces depends on the initial fraction of undecided agents P₀, and the sensitivity parameter Δ (see Fig 2).

When Δ is small, the situation is dominated by agents that change persuasions very little after pairwise interactions. Thus, many of these interactions are needed in order to produce a change in their opinions. Here, the fraction P₀ is important because it determines what will be the final state of this dynamics (all undecided ( 〈T₀〉 = 1) or none ( 〈T₀〉 = 0)).

Instead, when Δ increases, the preference for an agent to adopt extreme opinions (due to the asymmetry given by k, see Fig 1(b)) is more evident and is reflected in the fact that the more initially undecided agents are needed in order for the system to reach the final state with “all undecided”. As a consequence, the border of region T₀ grows monotonically with Δ until it encloses with P₀ = 1. When Δ is large, the final state is bi-polarization (region T_bp) independently of the initial density of undecided agents.

Consensus and Bi-Polarization

Given the global picture of the stable solutions, we look into the details at the dynamics which makes the systems reach these states. Furthermore, we restrict the discussion to regions T₊ T₋ and T_bp where the equilibrium states are consensus and bi-polarization, respectively, because region T₀ does not represent any interest neither from the social nor the individual point of view. For example, the statistics on undecided voters indicate that most individuals have pre-existing beliefs when it comes to politics, and relatively few ones remain undecided late into high-profile elections [39].

Let’s start with region T_bp, which deserves a closer look. Phase Diagram (see Fig 2) only gives information about this region regarding to whether is dominated by the coexistence of opposing opinions, but it does not clarify if one opinion prevails, or both exist in equal parts.

Fig 3 helps clarify these questions by showing that:

• For small values of Δ, both opinions exist, among which one is dominant and the other one is dominated. The degree of dominance depends on P₀. This can be seen in the diverging branches (Fig 3, black and red circles), which represent T₊ and T₋ for different simulations at different values of P₀ and Δ = 0.01. The distance between the branches follows a linear relation with P₀, meaning that in the final state, the undecided agents go to one or the other opinion with the same probability, given that initially they were equally distributed.
• For large values of Δ, the distribution of opinions is about 50% each, as it can be seen in Fig 3, panel (b).
• Given that initially neither of the two opinions is preferred, both are equally likely. This is reflected in the fact that, on average, either of 〈T₊〉 and 〈T₋〉 are 50% (Fig 3, solid lines in main panel), but their distributions are bimodal. An example of this bimodal distribution can be observed in Fig 3, panel (c) for P₀ = 0.20.

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Fig 3. Analysis of Bi-polarization Region T_bp.

Main Panel (a): Distribution of positive (T₊, black circles) and negative opinions (T₋, red circles) and their averages ( 〈T₊〉, 〈T₋〉 black and red thick lines respectively) as a function of P₀ for Δ = 0.01. The two branches in the bi-polarization regions mean the presence of dominant and dominated opinions for this value of Δ. Insets: Panel (b): Same quantities for Δ = 0.9. For large values of Δ, there does not exist a dominant opinion. Panel (c): Bimodal distribution of positive and negative final opinions for P₀ = 0.20 and Δ = 0.01.

https://doi.org/10.1371/journal.pone.0139572.g003

As was mentioned earlier, there are two competing mechanisms that act as persuasion forces and drive agents either towards the center (C = 0) or the extremes (C = ±C_max), giving place to the two symmetric solutions: all undecided (T₀) and bi-polarization (T_bp). However, there exist a region of space parameter, T₊ T₋, where all agents adopt, equally likely, either one of the two extreme opinions, involving a symmetry breaking in the dynamics. This kind of behaviour takes place for high values of P₀ and for intermediate values of Δ. In Fig 4, we plot the average fraction of agents with each opinion as a function of time, for a representative point of this region (P₀ = 0.80 and Δ = 0.55).

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Fig 4. Analysis of Consensus Region (T₊ T₋).

Averaged opinion dynamics for a representative point in region T₊ T₋ of the Fundamental Phase Diagram (P₀ = 0.80, Δ = 0.55). The plot shows the averaged evolution in time for each opinion dynamics (< S₊ >, black, < S₋ >, red and < S₀ >, green). It can be observed that undecided population grows until it reaches a value above 80% in the first time steps but finally one of the populations with defined opinions becomes dominant with the same probability. Evolutions are averaged over N_ev = 10000.

https://doi.org/10.1371/journal.pone.0139572.g004

In all the events the fraction of undecided agents, S₀, goes to zero after an initial transient time. On the contrary, the fraction of agents with opinion, S₊ and S₋, prevails with the same probability and therefore their averages, 〈S₊〉 and 〈S₋〉, go to 0.5 (see Fig 4 black and red lines respectively).

This behaviour can be understood if we retain that the change in agent’s opinion emerges from the interchanging dynamics of the persuasion variable C. In Fig 5, we plot the histograms of persuasion C, for different periods of the time dynamics of a single event, where all the agents reach consensus with opinion O = +1.

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Fig 5. Analysis of Consensus Region (T₊ T₋).

Evolution of Persuasion’s histograms for a single event. They show how undecided seems to win for times between 20 and 30, but the asymmetry in the persuasion distribution together with the mechanism described in the text makes all the agents adopt one of the opinions.

https://doi.org/10.1371/journal.pone.0139572.g005

In order to make clear that the final dominant opinion does not appear from any little difference in initial conditions, we choose, for this single event, a perfectly symmetrical initial condition: 8000 undecided agents (half with C ≥ 0 and the other half with the same values of C but with opposite sign), 1000 agents with O = +1 and 1000 with O = −1 and the same (but negative) values of persuasion variable C. The distribution of C within the interval corresponding to each opinion is constant, as can be seen in the histogram for t = 0. We can see that, after perfectly symmetrical initial conditions, some undecided agents with persuasion value close to threshold adopt ±1 opinions but then a vast majority of agents became undecided. This can be understood because, given the high proportion of undecided, these drive the agents toward the center (near C = 0). However these new distributions are not perfectly symmetrical, as can be appreciated from times t = 9 to t = 27. At this point of the evolution the dynamics is driven by interactions between decided (O = +1 in this case) and undecided agents near the threshold C_T. Here, we should remind that, when these types of agents interact, the decided agent changes its persuasion C by Δ while the undecided changes C by 2Δ. This asymmetry in interactions makes an initially undecided agent near the threshold to become “more decided” than an initially decided one near the threshold and the consequence after several interactions will be a drift from undecided to decided agents, producing the consensus in the population.

Summarising, the key of consensus in this model comes from this kind of interactions where an almost persuaded (but still undecided) agent interacts with a weakly decided one and, after the interaction, both adopt the same opinion but the initially undecided becomes more persuaded than the other one. This kind of mechanism describes situations where a new integrant of a group (in this case, those with O = +1) tries to act as a more authentic member of the group than genuine members of same group.

In order to analyse the role of the parameters on the solution of the model, we also explored the impact produced on the Phase Diagram when the threshold C_T and the asymmetry parameter k are modified for Δ = 0.01. For this value of Δ, the systems presents only two solutions: convergence of undecided agents or bi-polarization. Changing these two parameters does not modify qualitatively the solutions of the model.

Increasing the threshold makes the intermediate interval in the persuasion space, which defines agents to be undecided, to be larger. Less undecided are needed initially (at t = 0) in order to have more of them at the end. This moves the transition T_bp → T₀ slightly down (see Fig 6).

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Fig 6. Role of parameter C_T.

Regions of dominant solution as a function of P₀ and C_T for Δ = 0.01. This plot shows that in the low Δ region and with symmetric distribution of opposing opinions, the system evolves either to bi-polarization or convergence of undecided, depending on the initial fraction of undecided individuals, as have been seen in the Fundamental Phase Diagram (Fig 2).

https://doi.org/10.1371/journal.pone.0139572.g006

Instead, with increasing k, the tendency to adopt a definite opinion (+/ −1) after each interaction grows, and therefore, more undecided individuals are needed in order to achieve a final state where they predominate. Thus, this moves the transition slightly up (see Fig 7). When the asymmetry parameter is changed, the model moves between a pragmatic null “symmetric” model (k = 1) and a more extreme asymmetric case (k > 2).

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Fig 7. Role of parameter k.

Regions of dominant solution as a function of P₀ and k for Δ = 0.01. This plot shows that in the low Δ region and with symmetric distribution of opposing opinions, the system evolves either to bi-polarization or convergence of undecided, depending on the initial fraction of undecided individuals, as have been seen in the Fundamental Phase Diagram.

https://doi.org/10.1371/journal.pone.0139572.g007

How many undecided are relevant?

In real social situations, the value of P₀ depends on the underlying context, and it may be large or small. Clearly, if we consider examples of voting elections, then assuming very large values of P₀ is less feasible. It is hard to imagine a social system where there is a huge percentage of undecided voters. From the literature and web survey, this number is of orden 10–15%. In fact, voting must be considered carefully because the term “undecided” requires correct precision according to Gordon [40], and Galdi [41].

Instead, social examples of college decision (up to 50% of students enter college as “undecided” [40]) or the choice of major (an estimated 75% of students change their major at least once before graduation [40]) present situations where large values of P₀ are justified.

In the previous section we showed that, when the initial fraction of undecided agents, P₀ is large, the system presents three solutions as a function of Δ: convergence of undecided for small Δ, consensus of one of the two opinions for intermediate values of Δ and bi-polarization for large values of Δ. On the other side, when P₀ is small the equilibrium is only bi-polarization because it is driven mainly by the repulsive force (see Fig 2). Thus, we are interested in the next question: in systems with a relatively small fraction P₀ what should be undertaken in order to obtain the equilibria states observed previously.

We propose an alternative scenario for the interaction dynamics. Instead of implemented repulsive effect, allow the individuals with opposing opinions repel with some positive probability P_r, in a similar way it was treated in [32]. Also we analyse the effect of breaking the initial symmetry in the distribution of dissimilar opinions. Recall that we call P₊ the fraction of agents with O = +1 after the undecided are assigned. The concentration of undecided is fixed. The joint effect of these modifications produce an interesting result. The Phase Diagram for P₊ vs P_r (Repulsion probability) for initially low concentration of undecided agents and a small value of Δ (P₀ = 0.10 and Δ = 0.01) is presented in Fig 8.

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Fig 8. Alternative Phase Diagram.

Regions of dominant solution as a function of P₊ (Bias) and P_r (Repulsion probability) for initially low undecided concentration and small Δ (P₀ = 0.10 and Δ = 0.01). If P_r = 1, agents with opposing opinions always repel and the steady state is a polarized situation as we found in the Phase Diagram in Fig 2. When P_r < 0.5, it can happen that agents with opposite opinions get approached enough and the steady state depends on the bias to some opinion. If one of opinions initially prevails (P₊ > 0.68 or P₊ > 0.32), then the population will go to the consensus of this opinion. Otherwise, a convergence of undecided agents is reached.

https://doi.org/10.1371/journal.pone.0139572.g008

When P_r = 1, agents with opposing opinions always repel and the equilibrium is a polarized state as found in the Phase Diagram in Fig 2. When P_r < 0.5, agents sharing opposing opinions may get attracted with probability (1 − P_r), and the steady state depends on the bias to any opinion. If any of opinions initially prevails (P₊ > 0.68 or P₊ > 0.32), then the population will reach the consensus to this opinion. Otherwise, a convergence to “all undecided” for is reached.

Both scenarios for the interaction dynamics analysed here are interesting from the social point of view because, in turn, they correspond to the two different sides of the public debate: “Do we learn more from the people with opposing opinions of our own?”. On one side there is a view that we learn much more from people with similar opinions because we do learn more arguments fortifying that belief and take things as facts. On the other side, there is a view that we only learn if we look beyond: in a discussion with people with opposing thoughts we see the different points of view, the exchange of thoughts, etc.

Appendix: Theoretical Approach

In order to gain a deeper comprehension of the dynamics of this model, we present in this section the deduction of the master equations corresponding to the dynamics of the system.

Briefly, we present here a nonlinear system of nonlocal conservation laws, and let us remark that there are few models of this type even for a single equation. For instance, in the works of Deffuant, Neau, Amblard and Weisbuch, see [44, 45] in opinion dynamic models, among others, where only agents with similar opinions can interact, they obtained nonlocal terms involving a small neighbourhood of a given opinion and they simplify them performing Taylor expansions, recovering local equations of porous media or Fokker-Planck type. Of course, this is possible only in the frame of bounded confidence models, not enabling long range interactions.

Today, nonlinear equations are pervasive in theoretical and applied models, and there exist a growing literature on nonlocal problems. In opinion formation models, we can cite only the work of Aletti, Naldi and Toscani [46], where the mean value of the opinions m(t) appears in the transport term, here γ ∈ [−1, 1], and , although this term appears only as a mean field effect on the opinion. With a different motivation, inspired in the dislocation dynamic of crystals, Ghorbel and Monneau considered in [47] the following equation: and similar equations appeared in continuum mechanics in the theory of deformations and fractures, see for example [48] and the references therein.

It is worth noticing that there are few theoretical results and numerical methods for these problems, which are under active research. Let us mention that the limit as the nonlocal terms shrink to local ones gives an ill-posed problem, i.e., a porous media equation backward in time similar to the ones appearing in the Weisbuch-Deffuant type models. Recall that ill-posed problems lacks the continuity respect to small variations of the initial data, and this explains why the system is difficult to analyze from both the numerical and theoretical point of view. The study of the equations will appear in a separate work, due to the highly technical character of the analysis.

In order to obtain the partial differential equations describing the model, let us look at the system as composed by three populations, according the opinions of the agents. Lets recall than an agent with opinion O = +1 has a persuasion C ∈ [C_T, C_max], another with O = 0 has C ∈ [C_T, −C_T] and an individual with O = −1 has C ∈ [−C_max, −C_T]. We divide the interval corresponding to each opinion in M = Δ⁻¹ subintervals. Given the evolution of the persuasion according to the interaction-evolution rules detailed in previous sections, the best way to describe the dynamics of the model is in terms of the density of agents with a given persuasion. With this goal we define this density for 1 ≤ j ≤ M and t ≥ 0, which represent the fraction of agent of each opinion with persuasion in each interval of length Δ. In this way we can obtain a coupled system of 3M difference equations governing the evolution of the density of agents.

We call , , and recall that S_i(t) is the fraction of agents with opinion i, where i ∈ {+, −, 0}.

In the following, we omit the variable t in the right hand side of the equations for brevity. After some characteristic time τ, depending on the rate of the interactions, we have that the variation on the density of agents is given by the balance between gain and loss terms. For example, when O = −1, we have where the term G₋ corresponds to those agents located at j − 1 which interact with an agent with opinion O = +1 or an agent with opinion O = −1 and a stronger persuasion, plus those agents located at j + 1 which interact with an agent with neutral opinion O = 0 or an agent with opinion O = −1 and a weaker persuasion: On the other hand, the loss term corresponds to interactions between an agent located at j with another agent in any other location,

So, in this way we obtain the following system of equations: for 2 < j < M.

For j = 1,2 and M, the equations are slightly different, as for instance can be seen for s₋:

Up to here, we can see that the equations are rather difficult to study, but we can gain more perspective if we move from this discrete version to a continuous model. We can do that by introducing (smooth) functions u_i(x, t), i ∈ {0, +, −}, defined for (x, t) ∈ [0, 1] × [0,∞) such that and we can approximate the spatial partial derivative as (1) and the temporal partial derivative as

Also, We assume that τ = Δ, which corresponds to a time scaling of the rate of interactions.

After some algebra, the continuous version of the master equations reads as

The boundary conditions for s₋ are given by and the ones corresponding to u₊ are similar. The no flux boundary condition at x = 1 follows from the assumption that the persuasions are saturated at ±C_max. For u₀, there are two non zero Dirichlet boundary conditions similar to the one for u₋(0, t), reflecting the incoming agents with opinions O = ±1.

In this way we have obtained a nonlinear coupled system of first order differential equations of hyperbolic type including nonlocal terms and nonlocal boundary conditions.

Few remarks are in order:

• We have obtained a system of conservation laws, since the total mass of the solution is conserved, that is, for every t ≥ 0, Some mathematical properties of the solution, like positivity, seems difficult to prove, although the model clearly generates nonnegative solutions.
• The partial differential equation for each opinion has two competing terms: a coalescent one, depending on the own distribution u_i, which tends to concentrate the agents around the mean value of the opinion i; and the other one is a pure transport term, which drives the population to ±C_max, ±C_T depending on the densities of the other two populations, as was shown in previous sections.
• The coalescent terms changes signs, suggesting the existence of shocks (but perhaps they are smoothed by the transport term). Let us suppose that , and let us call μ_i(t) the median of the distribution u_i, i.e., We can rewrite the equation as and, for x ∈ (0, μ(t)), the characteristic curves travel from left to right, and for x ∈ (μ(t),1) they travel from right to left. Hence, we can expect the formation of a shock curve along the trajectory of μ_i(t).

We believe that they are out of the scope of this paper and deserve a lengthier discussion.