Depolarization of echo chambers by random dynamical nudge
Christopher Currin, Sebastian Vallejo Vera, Ali Khaledi-Nasab
DDepolarization of echo chambers by random dynamical nudge
Christopher B. Currin and Ali Khaledi-Nasab ∗ Department of Human Biology, Faculty of Health Sciences, University of Cape Town, South Africa and Department of Neurosurgery, Stanford University School of Medicine, Stanford, CA 94305 (Dated: January 12, 2021)Interactions among individuals in social networks leads to echo chambers where the distributionof opinions follows a bimodal distribution with two peaks at the opposite extremes. In issues withclear answers, such as global warming, one of the echo chambers reflects an inaccurate judgment,potentially from misinformation. However, in issues without clear answers such as elections, theneutral consensus is preferable for promoting discourse. In this letter we use an opinion dynamicsmodel to study the effect of a random dynamical nudge where we present random input to eachagent from the other individuals in the network. We show that random dynamical nudge disallowsthe formation of echo chambers and leads to a normal distribution of opinions centered around theneutral consensus. The random dynamical nudge relies on the collective dynamics and it does notrequire surveillance of every person’s opinions. Social media networks could implement a version ofthis self-feedback mechanism to prevent the formation of segregated online communities on pressingissues such as elections.
People have the propensity to form binary opinions(good or bad) quickly through social influences, with ma-jor implications [1–5]. This heterogeneous partisanship isobserved in pressing issues such as global warming, healthcare, gun control, presidential elections, street protests,and many other issues where the population’s opinionsdivide into two distinct clusters divergent from the neu-tral consensus [1–5]. If the segregation of opinions isreflected in users’ interaction, then social echo chambersemerge [6]. Echo chambers that show a deviation fromthe neutral consensus might contribute to the spread ofmisinformation where it spreads faster than actual infor-mation [7, 8]. Heterogeneity in a social group could helpto improve the collective accuracy [9].Motivated by activity driven models [10], a recent mod-elling study on echo chamber formation was validatedwith Twitter data from multiple issues such as Oba-macare and gun control [11]. The model gave rise toecho chambers as in social media, representing a dividein opinions on pressing issues.During circumstances without clear answers, havingnormal distributions of opinions is favorable. This meansmany opinions are around the neutral consensus. Radi-cal ideas on either end contain a degree of misinforma-tion, which might prove catastrophic as some individualsmight act based on these opinions and create tensions[12]. Here we propose a mechanism to avoid the for-mation of echo chambers by presenting each agent withinput from a random selection of other agents’ opinions.This mechanism, which we term the random dynamicalnudge (RDN), pushes a system of divergent opinions to-wards a normal distribution with most of the views closeto neutral.To study the effect of RDN, we use an opinion dynam-ics model. For a system of N agents, each agent i hasan evolving opinion x i ( t ) ∈ [ −∞ , ∞ ]. For a given issue, ∗ [email protected] agent i has a stance with sign σ ( x i ) and a convictionwith strength | x i | . Strong convictions correspond to oneof two extremes. Agent opinions change based on theirinteractions with other agents A ij ( t ), the strength of so-cial interactions K >
0, and the controversialness of theissue α > x i = − x i + K N (cid:88) j =1 A ij ( t ) tanh ( αx j ) + D √ n ( (cid:104) X n (cid:105) − (cid:104) X (cid:105) )(1)Where D is the RDN strength, (cid:104) X n (cid:105) is the mean from asample (of size n << N ) of opinions, and (cid:104) x (cid:105) is the truemean of all opinions. A ij ( t ) is the temporal adjacency matrix, which repre-sents interactions between agents [13]. A ij ( t ) = 1 whenthere is an input from agent j to i and A ij ( t ) = 0 other-wise. If an agent with a set activity level a i ∈ [ ε,
1] is ac-tive at time t , then they will interact with m other agents,weighted by the probability p ij that agent i would con-nect with agent j . These interactions are captured by thetemporal adjacency matrix. The probabilistic reciprocityfactor r ∈ [0 ,
1] determines the chance that a connectionis mutually influential, ( A ij ( t ) = A ji ( t ) = 1).If the interaction is reciprocated, both agents updatetheir opinions; otherwise, only one of the agents’ opinionsis updated. This can lead to the polarization of opinions,where most agents hold a moderate stance on a binaryissue and few, if any, agents have a neutral opinion.Ifactive agents have an equal chance of interacting with m other agents regardless of their stance, then the networkcan become radicalized, with all agents having the samestance.The probability distribution of activities follow a powerlaw decay F ( a ) = 1 − γ − ε − γ a − γ (2)where γ = 2 . a r X i v : . [ phy s i c s . s o c - ph ] J a n of the activity probability distribution and ε = 10 − isthe minimum activity. Most agents will have a low ac-tivity and a few agents will have a high activity. Mostagents with low activity have little conviction; in con-trast, active agents have greater conviction [11].We define the connection probabilities as a function ofthe magnitude between two agents’ opinions: p ij = | x i − x j | − β (cid:80) j | x i − x j | − β (3)Where β is the homophily factor, the tendency foragents with similar opinions to interact with each other: β = 0 refers to no interaction preference, and β > ε, γ, m, β ). Together withthe issue parameters ( K, α ), the network of agents tendsto a transient polarization and forms echo chambers. Forcontroversial issues with strong proponents on either side,the network eventually goes into a radicalized state. Thecritical controversialness α c (cid:39) ((1+ r ) · K · m ·(cid:104) α (cid:105) ) − is thetransition value above which a network will be polarizedor radicalized [11] .In this letter, we consider a random dynamical nudge (RDN) term in Eq.(1), which acts as a self-feedbackmechanism. The RDN emulates a social network showingeach person how a random sample of peers’ opinions dif-fered from the overall mean of opinion. That is, at eachtime step dt , we randomly sample n opinions, find themean (cid:104) X n (cid:105) , and subtract it from the overall mean of opin-ions, (cid:104) x (cid:105) . The term is based on the Central Limit Theo-rem, which tends towards a normal distribution with in-creasing sample size. In a dynamical system, the discretesample size n is surrogated by repeatedly computing theterm at each dt . We show that variations of this termproduce similar depolarization of echo chambers; henceone can select the proper RDN given the issue.The motivation for RDN comes from research in be-havioral economics that showed how subtle nudges couldsignificantly affect individuals’ behavior even when theyknow that they are receiving a random nudge. One exam-ple is the Anchoring Effect, where a random input skewsthe individuals’ judgment toward itself [15–18]. Integrat-ing epistemic cues - an educative nudge - can lead to moreinformed choices [18].The addition of an RDN term shows a shift in opinions’from a bimodal to normal distribution. Fig. 1 clearly in-dicates that even with an RDN strength of 1, there aremany more neutral opinions than without RDN ( D = 0).With an RDN of D = 3, the opinions are normally dis-tributed, with a mean of 0. However, stronger RDNscreate an undesirable paradoxical effect of increasing the P ( x ) Λ x Λ x P ( x ) P ( x ) Λ x −5 x P ( x ) n −5 x −5 ⟨ x ⟩ NN D = 0D = 1D = 5D = 3 −5 ⟨ x ⟩ NN (a) −5 ⟨ x ⟩ NN −5 ⟨ x ⟩ NN −5 ⟨ x ⟩ NN (b) n = 10 Λ x FIG. 1.
Densities of opinions showing the depolariza-tion effect of the random dynamical nudge
Each rowshows the distributions of opinions (a) or the agent’s opin-ions versus the mean of the agent’s nearest neighbours (b) fordifferent RDN strengths ( D ∈ { , , , } ). Varying the sam-ple size n did not significantly change distributions.The RDNcauses the opinions to be predominantly neutral ( ≈
0) andprevented the formation of echo chambers, but large RDNvalues (e.g. D = 5) also have more agents with extreme opin-ions. The heat maps in (b) are for a n of 10 only. Otherparameters were N = 1000, K = 3, β = 3, α = 3, m = 10, r = 0 . T = 5. number of opinions with strong convictions, i.e., increas-ing the standard deviation of the opinions, which effec-tively flattens the curve. Hence the strength RDN, D ,is an essential factor in the effectiveness of RDN. Fortu-nately, the effect of sample size n is negligible.To quantify the effect of the RDN on the opinion distri-bution, we use the distance between the polarized peaksof the distribution, Λ x . Formally,Λ x = argmax x> fw − argmax x< fw (4)Where f is the frequency of opinions in a bin width of w . w was determined from the minimum of the Sturges[19] and Freedman-Diaconis [20] bin estimation methods w = min( max x R − min x R log N +1 , IQR N ) where x R is an opinionsubset ( x > x <
0) and IQR is the interquartilerange of the subset. The peak distance Λ x can be intu-itively understood as the degree of polarization of echochambers. For Λ x close to 0, the distribution of opinionsis normal and depolarized, and for larger Λ x , the opinionsof agents are polarized.Given this metric, we ran an extensive number of simu-lations to determine the influence of sample size n on Λ x .We found that Λ x is largely unaffected by n beyond theusual trial-by-trial variation (Fig. 2). This finding wasrobust to different α , β , K , and D . Next, we sought todetermine how sensitive the dynamics were to the RDNstrength of D . FIG. 2.
The RDN sample size n has a negligible effecton the polarized peak distance Λ x . Although systemat-ically varying the RDN strength D had a profound effect onpolarized peak distance Λ x , the sample size n did not. Eachpanel varies either α , β , or K from the default parameters of α = 3, β = 3, K = 3, N = 1000, m = 10, r = 0 .
5, and T = 5.Each panel consists of 50 x 50 (1 < = n < = 50, 0 < = D < = 5)simulations which were up-sampled and smoothed for visual-isation.
Overall, increasing RDN strength D tends to decreaseΛ x up to an optimal strength that depends on α and K (Fig. 3). More controversial systems and those withstronger social interactions require a stronger RDN; D therefore needs to be considered on a case-by-case basis.Note that homophilic networks β > x .The current RDN formulation in Eq.1 relies on the truemean (cid:104) x (cid:105) , something that would be hard to measure inpractice. Furthermore, n appears to be a potential areaof optimization due to its minor influence on RDN dy-namics, see Fig.2. To investigate a more practical imple-mentation of RDN by stripping it to its bare essentials,we ran extensive simulations with different RDN terms(Table I).In all setups, adding the RDN was beneficial in de-polarizing the echo chambers, and given the optimumvalue of D , leads to a normal distribution of opinions.The critical part of the RDN was that each agent x i wasshown another random agent’s opinion with a different FIG. 3.
Assessing the sensitivity of opinions to aRDN using the trial-averaged polarized peak distance (cid:104) Λ x (cid:105) . The scatter plots in each panel shows the polarizedpeak distance Λ x averaged over sample size (1 < = n < =20). Columns indicate different values of α . Rows indicatedifferent values of K . Colors indicate different values of β .Other parameters were N = 1000, m = 10, r = 0 .
5, and T = 5. weighting to the usual interaction specified in the origi-nal equation. The effects of a homophilic system ( β > K = 3 fora controversial topic ( α = 3). Remarkably, no extra re-strictions were required, such as requiring reciprocity orbeing of the opposite stance to depolarize the agents’opinions.The results so far have shown how an RDN can pre-vent the polarization of systems that start with randomuniformly distributed opinions. Hence, we assessed theefficacy of an RDN in depolarizing a system that has al-ready formed echo chambers (Fig. 4). We allowed a sys-tem to become polarized until t = 10 and then appliedan RDN strength of 3 for an equal time (Fig. 4a). Wealso examined the after-effects of applying an RDN byremoving it again. By examining the population opiniondistribution at different time points (Fig. 4b), we foundthat adding the RDN after a system has become polar-ized does depolarize it. Removing the RDN does causethe system to regress, so RDN needs to be present. RDNwith adequate parameters depolarizes the echo chambersand leads to a normal distribution of the opinions.We argue that people form echo chambers due to thelimited connection they develop early on. RDN ex-pose individuals to new insights and has the potentialto nudge the collective opinion dynamics toward modera-tion [16, 18, 21]. RDN is agnostic to each agent’s opinion,and it adds a new perspective by providing input fromout of the immediate circle. This is like presenting a sam-ple of exotic and unfamiliar food to an individual; theymight like it or at least a particular flavor. Based on our TABLE I. Different RDN terms yielding similar resultsRDN Meaning D √ n ( (cid:104) X n (cid:105) − (cid:104) X (cid:105) ) sample mean of opinions compared against true population mean of opinions and scaled by sample size D ( (cid:104) X n (cid:105) − (cid:104) X (cid:105) ) sample mean of opinions compared against true population mean of opinions D ( X − (cid:104) X n (cid:105) ) sample agent’s opinion compared against sample population mean of opinions D ( X − X (cid:48) ) sample agent’s opinion compared against another sample agent’s opinion ± D · X a single sample agent’s opinionFIG. 4. The depolarization of existing echo chambersby RDN . The opinions of 1000 agents over time (a) withoutthe RDN until t = 10, with the RDN from t = 10 until t = 20( D = 3 , n = 50, black bar), and finally without the RDNagain from t = 20. Densities of agent opinions at momentsin time (b) at 10 (pink), 20 (gold), and 30 (green). The peakdistance (Λ x ) is smallest at t = 20. Other parameters were m = 10, K = 2, α = 2, β = 3, r = 0 . results, we suggest that social media implement a contin-uous RDN on pressing issues where a clear answer doesnot exist. This will assist users in being more informedand also limit the formation of the echo chambers. Oncethe echo chambers are formed, and people identify withone of the extremes, they even ignore facts in their fa-vor if the source is not their affiliated group. This meansthat once people are affiliated with one side, they assumetheir side is objective and the other side is biased, a phe-nomenon known as objectivity illusion [5, 22, 23]. In theUS 2016 presidential election, the objectivity illusion wasfirmly at play and led to polarization [23]. The RDN strength ( D ) plays a similar role to the in-teraction strength ( K ) but relates to a random sample ofagents instead of the connections determined by agents’opinions. It is then reasonable that D would naturallybe a function of the issue at hand and the network’s in-teraction strength. The content of RDN would also playa role and would be an integral part of determining ad-equate RDN strength. For example, one can show anaggregation of some users’ opinions using numerical val-ues of likes, retweets, engagement, etc. Relevant to RDN,adding randomness into each user’s feed on a given topicmay be a reasonable real-life approximation of the RDN.The frequency of these random posts, among other fac-tors, could be a possible modulator of D . We note thatunderstanding the nuances of user interface design ap-proaches is out of this article’s scope.A limitation of this formulation compared to a socialnetwork is that the RDN is influential at every time point dt . It may be more realistic to consider a scenario where D >
Acknowledgment:
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