Modeling confirmation bias and polarization

Online users tend to select claims that adhere to their system of beliefs and to ignore dissenting information. Confirmation bias, indeed, plays a pivotal role in viral phenomena. Furthermore, the wide availability of content on the web fosters the aggregation of likeminded people where debates tend to enforce group polarization. Such a configuration might alter the public debate and thus the formation of the public opinion. In this paper we provide a mathematical model to study online social debates and the related polarization dynamics. We assume the basic updating rule of the Bounded Confidence Model (BCM) and we develop two variations a) the Rewire with Bounded Confidence Model (RBCM), in which discordant links are broken until convergence is reached; and b) the Unbounded Confidence Model, under which the interaction among discordant pairs of users is allowed even with a negative feedback, either with the rewiring step (RUCM) or without it (UCM). From numerical simulations we find that the new models (UCM and RUCM), unlike the BCM, are able to explain the coexistence of two stable final opinions, often observed in reality. Lastly, we present a mean field approximation of the newly introduced models.


Introduction
2][3][4][5] The wide availability of content on the web fosters the aggregation of likeminded people where the discussion tends to enforce group polarization. 6,7 onfirmation bias, indeed, plays a pivotal role in viral phenomena. 8Under such conditions public debates, in particular on social relevant issues, tend to further fragment and polarize the public opinion. 9,10  better understand this process, in this paper we provide a mathematical model mimicking polarization in online social dynamics.
Opinion dynamics, have been widely investigated in recent years, using different approaches from statistical physics and network science. 119][20] Besides the different assumptions and dynamics rules, for all the cited models the consensus state, in which all agents share the same opinion, is reached under the right conditions.However, consensus in far from common in real world and Internet based opinion exchanges.A recent study showed the emergence of polarized communities, i.e., echo chambers, in online social networks. 8Inside these communities, homogeneity appears to be the primary driver for the diffusion of contents.Both polarization and homogeneity might be the result of the conjugate effect of confirmation bias and social influence.Confirmation bias is the tendency to acquire or process new information in a way that confirms one's preconceptions and avoids contradiction with prior belief. 21Social influence is the process under which one's emotions, opinions, or behaviors are affected by others.In particular, informational influence occurs when individuals accept information from others as evidence about reality. 22,23 evious studies 24,25 proposed a non consensus opinion model (NCO) that allowed for the stable coexistence of two opinions by also considering the opinion of the user herself when applying the majority rule update, 24 while in 25 the competition between two groups is investigated by the introduction of a set of contrarians in one of the two.The survival of a two-opinions state is studied in 26 from a different point of view, considering the emergence of spontaneous recovery of failed nodes and the majority rule update.Both these models assume only two opinion states (±1) and a majority rule update, with the novelty of accounting for the individual opinion 24,25 and for an external source of influence. 26or the purpose of this study, we are more interested in capturing the emergence of two (or more) stable opinions out of a wide range of possibilities rather than the predominance of one over another.Hence, we will assume a continuous interval of opinions.Also we are inclined to assume peer-to-peer interactions rather than majority rule as a better proxy of face-to-face and online social interactions.A model grounded on the BCM and accounting for the interconnection and complexity of the online environment as well as the competition among sources of information is presented in. 27eople shape their opinions on the basis of both confirmation bias and social influence, a combination of these two forces generates the observed polarization of communities and homogeneous links. 8Accounting for this phenomenon, we build a model of opinion dynamics and network's evolution that considers both mechanisms and expands itself from the classical Bounded Confidence Model (BCM). 18We consider two variations of the model: the Rewire with Bounded Confidence Model (RBCM), in which discordant links are broken until convergence is reached; and the Unbounded Confidence Model, under which interaction among discordant pairs of users is allowed and a negative updating rule is introduced, either with the rewiring step (RUCM) or without it (UCM).
The paper is structured as follows.In the first section we provide references to the methods employed and give a brief overview of the BCM and its convergence results.In the Results and Discussion section, we first present the new models and give an account of the simulation results, then we present a mean field approximation of the newly introduced models.
The Bounded Confidence Model (BCM) The Bounded Confidence Model (BCM) 18,20 considers a set of N agents arranged on a complex network G. 1 Each agent holds an opinion x i , i ∈ {1, . . ., N}, uniformly distributed in [0, 1].Two agents interact if and only if they are connected in G and their present opinions are close enough, i.e. iff j ∈ N G (i) and 2 If these conditions hold, the two agents change their opinions according to rule (2), otherwise they do not interact at all: where µ is taken in the interval (0, 0.5).It is known from previous studies 31,32 that for ε big enough consensus is reached.The time rate change of P(x,t)dx, the fraction of agents whose opinion at time t lies in the interval [x, x + dx], is given by: The first two moments are given by M 0 = P(x,t)dx = 1 and M 1 = xP(x,t)dx = 0, i.e. the total mass and the mean opinion, are conserved. 31Let P(x, 0) = 1 be a flat initial condition, with x ∈ [0, 1].We are interested in the final state of the system P(x, ∞).When all agents interact, i.e., when ε ≥ 1 the rate equation is integrable. 3The second moment obeys Ṁ2 + M 0 M 2 /2 = M2 1 , and using M 1 = 0 and M 0 = 1 we find that M 2 (t) = M 2 (0)e −t/2 , hence the second moment vanishes exponentially in time, all agents approach the center opinion and the system eventually reaches consensus: 31 When ε ≥ 1 the final state is a single peak located in the middle and, as long as ε ≥ 1/2, this situation persists. 4For smaller values of the threshold ε, it has been shown, by numerical simulations, that consensus is not reached and the opinion evolves into clusters that are separated by a distance larger than ε.Once each cluster is isolated it evolves into a Dirac delta function as in the case ε ≥ 1.The final distribution consists of a series of non interacting clusters at locations x i with masses m i : where r is the number of evolving opinion clusters. 31All clusters must fulfill the conservation laws ∑ m i = M 0 = 1, and ∑ x i m i = M 1 = 0 is equal to the conserved mean opinion.All different clusters i = j must also fulfill |x i − x j | > ε.

Models
Starting from the BCM we introduce three new models of opinion dynamics and network evolution.The first model we consider is the Rewire with Bounded Confidence Model (RBCM) that considers the same framework as in BCM and involves two phases.In phase one we run the rewiring steps in which each agent i interacts with a randomly chosen neighbor j and, if the distance between the two opinions is above the tolerance ε, i.e. if 5 then their link is broken and i is rewired to a randomly chosen agent k ∈ {1, . . ., N}/(N G (i) ∪ {i}).Phase one ends when all links have an opinion distance below the tolerance ε.In phase two we run the BCM on the rewired network.As all the couples in the new network are concordant, all the randomly chosen pairs will interact and readjust their opinion according to the rule (2).
The Unbounded Confidence Model (UCM) is the second of the models that we introduce and its novelty is to allow the interaction for every randomly chosen pair of neighbors (i, j).To be specific, if two agents have concordant opinions, i.e. if |x i − x j | τ < ε, as for the previous model, we adjust x i and x j by rule (2).However, if their opinions are discordant, i.e. if |x i − x j | τ ≥ ε, we use a new updating rule (3) that enables us to replicate the empirically observed repulsion of discordant opinions: where µ is taken in the interval (0, 0.5) and ρ(.) is defined in (1).The adjustment ρ(.) ensures the PBC by maintaining the opinions inside the interval [0, 1].
The last model that we introduce is the Rewire with Unbounded Confidence Model (RUCM) that again allows the interaction for every randomly chosen pair of users (i, j) but at the same time allows for the random rewiring of discordant pairs.Specifically, if |x i − x j | τ < ε, then we adjust x i and x j by rule (2).If |x i − x j | τ ≥ ε, then we adjust x i and x j by rule (3), the link between nodes i and j is broken, and a new link between i and a randomly chosen user k ∈ {1, . . ., N}\(N G (i) ∪ {i, j}) is created.

Simulation Results
We perform Monte Carlo simulations of the BCM and the three new models on a scale-free network (SF) of 2000 nodes with the parameters (ε, µ) varying in the parameter space [0, 0.5] × [0, 0.5], for a maximum of 105 times steps or until convergence is reached.We averaged our results over 5 repetitions.Figure 1 shows the probability density functions (PDFs) of final opinion, after a maximum of 10 5 time steps, for four different combinations of the pair of parameters (ε, µ): (ε, µ) ∈ {(0, 0.05), (0, 0.1), (0.2, 0.05), (0.2, 0.1)}.The blue solid and the green dot-dashed curves refer to the newly introduced RUCM and UCM respectively, while the violet dotted curve is for BCM and the orange dashed for RBCM.For all the parameter choices we observe a bimodal opinion distribution in the cases of RUCM and UCM (note that we assume periodic boundary conditions).It is interesting to note that for UCM and RUCM there are two polarized opinions also for ε = 0, while in that case BCM and RBCM show no changes with respect to the initial uniform distribution. .In all figures the blue solid curve is for RUCM, the green dot-dashed one for UCM, the violet dotted one for BCM, and the pale orange dashed one for RBCM.We observe a bimodal distribution for RUCM and UCM, representing the coexistence of two polarized stable opinions.
Figure 2 reports a collection of summary statistics (mean, standard deviation, 1st quantile, and 3rd quantile) of the final opinion distributions for varying ε and three different values of µ (violet is for µ = 0.05, blue for µ = 0.25, and orange for µ = 0.5).The left column is for BCM, the central one for UCM, and the right one for RUCM.We omit the results for RBCM as we observe from the simulations that, after the rewiring steps, the dynamics are similar to the BCM case but with a faster convergence, refer to the Appendix for an in depth analysis of the RBCM model.We observe different mechanisms for the two newly introduced models, such as a faster convergence to the consensus state for RUCM.However, we need to study the final number of peaks to better characterize the differences between UCM and RUCM, and to relate them with the results for the classical BCM.

Final Distribution of Peaks
We perform Monte Carlo simulations of the BCM, UCM, and RUCM on a scale-free network of 2000 nodes with (ε, µ) ∈ [0, 0.5] × [0, 0.5], for a maximum of 10 5 steps or until convergence is reached (the results are averaged over 5 repetitions).

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Given the final distributions of opinions obtained by the simulations, we compute the number of peaks of opinions as the local maxima in the distribution of frequencies of opinions.To be specific, we divide the interval [0, 1] in 100 bins of length 0.01 and consider the frequencies of values falling in each interval.We regard two peaks to be separate if the distance between the middle points of the respective bins is smaller than 0.1.All the results are averaged over 5 repetitions.
Figure 3 shows the final distribution of peaks of BCM for varying (ε, µ) ∈ [0, 0.5] × [0, 0.5]. 6The final peaks distribution complies with theoretical 31,32 and simulation 18 results from previous work.Figure 4 shows the final peaks distribution of UCM (left) and RUCM (right) for varying (ε, µ) ∈ [0, 0.5] × [0, 0.5].For both models we observe a large area of the parameter space for which two final opinions coexist.We register a faster convergence to the consensus state for the RUCM (w.r.t UCM), that is due to the rewiring rule.Also, we observe that for the RUCM there is a direct transition from many opinions to two opinions, as well as from two opinions to consensus, while for the UCM there is an intermediate area where 3 or 4 opinions emerge, respectively shown in yellow and pale orange.
Comparing Figs. 3 and 4, we see that the new models, unlike the BCM, are able to explain the coexistence of two stable final opinions, often observed in reality.Another important difference with respect to the BCM is that the µ parameter assumes an important role in tuning the number of final opinions peaks.The dependence of the number of final peaks on the µ parameter is stronger for the RUCM, where we observe a clear transition from many opinions to exactly two on the diagonal.

Mean Field Approximation
For the RBCM, after the rewiring steps, all connected agents have an opinion distance below ε, meaning that they will always interact.The time rate of change of P(x,t) is equal to: Considerations analogous to the BCM case hold (see the Section Material and Methods).A faster convergence scale is also observed in the simulations.In the UCM and RUCM case we consider two updating rules: rule (2) if the opinions (x i , x j ) of the agents are close enough (|x i − x j | τ < ε) and rule (3) if they are not (|x i − x j | τ ≥ ε),.Thus the opinions will change according to (x i , x j ) → ( xi , x j ): where is the Heaviside theta function that equals 1 if ε − |x i − x j | τ < 0, 0 otherwise, and ρ is defined in (1).There are two ways in which the density of opinion x changes at every time step t: either an agent moves away from x after an interaction (I − ) or she arrives in x after an interaction (I + ).Let P(x,t)dx be the fraction of agents whose opinion at time t lies in the interval [x, x + dx], then its time rate of change is: The negative part is defined as in the BCM case but for a wider interval: as I − (x,t) is simply the probability that an agent with opinion x interacts with some other agent and thus moves away from x.For I + (x,t) we have two terms depending on the distance of the initial opinions: for the first term we get the same expression as in the BCM case:

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For I + 2 (x,t) we have to consider the negative update in (3), and the integrals are over the interval for which |x 1 − x 2 | τ ≥ ε: where ρ = ρ x 2 −x 1 .Hence we obtain: When all agents interact positively, i.e. when ε ≥ 1/2, the third term of the rate equation disappears and we are again in the BCM case, where consensus is reached asymptotically and: From simulations results, we notice that the final state is a single peak as long as ε ∈ (0.45, 0.5) for the UCM, or ε ∈ (0.3, 0.5) for the RUCM (with the exception of those points for which µ is near to zero).Unlike for BCM, in the new models the parameter µ plays an important role in the evolution of the distribution of opinions.For both UCM and RUCM we have the coexistence of two opinions in the final state for a wide region of the (ε, µ)-plane, this region varies for the two models, in particular the faster convergence to the consensus state for the RUCM is due to the rewiring rule.For smaller values of ε, and outside the two opinions region, we showed by numerical simulations that consensus is not reached, and many opinions at distance larger than ε coexist.

Conclusions
In recent years opinion dynamics has attracted much interest from the fields of both statistical physics and social science.In classical models such as the Sznajd model, the voter model, the majority rule model, and the bounded confidence model, consensus is eventually reached, under the correct conditions.However, in face-to-face and online opinion exchanges, consensus is not commonly achieved, and classical models fail to explain this empirically observed fact.
We propose a model of opinion dynamics capable of reproducing the empirically observed coexistence of two stable opinions.We assume the basic updating rule of the BCM and we develop two variations of the model: the Rewire with Bounded Confidence Model (RBCM), in which discordant links are broken until convergence is reached; and the Unbounded Confidence Model, under which the interaction among discordant pairs of users is allowed and a negative updating rule is introduced, either with the rewiring step (RUCM) or without it (UCM).
From numerical simulations we find that the new models (UCM and RUCM), unlike the BCM, are able to explain the coexistence of two stable final opinions, often observed in reality.Another important difference with respect to the BCM is that the convergence parameter µ assumes an important role in tuning the number of final opinions peaks; hence, in our model the speed at which opinions converge/diverge allows to change the final opinion landscape.Lastly, we derive a mean field approximation of all the three new models.12/12

Figure 1 .
Figure 1.Probability density functions (PDFs) of final opinion, after a maximum of 10 5 time steps or until convergence is reached, for four different combinations of the parameters (ε, µ).In the upper left figure we have (ε, µ) = (0, 0.05), in the upper right (ε, µ) = (0, 0.1), in the lower left (ε, µ) = (0.2, 0.05), and in the lower right (ε, µ) = (0.2, 0.1).In all figures the blue solid curve is for RUCM, the green dot-dashed one for UCM, the violet dotted one for BCM, and the pale orange dashed one for RBCM.We observe a bimodal distribution for RUCM and UCM, representing the coexistence of two polarized stable opinions.

Figure 2 .
Figure 2. Summary statistics (mean, standard deviation, 1st quantile, and 3rd quantile) of the final opinion distributions for varying ε and three different values of µ: violet denotes µ = 0.05, blue denotes µ = 0.25, and orange denotes µ = 0.5.The left column is for BCM, the central one for UCM, and the right one for RUCM.

Figure 3 .
Figure 3. Final distribution of peaks for the BCM, with varying (ε, µ) ∈ [0, 0.5] × [0, 0.5].The Monte Carlo simulations are carried on a Scale-Free network with 2000 nodes for a maximum of 10 5 steps or until convergence is reached (all results are averaged over 5 repetitions).

Figure 4 .
Figure 4. Final distribution of peaks for the UCM (left) and RUCM (right), with varying (ε, µ) ∈ [0, 0.5] × [0, 0.5].The Monte Carlo simulations are carried on a Scale-Free network with 2000 nodes for a maximum of 10 5 steps or until convergence is reached (all results are averaged over 5 repetitions).

Figure 6 .
Figure 6.Final distribution of peaks for the RBCM, with varying (ε, µ) ∈ [0, 0.5] × [0, 0.5].The Monte Carlo simulations are carried on a Scale-Free network with 2000 nodes for a maximum of 10 5 steps or until convergence is reached (the results are averaged over 5 repetitions).