When facts fail: Bias, polarisation and truth in social networks
Orowa Sikder, Robert E. Smith, Pierpaolo Vivo, Giacomo Livan
WWhen facts fail: Bias, polarisation and truth in social networks
Orowa Sikder, Robert E. Smith, Pierpaolo Vivo, and Giacomo Livan
1, 3 Department of Computer Science, University College London, Gower Street, London WC1E 6EA, UK Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK Systemic Risk Centre, London School of Economics and Political Sciences, Houghton Street, London WC2A 2AE, UK (Dated: January 22, 2020)Online social networks provide users with unprecedented opportunities to engage with diverseopinions. At the same time, they enable confirmation bias on large scales by empowering individualsto self-select narratives they want to be exposed to. A precise understanding of such tradeoffs isstill largely missing. We introduce a social learning model where most participants in a networkupdate their beliefs unbiasedly based on new information, while a minority of participants rejectinformation that is incongruent with their preexisting beliefs. This simple mechanism generatespermanent opinion polarization and cascade dynamics, and accounts for the aforementioned tradeoffbetween confirmation bias and social connectivity through analytic results. We investigate themodel’s predictions empirically using US county-level data on the impact of Internet access on theformation of beliefs about global warming. We conclude by discussing policy implications of ourmodel, highlighting the downsides of debunking and suggesting alternative strategies to contrastmisinformation.
Introduction
We currently live in a paradoxical stage of the information age. The more we gain access to unprecedented amountsof knowledge thanks to digital technologies, the less our societies seem capable of discerning what is true from what isfalse, even in the presence of overwhelming evidence in support of a particular position. For example, large segmentsof our societies do not believe in the reality of climate change [1] or believe in the relationship between vaccinationsand autism [2].As recent studies indicate, over two-thirds of US adults get information from online and social media, with theproportion growing annually [3, 4]. Hence, the impact such media have in shaping societal narratives cannot beunderstated. Online media empower their users to choose the news sources they want to be exposed to. This, inturn, makes it easier to restrict exposure only to narratives that are congruent to pre-established viewpoints [5–8],and this positive feedback mechanism is further exacerbated by the widespread use of personalized news algorithms[9]. In other words, confirmation bias [10, 11] is enabled at unprecedented scales [12].Another major impact of digital technologies has been the increase in connectivity fostered by the growth of onlinesocial networks, which plays a double-edged role. On the one hand, it can compound the effects of confirmation bias,as users are likely to re-transmit the same information they are selectively exposed to, leading to fragmented societiesthat break down into online “echo chambers” where the same opinions keep being bounced around [12, 13]. On theother hand, it also translates into a potentially increased heterogeneity of the information and viewpoints users areexposed to [14, 15].Online social networks can therefore both improve and restrict the diversity of information individuals engagewith, and their net effect is still very much debated. Empirical research is still in its infancy, with evidence forboth positive and negative effects being found [15–17]. The theoretical literature is lagging somewhat further behind.While there exist a plethora of models related to information diffusion and opinion formation in social networks, asound theoretical framework accounting for the emergence of the phenomena that are relevant to modern informationconsumption (rather than explicitly introducing them ad hoc), is still largely lacking.In bounded confidence models [18–20] agents only interact with others sharing similar opinions, and thus arecharacterized by a form of confirmation bias. In such models polarization is a natural outcome assuming agentsare narrow enough in their choice of interaction partners [21]. However, these models tend to lack behaviouralmicro-foundations and a clear mechanism to link information diffusion to opinion formation, making it hard to drawconclusions about learning and accuracy amongst agents.Social learning models [22–24] provide a broader, empirically grounded, and analytically tractable framework tounderstand information aggregation [25]. Their main drawback, however, is that by design they tend to produce longrun population consensus, hence fail to account for any form of opinion heterogeneity or polarization [26]. Polarizationcan be generated by introducing “stubborn” agents that remain fully attached to their initial opinions rather thaninteracting and learning from their neighbors [27, 28], a mechanism reminiscent of confirmation bias. However,the conditions under which polarization occurs are very strict, as populations converge towards consensus as soonas stubborn agents accept even a negligible fraction of influence from their neighbors [26]. A similar phenomenonis explored in social physics literature where it is referred to as networks with “zealots”, which similarly impede a r X i v : . [ phy s i c s . s o c - ph ] J a n consensus, such as in [29]. A key distinction in the model we introduce in the following is that all agents are free tovary their opinions over time, resulting in cascade dynamics that separate consensus and polarization regimes.Overall, while it is clear from the literature that some notion of “bias” in networks is a key requirement for us toreproduce realistic dynamics of opinion formation, it is still difficult to provide a unified framework that can accountfor information aggregation, polarization and learning. The purpose of the present paper is to develop a frameworkthat naturally captures the effect of large-scale confirmation bias on social learning, and to examine how it candrastically change the way a networked, decentralized, society processes information. We are able to provide analyticresults at all scales of the model. At the macroscopic scale, we determine under what conditions the model ends upin a polarised state or cascades towards a consensus. At the mesoscopic scale, we are able to provide an intuitivechracterization of the trade-off between bias and connectivity in the context of such dynamics, and explain the roleecho chambers play in such outcomes. At the microscopic scale, we are able to study the full distribution of eachagent’s available information and subsequent accuracy, and demonstrate that small amounts of bias can have positiveeffects on learning by preserving information heterogeneity. Our model unveils a stylized yet rich phenomenologywhich, as we shall discuss in our final remarks, has substantial correspondence with the available empirical evidence. ResultsSocial learning and confirmation bias
We consider a model of a social network described by a graph G = ( V, E ) consisting of a set of agents V (where | V | = n ), and the edges between them E . Each agent seeks to learn the unobservable ground truth about a binarystatement X such as, e.g., “global warming is / is not happening” or “gun control does / does not reduces crime”.The value X = +1 represents the statement’s true value, whose negation is X = − t = 0), each agent i ( i = 1 , . . . , n )independently receives an initial signal s i = ±
1, which is informative of the underlying state, i.e. p = Prob( s i =+1 | X = +1) = 1 − Prob( s i = − | X = +1) > /
2. Signals can be thought of as news, stories, quotations, etc., thatsupport or detract from the ground truth. The model evolves in discrete time steps, and at each time step t > t = 1 information set of an agent i with two neighbors j and (cid:96) will be s i ( t = 1) = { s i , s j , s (cid:96) } , theirtime t = 2 set will be s i ( t = 2) = { s i , s i , s i , s j , s j , s (cid:96) , s (cid:96) , s d =2 } , where s d =2 denotes the set of all signals incoming fromnodes at distance d = 2 (i.e., j ’s and (cid:96) ’s neighbors), and so on. Furthermore, we define the following: x i ( t ) = N + i ( t ) N + i ( t ) + N − i ( t ) , (1)where N + i ( t ) and N − i ( t ) denote, respectively, the number of positive and negative signals accrued by i up to time t .We refer to this quantity as an agent’s signal mix , and we straightforwardly generalize it to any set of agents C ⊆ V ,i.e., we indicate the fraction of positive signals in their pooled information sets at time t as x C ( t ). The list of allagents’ signal mixes at time t is vectorised as x ( t ).Each agent forms a posterior belief of the likelihood of the ground truth given their information sets using Bayes’ rule.This is done under a bounded rationality assumption, as the agents fail to accurately model the statistical dependencebetween the signals they receive, substituting it with a naive updating rule that assumes all signals in their informationsets to be independent (Prob( X | s i ( t )) = Prob( s i ( t ) | X )Prob( X ) / Prob( s i ( t )), where Prob( s i ( t ) | X ) is computed as afactorization over probabilities associated to individual signals, i.e. Prob( s i ( t ) | X ) = (cid:81) c Prob (cid:16) s ( c ) i ( t ) | X (cid:17) , where s ( c ) i ( t ) denotes the c th component of the vector), which is a standard assumption of social learning models [25].Under such a framework (and uniform priors), the best guess an agent can make at any time over the statement X given their information set is precisely equal to their orientation y i ( t ), where y i ( t ) = +1 for x i ( t ) > / y i ( t ) = − x i ( t ) < / x i ( t ) = 1 / n agents at time t are vectorized as y ( t ); thefraction of positively oriented agents in a group of nodes C ⊆ V is denoted as y C ( t ).The polarization z C ( t ) = min( y C ( t ) , − y C ( t )) of the group C is then defined as the fraction of agents in that groupthat have the minority orientation. Note that polarization equals zero when there is full consensus and all agents areeither positively or negatively oriented. It is maximized when there are exactly half the group in each orientation.It is useful to think of x ( t ), y ( t ) and z V ( t ) as respectively representing the pool of available signals, the conclusionsagents draw on the basis of the available signals, and a summary measure of the heterogeneity of agents’ conclusions.In the context of news diffusion, for example, they would represent the availability of news of each type across agents,the resulting agents’ opinions on some topic, and the extent to which those opinions have converged to a consensus.We distinguish between two kinds of agents in the model: unbiased agents and biased agents. Both agents sharesignals and update their posterior beliefs through Bayes’ rule, as described in the previous section. However, theydiffer in how they acquire incoming signals. Unbiased agents accept the set of signals provided by their neighbourswithout any distortion. On the other hand, biased agents exercise a model of confirmation bias [11, 30], and are ableto distort the information sets they accrue. We denote the two sets of agents as U and B , respectively.To describe the behaviour of these biased agents we use a slight variation of the confirmation bias model introducedby Rabin and Shrag [31]. We refer to an incoming signal s as congruent to i if it is aligned with i ’s current orientation,i.e. if s = y i ( t ), and incongruent if s = − y i ( t ). When biased agents are presented with incongruent signals, theyreject them with a fixed probability q and replace them with a congruent signal, which they add to their informationset and propagate to their neighbors. We refer to q as the confirmation bias parameter. Denote the set of positively(negatively) oriented biased agents at time t as B + ( t ) ( B − ( t )), and the corresponding fraction as y B ( t ) = |B + ( t ) | / |B| .Note that this is an important departure from “stubborn agent” models, as such biased agents do have a non-zeroinfluence from their neighbours, and they can change their beliefs over time as they aggregate information.An intuitive interpretation of what this mechanism is intended to model is as follows: biased agents are empoweredto reject incoming signals they disagree with, and instead refer to preferred sources of information to find signalsthat are congruent with their existing viewpoint (see Fig. 1). This mechanism models both active behaviour, whereagents deliberately choose to ignore or contort information that contradicts their beliefs (mirroring the “backfireeffect” evidenced both in psychological experiments [32, 33] and in online social network behaviour [34]), and passive behaviour, where personalized news algorithms filter out incongruent information and select other information whichcoheres with the agents’ beliefs [15].In the following, we shall denote the fraction of biased agents in a network as f . We shall refer to networks where f = 0 as unbiased networks, and to networks where f > biased networks.For the bulk of the analytic results in the paper, we assume that the social network G is an undirected k -regularnetwork. The motivation for this is two-fold. Firstly, empirical research [35] suggests that for online social networkssuch as Facebook (where social connections are symmetric), heterogenous network features such as hubs do not play adisproportionately significant role in the diffusion of information. Intuitively, while social networks themselves mightbe highly heterogenous, the network of information transmission is a lot more restricted, as individuals tend to discusstopics with a small group such as immediate friends and family. Secondly, utilizing a simple k -regular network allowsfor considerable analytical tractability. However, one can show that our main results can be easily extended to holdunder a variety of network topologies characterized by degree heterogeneity.The assumption of regular network structure (coupled with the aforementioned synchronous belief update dynamics)allows the information sets of all agents to grow at the same rate, and as a result the evolution of the signal mix x ( t )can be mapped to a DeGroot averaging process for unbiased networks [36]: x ( t ) = A x ( t − A is an n × n matrix with entries a ij = 1 / ( k + 1) for each pair ( i, j ) of connected nodes.For biased networks, one can demonstrate (see Section S1 of the Supplementary Materials) that the above con-firmation bias mechanics can be reproduced by introducing a positive and negative “ghost” node which maintainrespective signal mixes of 1 and 0. Biased agents sample each signal from their orientation-aligned ghost node nodeswith probability q , and from their neighbourhood with probability (1 − q ).Furthermore, while the process is stochastic, we also show that it converges to a deterministic process with asimple update matrix described as follows. In Section S3 of the Supplementary Materials we discuss in detail thecorrespondence and convergence between the stochastic and deterministic processes. Biased agents down-weight theirconnections to neighbours by a factor (1 − q ) and place the remaining fraction kq/ ( k + 1) of their outgoing weight onthe corresponding ghost node. With these positions the updating process now simply reads ˆ x ( t ) = ˆ A ( t ) ˆ x ( t − A ( t ) is an ( n + 2) × ( n + 2) asymmetric matrix whose entries in its n × n upper-left block are as those in A , exceptˆ a ij = (1 − q ) / ( k + 1) when i ∈ B = B + ∪ B − . The positive (negative) ghost node corresponds to node n + 1 ( n + 2)of the augmented matrix ˆ A ( t ), and we shall label it as + ( − ) for convenience, i.e., we shall have ˆ a i + = kq/ ( k + 1)(ˆ a i − = kq/ ( k + 1)) for i ∈ B + ( i ∈ B − ) and ˆ a ++ = ˆ a −− = 1. Similarly, ˆ x ( t ) denotes an augmented signal mix vectorwhere ˆ x + ( t ) = ˆ x n +1 ( t ) = 1, and ˆ x − ( t ) = ˆ x n +2 ( t ) = 0.The time dependence of the matrix ˆ A ( t ) is due to the fact that whenever a biased agent switches orientation itslinks to the ghost nodes change. This happens whenever the agent’s signal mix x i ( t ) (see Eq. 1) goes from belowto above 1 / i will change its links as follows:ˆ a i + = kq/ ( k + 1) → ˆ a i + = 0, and ˆ a i − = 0 → ˆ a i − = kq/ ( k + 1). In the following, all quantities pertaining to biasednetworks will be denoted with aˆsymbol.We provide a sketch of the above mapping in Fig. 1. There is an appealing intuition to this interpretation: biasedagents have a “preferred” information source they sample from in lieu of incongruent information provided from theirpeers. If their beliefs change, their preferred information source can change. + k +1 qk +1 k +1 qk +1 k +11 k +1 1 k +11 k +1 kqk +11 k +1 1 k +1 1 k +1 1 k +1 A = k +1
We begin by studying the signal mix of unbiased agents in biased networks (ˆ x ∗U ) to provide a like for like comparisonwith the fully unbiased networks. In the context of the diffusion of news, the global signal mix can be thought of asa model of the long term balance of news of different types that survive following the diffusion dynamics.For unbiased networks, it is demonstrated in [37] that ˆ x ∗U = ¯ x (0). That is, the steady state signal mix in unbiasednetworks precisely reflects the original, unbiased informative signals injected into the network. Determining thesteady state signal mix of biased networks entails considering the interactions between three subpopulations - theunbiased agents U , positively biased agents B + , and negatively biased agents B − . One can show (see Section S4 ofthe Supplementary Materials) that this can be approximated as: ˆ x ∗U ˆ x ∗B + ˆ x ∗B − = ˆ y B ( t ∗ )(1 − q )ˆ y B ( t ∗ ) + q (1 − q )ˆ y B ( t ∗ ) . (2)Let us now consider the situation under which t ∗ = 0, i.e. where the initial orientation of each biased agent does notchange, and is therefore equal to the initial signal it receives. Intuitively, this will occur for large q which allows forbiased agents to reject the majority of incongruent signals they receive (shortly we demonstrate in fact this generallyoccurs for q > / x ∗U of unbiased agents is determined by the initial proportion of positivelyoriented biased agents y B + (0), which is the mean of f n i.i.d. Bernoulli variables with probability p (which, we recall,denotes the probability of an initially assigned signal being informative). One can compare this to unbiased networks( f = 0), where the long run average signal mix is x ∗ V = ¯ x (0), and is hence the mean of n i.i.d. Bernoulli variableswith probability p . Applying the central limit theorem we see that injecting a fraction f of biased agents thereforeamplifies the variance of the long run global signal mix by a factor of f − with respect to the unbiased case: x ∗ V ∼ N (cid:18) p, p (1 − p ) n (cid:19) → ˆ x ∗U ∼ N (cid:18) p, p (1 − p ) f n (cid:19) . (3)This means that the “wisdom of unbiased crowds” is effectively undone by small biased populations, and the unbiasednetwork’s variability is recovered for f →
1, and not for f → + , as one might intuitively expect.Consider now the general case where biased agents can, in principle, switch orientation a few times before settlingon their steady state orientation. Using mean-field methods one can determine the general conditions under whicha cascade in these orientation changes can be expected (see Section S4 of the Supplementary Materials) but herewe only provide some intuition. As q is lower, it is easier for an initial majority camp of biased agents to convertthe minority camp of biased agents. As the conversion of the minority camp begins, this triggers a domino effect asnewly converted biased agents add to the critical mass of the majority camp and are able to overwhelm the minorityorientation.This mechanism allows us to derive analytic curves in the parameter space to approximate the steady state outcomeof the unbiased agent population’s average signal mix based on the orientations of the biased agents at time 0:ˆ x ∗U = ˆ y B (0) for − q − q ) ≤ ˆ y B (0) ≤ − q ) y B (0) > − q − q ) y B (0) < − q ) . (4)The above result is sketched in Fig. 2, and we have verified that it matches numerical simulations even forheterogenous networks. For 1 / < q ≤ q , on the other hand, small variationsin the initial biased population translate to completely opposite consensus, and only by increasing the confirmationbias q , paradoxically, the model tends back to a balance of signals that resembles the initially available information.Putting the above results together, we note that biased networks with small f and q are, surprisingly, the mostunstable. Indeed, such networks sit on a knife-edge between two extremes where one signal type flourishes and theother is totally censored. In this context, the model indicates that confirmation bias helps preserve a degree ofinformation heterogeneity, which, in turn, ensures that alternative viewpoints and information are not eradicated. Insubsequent sections we consider a normative interpretation of this effect in the context of accuracy and learning. FIG. 2: Average steady state signal mix of unbiased agents (ˆ x ∗U ) as a function of the time t = 0 fraction of positively orientedbiased agents (ˆ y B (0)) and confirmation bias q . The color gradient denotes the average long run signal mix for unbiased agentsfrom 0 to 1. The top-left and bottom-left regions are characterized, respectively, by a global signal mix of 1 and 0 respectively,and are separated by a discontinuous transition from a region characterized by a steady state that maintains a mixed set ofsignals. That is, in the top-left region almost all negative signals have been removed from the network, leaving almost entirelypositive signals in circulation (and vice versa for the bottom-left region). In the remaining region, signal mixes of both typessurvive in the long run, and the balance between positive and negative signals reflects the fraction of positively oriented biasedagents. The lower q falls, the easier it is to tip the network into a total assimilation of a single signal type. Results are shownfor simulations on k -regular (top left), Erd˝os-R´enyi (top right), Barabasi-Albert (bottom left) and Small-world (bottom right)networks. Analytic predictions (given by Eq. (4)) are denoted by solid red lines. The parameters used in the simulations were n = 10 , p = 0 . k = 6 (which corresponds to an average degree in all cases), f = 0 . Polarization, echo chambers and the bias-connectivity trade-off
So far we have derived the statistical properties of the average steady state signal mix across all unbiased agents.We now aim to establish how these signals are distributed across individual agents. Throughout the following, assumethe global steady state signal mix ˆ x ∗U has been determined.In the limit of large n and k , x ∗ i for i ∈ U is normally distributed with mean ˆ x ∗U and variance σ (ˆ x ∗U ) that can beapproximated as follows (see Section S4 of the Supplementary Materials): σ (ˆ x ∗U ) ≈ f q k (ˆ x ∗U (1 − ˆ x ∗U )) , (5)and this result is quite accurate even when compared with simulations for small n and k , as demonstrated in Figure3. This result further shows that the presence of biased agents is effectively responsible for the polarization of unbiasedagents in the steady state. Indeed, both a larger biased population and higher confirmation bias - i.e. higher f or q ,respectively - result in an increased variance and steady state polarization ˆ z ∗U , since a larger variance σ ( x ∗U ) implieslarger numbers of agents displaying the minority orientation. This is illustrated in the top left panel of Fig. 5. FIG. 3: Distribution of individual unbiased agents’ steady state signal mixes (points) vs analytic predictions (dashed lines).As discussed in the main text, the model predicts such distribution to be a Gaussian (for both large n and k ) with mean equalto ˆ x ∗U and variance given by Eq. (5). ˆ x ∗U is kept fixed to demonstrate the effect of varying f and k . As shown in the case for k/f = 120, f and k trade off, and scaling both by the same constant results in the same distribution. The parameters used inthe simulations were n = 10 , ˆ x ∗U = 0 . q = 0 . On the other hand, a larger degree k contrasts this effect by creating more paths to transport unbiased information.It is worth pointing out that the variance in Eq. 5 does not decay with n , showing that steady state polarizationpersists even in the large n limit. We refer to this as the bias-connectivity trade-off, and the intuition behind thisresult is illustrated in Figure 4.Further intuition for this result can be found at the mesoscopic level of agent clusters, where we see the emergenceof natural “echo chambers” in the model. We define an echo chamber C as a subset of unbiased agents such that: C = { i ∈ U : ∂ i ∈ B ∪ C ∧ ∂ i ∩ C (cid:54) = ∅} , where ∂ i denotes the neighbourhood of agent i . In other words, an echochamber is a set of connected unbiased agents such that all nodes are either connected to other nodes in the echochamber or to biased agents. Therefore, biased agents form the echo chamber’s boundary, which we refer to as ∂ C .Echo chambers in our model represent groups of unbiased agents that are completely surrounded by biased agentswho effectively modulate the information that can flow in and out of these groups.Echo chambers allow us to examine the qualitative effect of confirmation bias ( f, q ) and connectivity k . Let us labelthe fraction of unbiased agents enclosed in an echo chamber as η C . Leveraging some simple results from percolationtheory[38] we can show that η C increases with f and decreases with k , as the creation of more pathways that bypassbiased agents effectively breaks up echo chambers. Furthermore, the equilibrium signal mix of unbiased agents insideecho chambers is well approximated by a weighted average between the signal mix of the biased agents surroundingthem ( x ∗ ∂ C ) and the signal mix x ∗U of the whole population: x ∗ C = q x ∗ ∂C + (1 − q ) x ∗U . The confirmation bias parameter q therefore determines the “permeability” of echo chambers to the information flow from the broader network. Hence,unbiased agents enclosed in echo chambers are likely to be exceedingly affected by the views of the small set of biasedagents surrounding them, and, as such, to hold information sets that are unrepresentative of the information availableto the broader network. In doing so, we can envision these echo chambers as effective “building blocks” of the overallpolarization observed in the network. Increasing f
Up until now, we have not attempted to make any normative interpretations of the ground truth X = +1. Inthe following, we shall refer to unbiased agents whose steady state orientation is positive (negative) as accurate(inaccurate) agents, and we shall define the overall accuracy A ( G ) of a network G as the expected fraction of accurateagents in the steady state. This allows us to investigate how biased and unbiased networks respond to changes inthe reliability of the available information, which ultimately depends on the prevalence of positive or negative signals(modulated by the parameter p = Prob( s = +1 | X = +1)), which, loosely speaking, can be interpreted as “real” and“fake” news.The accuracy of unbiased networks obtains a neat closed form that can be approximated as A ( G | f = 0) ≈ erfc((1 − p ) (cid:112) n/ / f >
0, we compute the expected accuracy as the expected fraction of accurate agentswith respect to a certain global signal mix. This reads: A ( G | f >
0) = 12 (cid:90) d x ∗U P ( x ∗U ) erfc (cid:32) / − x ∗U √ σ x ∗U (cid:33) , (6)where P ( x ∗U ) is the distribution of the average signal mix across unbiased agents (see Eq. 3) (we take the simplifyingcase of q > /
2, but this can easily be extended to the case for q ≤ / σ x ∗U isgiven by Eq. 5).The top right panel in Fig. 5 contrasts biased and unbiased networks, and shows how the former remain veryinefficient in aggregating information compared to the latter, even as the reliability of the signals ( p ) improve. However, -2 -1 FIG. 5: polarization ˆ z ∗U = 1 − ˆ y + ∗U of the unbiased agent population as a function of the average signal mix ˆ x ∗U in the steadystate calculated as erfc(ˆ x ∗U − / / ( √ σ (ˆ x ∗U ))) /
2, with σ (ˆ x ∗U ) given by Eq. 5 (top left panel). Expected accuracy (Eq. 6) asa function of the initial signals’ informativeness p (top right panel) and of the fraction f of biased agents (bottom left panel).behaviour of the accuracy-maximizing value f ∗ and of the corresponding accuracy A ( G | f = f ∗ ) as functions of k (bottomright panel). In the first three panels the model’s parameter n = 10 , q = 1, k = 8, while the parameters in the last panel are n = 10 , q = 1, p = 0 .
53. In all cases we assume X = +1 without loss of generality. accuracy in biased networks is non-monotonic with respect to f . As shown in the bottom left panel in Fig. 5, accuracyreaches a maximum in correspondence of an optimal value f ∗ (see Section S5 of the Supplementary Materials for acomparison with numerical simulations). Intuitively, this is because for small values of f , as already discussed, themodel can converge to the very inaccurate views of a small set of biased agents. As f grows, the views of the twobiased camps tend to cancel each other out, and the signal set will match more closely the balance of the originaldistribution of signals (Eq. 3). However, in doing so large values of f lead to increased polarization (Eq. 5), whereaccurate and inaccurate agents coexist. The trade-off between balance and polarization is optimised at f ∗ .It is also interesting to note that, as shown in the bottom right panel of Fig. 5, the optimal fraction of biasedagents f ∗ and the corresponding maximum accuracy A ( G | f = f ∗ ) both increase monotonically with the degree.This indicates that as networks are better connected, they can absorb a greater degree of confirmation bias withoutaffecting accuracy. Internet access, confirmation bias, and social learning
We now seek to test some of the model’s predictions against real world data. Clearly, a full validation of the modelwill require an experimental setup, but a simple test case on existing data can demonstrate the utility of the frameworkin disambiguating the competing effects of bias and connectivity. We employ the model to investigate the effect ofonline media in the process of opinion formation using survey data. Empirical literature on this phenomenon has beenmixed, with different analyses reaching completely opposite conclusions, e.g., showing that Internet access increases[39], decreases [40] and has no effect [41] on opinion polarization. In Section S6 of the Supplementary Materials webriefly review how our model can help better understand some of the inconsistencies between these results.Our position is that the effect of Internet access can be split into the effect it has on social connectivity and social0discussion ( k ) and the residual effect it has on enabling active and passive confirmation bias behaviours ( f ). As perEq. 5, assuming the majority of the population accurately learns the ground truth ( x ∗U > / k . We combine this with FCC reports on county level broadbandinternet penetration, which proxies for f after controlling for the considerable effect this has on social connectivity.We also account for a range of covariates (income, age, education, etc) and make use of an instrumental variableapproach to account for simultaneous causality. We then attempt to predict the fraction of each county’s populationthat correctly learns that “global warming is happening” (see Section S6 of the Supplementary Materials for detailson assumptions and results).As predicted by our model, we find the accurate fraction of the population to have statistically significant positiverelationships with k , and a negative relationship with f . We find such relationships to account for 65% of the variance inthe data. This indicates that, after controlling for the improvements on social connectivity, Internet access does indeedincrease polarization and reduces a population’s ability to accurately learn. While simple, this analysis illustrates thevalue of our model: by explicitly accounting for the separate effects of large-scale online communication (confirmationbias and connectivity), it can shed light on the mixed empirical results currently available in the literature. In SectionS6 of the Supplementary Materials we explore this further by reviewing some of these empirical results and showinghow our model provides useful further interpretations of available findings.It should be emphasized that this result is merely an initial exploration of how our model can provide some testablepredictions to empirical data, as opposed to a detailed effort to understand the effect of Internet access on globalwarming beliefs. Having said that, the initial results are encouraging, and we hope the clarity of the analytic resultsof our model pave the way for testing variations of the idea of biased information aggregation in a range of outcomesand settings. Discussion
We introduced a model of social learning in networked societies where only a fraction of the agents update beliefsunbiasedly based on the arrival of new information. The model only provides a stylized representation of the real-world complexity underpinning the propagation of information and the ensuing opinion formation process. Its valuestands in the transparency of the assumptions made, and in the fact that it allows us to “unpack” blanket termssuch as, e.g., social media and Internet penetration, by assigning specific parameters to their different facets, suchas connectivity ( k ) and the level of confirmation bias it enables in a society ( f, q ). This, in turn, yields quantitative testable predictions that contribute to shed light on the mixed results that the empirical literature has so far collectedon the effects online media have in shaping societal debates.Our model indicates the possibility that the “narratives” (information sets) biased societies generate can be en-tirely determined by the composition of their sub-populations of biased reasoners. This is reminiscent of the over-representation in public discourse of issues that are often supported by small but dedicated minorities, such as GMOopposition [43], and of the domination of political news sharing on Facebook by heavily partisan users [15]; it alsoresonates with recent experimental results showing that committed minorities can overturn established social conven-tions [44]. The model indicates that societies that contain only small minorities of biased individuals ( f → + ) may bemuch more prone to producing long run narratives that deviate significantly from their initially available informationset (see Eq. 3) than societies where the vast majority of the agents actively propagate biases. This resonates, forexample, with Gallup survey data about vaccine beliefs in the US population, where only 6% of respondents reporttheir belief in the relationship between vaccines and autism, but more than 50% report to be unsure about it andalmost 75% report to have heard about the disadvantages of vaccinations [45]. Similarly, the model suggests that mildlevels of confirmation bias ( q (cid:28)
1) may prove to be the most damaging in this regard, as they cause societies to liveon a knife-edge where small fluctuations in the information set initially available to the biased agent population cancompletely censor information signals from opposing viewpoints (see Fig. 2). All in all, the model suggests that a lack of confirmation bias can ensure that small biased minorities much more easily hijack and dictate public discourse.The model suggests that as the prevalence of biased agents grows, the available balance of information improvesand society is more likely to maintain a long term narrative that is representative of all the information available.On the other hand, it suggests that such societies may grow more polarised. When we examine the net effect of thistrade off between bias and polarization through an ensemble approach, our model suggests that the expected accuracyof a society may initially improve with the growth of confirmation bias, then reach a maximum at a value f ∗ before1marginal returns to confirmation bias are negative, i.e. confirmation bias experiences an “optimal” intermediate value.The model suggests that such value and its corresponding accuracy should increase monotonically with a society’sconnectivity, meaning that more densely connected societies can support a greater amount of biased reasoners (andhealthy debate between biased camps) before partitioning into echo chambers and suffering from polarization. SUPPLEMENTARY MATERIALSI. SECTION S1: UPDATE DYNAMICSA. Update dynamics as random matrix A ( t ) . Consider the set of signals s i ( t ) possessed by a positively oriented agent i at time t (i.e., i ∈ B + ). This will consistof a set of signals retained from the previous time step, s i ( t − s (cid:48) i ( t ) constructed fromthe signals available from the nodes j ∈ ∂ i at the end of time ( t − s ∗ i ( t ) = (cid:83) j s j ( t −
1) be the set of the unbiased signals available to node i at time t , i.e. the set of nodes i willreceive from her neighbors before applying the confirmation bias function. Let s ∗ ( a ) i ( a = 1 , . . . , k ( k + 1) t − ) be ageneric signal in the set s ∗ i ( t ). After the application of the confirmation bias function, this will be turned into a signal s (cid:48) ( a ) i ( t ) ∈ s (cid:48) i ( t ) such that s (cid:48) ( a ) i ( t ) = ± s ∗ ( a ) i ( t ) according to the following probabilities:Prob( s (cid:48) ( a ) i = +1 | s ∗ ( a ) i = −
1) = q Prob( s (cid:48) ( a ) i = − | s ∗ ( a ) i = −
1) = (1 − q )Prob( s (cid:48) ( a ) i = +1 | s ∗ ( a ) i = +1) = 1Prob( s (cid:48) ( a ) i = − | s ∗ ( a ) i = +1) = 0 . According to the above rules, agent i checks the value of the new incoming signal, and flips it with probability q if itis incongruent with respect to her current orientation. This is entirely equivalent to node i sampling with probability q from the set s ∗ i ( t ), and with probability 1 − q from an equally sized set of positive signals belonging to a positivelyoriented “ghost” node.Let us consider the number N + i ( t ) of positive signals possessed by agent i at time t . Due to the above rules, itstime evolution will be such that N + i ( t ) = N + i ( t −
1) + (cid:88) j ∈ ∂ i (cid:0) N + j ( t −
1) + w i ( t ) N − j ( t − (cid:1) , where w i ( t ) ∈ [0 ,
1] is a random variable denoting the fraction of negative signals successfully distorted by i of thosereceived by its neighbours at time t , with distribution such that w i ( t ) N − ∂ i ( t ) ∼ Bin( N − ∂ i ( t ) , q ), where N − ∂ i ( t ) is thenumber of negative signals received by i from her neighbourhood at time t . When considering agent i ’s signal mix ,the above translates to x i ( t ) = 1 k + 1 x i ( t −
1) + (1 − w i ( t )) (cid:88) j ∈ ∂ i x j ( t −
1) + w i ( t ) k . (7)Similarly, for a negatively oriented biased agent (i.e., i ∈ B − ) we have x i ( t ) = 1 k + 1 x i ( t −
1) + (1 − w i ( t )) (cid:88) j ∈ ∂ i x j ( t − , (8) We recall that the signal mix, as per Eq. (1) of the main paper, is defined as the fraction of positive signals possessed by an agent at acertain time, i.e., x i ( t ) = N + i ( t ) / ( N + i ( t ) + N − i ( t )) = N + i ( t ) / ( k + 1) t . w i ( t ) N + ∂ i ( t ) ∼ Bin( N + ∂ i ( t ) , q ).Combining Eqs. (7) and (8) with the time evolution for the signal mix of unbiased agents, which reads x i ( t ) = 1 k + 1 x i ( t −
1) + (cid:88) j ∈ ∂ i x j ( t − , we can see that the time evolution for the vector of signal mixes x ( t ) can be written as ˆ x ( t ) = ˆ A ( t ) ˆ x ( t − , (9)where ˆ x = [ x T , , T where the latter terms represent the (fixed) signal mixes of the ghost nodes and the ˆ x ( t ) thesignal mixes of the original set. ˆ A ( t ) is an ( n + 2) × ( n + 2) random matrix with entries with a block structure asfollows: ˆ A ( t ) = (cid:20) Q ( t ) R ( t )0 I (cid:21) , where Q ( t ) is an ( n × n ) matrix representing the original graph structure with Q ii ( t ) = k +1 , Q ij ( t ) = k +1 where i is a unbiased agent connected to j , Q ij ( t ) = − w i ( t ) k +1 where i is a biased agent connected to j , and 0 otherwise. R ( t )is an ( n ×
2) matrix representing connections from biased agents to their preferred ghost node (which we index by+ and − ). R i + ( t ) = w i ( t ) kk +1 if i ∈ B + and 0 otherwise. Analogous weights exist for negatively biased agents to thenegative ghost node. I is a (2 ×
2) identity matrix representing the weights of ghost nodes to themselves. 0 is the(2 × n ) block of zeros representing the (lack of) edges outbound from the ghost nodes.Finally, it is worth noting that the above formulation consisting of two ghost nodes is fully equivalent to a formu-lation where each biased agent has a “personalized” ghost node that reflects their positive or negative orientationappropriately. In this case ˆ A ( t ) is an ( n + f n ) × ( n + f n ) matrix with an extra f n ghost nodes added, one for eachbiased agent. However, while this formulation has a more favourable interpretation in terms of “content personal-ization”, it is less convenient analytically, so for the reminder of the Supplementary Information the simplified ghostnode formulation will be utilised. B. Almost sure convergence of ˆ A ( t ) . We now proceed to show that stochastic weights w i ( t ) appearing in the matrix ˆ A ( t ) of (9) converge almost surelyto q when t → ∞ as long as at least one signal of each type is held by at least one node in the network. As such therandom matrix ˆ A ( t ) converges almost surely to a fixed matrix ˆ A = E ( ˆ A ( t )).Let us consider i ∈ B + . As established in the previous section, w i ( t ) is simply the fraction of negative signals heldby node i ’s neighbours that i successfully flips to positive at time t . Let us also recall that N − ∂ i ( t ) represents this setof negative signals available from all j ∈ ∂ i , and that each one is independently flipped to positive with probability q .If we can establish that N − ∂ i ( t ) grows indefinitely as t → ∞ , the Strong Law of Large Numbers (SLLN) can then beinvoked to establish the desired result. Since N − ∂ i ( t ) = (cid:80) j ∈ ∂ i N − j ( t ), then if i ’s neighbours possess an increasing andunbounded number of negative signals over time, then N − ∂ i ( t ) will also be increasing and unbounded. As such, each w i ( t ) will converge almost surely to q .Consider an arbitrary j ∈ ∂ i . Note that since information sets are retained by agents at every time step, we canimmediately rule out the possibility of that the number of negative signals held by agent j shrinks over time, and wemerely need to show that her set of negative signals does not remain constant over time.Let us assume that at least one negative signal has been injected into the network at t = 0, and that one agent (cid:96) possesses such negative signal. In a strongly connected network (such as the k -regular network we consider in themain paper), there exists at least one directed path from k to j of length d . Let us indicate the probability of anegative signal successfully being transmitted from an agent a to an agent b along such path as p ab . We note that p ab = 1 − q if b ∈ B + and p ab = 1 otherwise. Therefore, the probability of the signal successfully reaching j in d timesteps is: p (cid:96)j = (cid:89) ( a,b ) p ab ≥ (1 − q ) d > , t > d there exists a strictly positive probability that a negativesignal is added to j ’s information set. This, in turn, implies that the set of negative signals obtained by j will growwithout bound for t → ∞ , which establishes our result. Since this occurs for each w i , we can conclude also thatˆ A ( t ) a.s. −−→ ˆ A , as well as the block submatrices Q ( t ) a.s. −−→ Q and R ( t ) a.s. −−→ R . The edges of these fixed matrices areidentical to the structure outlined in the previous section except w i is replaced with q .Finally it is worth noting that this convergence result depends only on the strong connectedness of G and not onthe edges from the biased agents to the ghost nodes. This is important as this means that even as the orientations ofthe biased agents change (which is reflected in the rewiring of these ghost node edges), the almost sure convergenceis not interrupted. II. SECTION S2: BIASED AGENTS SETTLE IN THEIR ORIENTATION
In this section, we show that biased agents cannot continue to switch orientation indefinitely, and instead settleinto a fixed set of orientations given sufficient time. Recall that a biased agent i ∈ B switches her orientation y i ( t )when her information sets switches from a majority of positive signals ( x i ( t ) > /
2) to a majority of negative signals( x i ( t ) < / t after which biased agents cease switchingtheir orientation. For convenience, we define a network as settled at t ∗ if for all t > t ∗ , y i ( t ∗ ) = y i ( t ) for all i ∈ B .To do this, we first consider an “adversarial” toy example designed to maximise the likelihood of indefinite switching,and show that assuming perpetual switching leads to a contradiction even in this case. We then go on to show howother, more complex, network topologies are also guaranteed to settle. We limit to two topologies for brevity butthese results can be extended. Alongside the extensive evidence from numerical simulations, we argue that the modelis likely to settle for any arbitrary graph. A. Two node network.
Consider a network with two nodes, labeled 1 and 2 respectively, both of which are biased agents. Each node hasa self-weight of y and a weight of (1 − y ) on its sole neighbour . This schematic is illustrated in Figure 6. As hasbeen established in I, the signal distortion dyanmics can be mimicked by introducing two ghost nodes that representa source of positive and negative signals respectively. The weights associated with these ghost nodes are randomvariables that converge almost surely to q as t → ∞ . FIG. 6: Left: A schematic of the two node symmetric network. Right: A schematic of the two node symmetric network whereghost nodes are introduced to mimic the effect of the biased signals.
In what follows, we show that this simplified model settles (i.e., both biased agents settle at a finite time on a pairof orientations that they do not thereafter change). For the purposes of illustration, for the moment let us considerthe asymptotic case where the random weights have converged to a deterministic set of weights ( q ).The outline of this proof (and subsequent ones on alternative network structures) is to establish that in order fora biased agent i to switch orientation, their neighbours must have signal mixes sufficiently far from i ’s that they This setting generalizes the one introduced in Eqs. (7) and (8), which is recovered for k = 1 and y = 1 / i to switch orientation despite the fact that i ’s ghost node biases her learning to maintain “inertia” in thecurrent orientation. However, at the same time, the network structure ensures that nodes tend to converge closely totheir neighbourhood, which eventually prevents switching from occurring.The proof follows by contradiction. Suppose that the model never stabilizes, i.e., that at least one of the biasedagents keeps switching perpetually. Suppose node 1 switches at arbitrary times { T } = . . . < t n − < t n − < t n < . . . .We do not assume for now that times in T are over consecutive time steps, the gap between them can be as large asintended (see Fig. 7). FIG. 7: By assumption, node 1 continues to switch orientation at arbitrary time steps t n − , t n − , t n by crossing the thresholdsignal mix x i = . The threshold is denoted by a dashed line. If node 1 switches, its neighbour (node 2, white) must crosssufficiently distant thresholds, denoted by the red dashed lines. Furthermore, the switches must be simultaneous, or else theswitching terminates perpetually. The regions O u , I u , I l , O l are outlined. Consider some arbitrary t n , where x switches from x ( t n − < / x ( t n ) > /
2. Using the model’s updaterule (see (8)) we can note: x ( t n ) = yx ( t n −
1) + (1 − q )(1 − y ) x ( t n − > , (10)and using the fact that x ( t n − < / y − q )(1 − y ) x ( t n − ≥ x ( t n − ≥ − q ) > . By following the same reasoning one can show that an x switch in the opposite direction would imply x ( t n − ≤ − q − q ) = 12 − q − q ) < . Therefore, for node 1 to switch endlessly, then node 2 must also do so, and cannot start from an arbitrary point,but rather has to either be above 1 / (2(1 − q )) or below (1 − q ) / (2(1 − q ) at time t n − x to cross the 1 / t n from below or above, respectively. For the sake of convenience we introduce the following regions O l = (cid:20) , − q − q ) (cid:19) I l = (cid:20) − q − q ) , (cid:19) I u = (cid:20) , − q ) (cid:19) O u = (cid:18) − q ) , (cid:21) , / u and l ). We also denote the “inner” region I = I l ∪ I u and the “outer” region O = [0 , / I defined by the above boundaries.According to the above considerations, for node 1 to switch orientation to negative at t n , then x ( t n − ∈ O l ,and for node 1 to switch to positive at t n , then x ( t n − ∈ O u . These regions are highlighted in 7. Clearly, the sizeof I grows with q (and O shrinks with q ). It is worth noting that for q > /
2, the inner region’s boundaries exceed[0 , q , thereis no way for either node to accumulate sufficient incongruent signals to switch orientation. All in all, it follows thatboth node 1 and node 2 must switch at the same time step whenever a switch occurs. This result is illustrated in 7.We now show that if x lies in the outer region O , it will converge to the inner region I . Furthermore, once itenters the inner region, it cannot leave it. Also, this ceases the switching of the node 1, since its switching requires x to alternate between the upper and lower hemispheres of the outer region.As proved above, at any given time step node 1 and its neighbour 2 can either both switch orientation, or bothmaintain their current orientation. We will consider both possibilities. Assume the former first, in which case we canshow the two nodes must grow closer together. Suppose that at time t n , x ( t n ) > /
2, and x ( t n ) < /
2. At t n + 1,this orientation switches so x ( t n + 1) > x ( t n + 1). Making use of Eqs. (7) and (8), we can write x ( t n + 1) − x ( t n + 1) = [(1 − y )(1 − q ) − y ] ( x ( t n ) − x ( t n )) − (1 − y ) q (11) < δ ( x ( t ) − x ( t )) , where δ = [(1 − y )(1 − q ) − y ] <
1. Therefore, when the node switches orientation with their neighbour, they mustconverge strictly closer. We now consider the logical disjunct. Suppose instead that a switch does not occur, and at times t n , t n + 1 wehave x ( t n ) , x ( t n + 1) > /
2, and x ( t n ) , x ( t n + 1) < /
2. Therefore, we can write x ( t n + 1) − x ( t n + 1) = [ y − (1 − y )(1 − q ))] ( x ( t n ) − x ( t n )) + (1 − y ) q. If the two nodes are to move closer in this time step, then we must have x ( t n +1) − x ( t n +1) < (1 − µ )( x ( t n ) − x ( t n ))for some µ ∈ (0 , x ( t n ) − x ( t n ) > q − q − µ − y . (12)Finally, note that if x ( t n ) > / x ( t n ) ∈ O l , then: x ( t n ) − x ( t n ) > − − q − q ) > q − q − µ − y (13)for an arbitrarily small µ . Thus, if x ( t n ) ∈ O l , then the two nodes are sufficiently far apart that the condition in(12) holds, and the two nodes must converge closer together. The parallel argument can be made for the oppositestarting orientations.Even if a switch does not occur, then the nodes will converge strictly closer. Indeed, We have established that if x ( t ) ∈ O , then at each time step the distance | x ( t ) − x ( t ) | must strictly shrink. As such, the nodes will eventuallybecome close enough that x ( t ) ∈ I , and switching of node 1 ceases.We complete the proof by showing that once node 2’s signal mix has entered the inner region I , it cannot leave it.We have established already that nodes must have opposing orientations at all times. Let us consider the case where x ( t n ) ∈ I u and x ( t n ) < /
2. Suppose by contradiction that in time step t n + 1 node 2 is able to “escape” I frombelow, going from below 1 / (2(1 − q )) to above such value (i.e., to O u ). This implies We require δ < | δ | <
1. While δ < − x ( t + 1) < x ( t + 1),leading to a contradiction. − q ) < x ( t n + 1) = yx ( t n ) + (1 − y )(1 − q ) x ( t n ) + (1 − y ) q< y − q ) + 12 (1 − y )(1 − q ) + (1 − y ) q , which leads to 1 + q < (1 − q ) − , i.e. to the impossible result q <
0. Therefore, node 2 cannot go from I to O u .Finally we also know that it cannot go from I to O l as this would require both nodes to switch orientation, which isruled out because x ( t ) ∈ I . A parallel argument can be made if the orientations are reversed. Thus, the two nodesymmetric network will always converge to a region of the signal mix space where the nodes’ signal mixes are tooclose to support any switch of orientation, arriving at the desired result.The above proof can be easily replicated after relaxing the simplifying asymptotic assumption that q is fixed. Thiscan be done by reintroducing the time-dependent random weights w i ( t ) ( i = 1 , q , for any (cid:15) > t ∗ such that for all t > t ∗ and for all iq − (cid:15) < w i ( t ) < q + (cid:15) . Adjusting the bounds used in the convergence proof to include the above time evolution allows to obtain the sameresult.
B. Star network.
Let us now consider a k -star network of biased agents, with the central node labeled as 0 and branch nodes labeledas 1 , . . . , k . As before, allow y to be the self-weight of each node and q the confirmation bias parameter. Assume forsimplicity that the central node has a weight of (1 − y ) /k on each branch node.Firstly, note that if any branch node switches indefinitely, then the central node 0 must also switch indefinitely (orelse there would be no “driving force” causing the branch nodes to switch). So, let us focus on showing that it isimpossible for the central node to do so. The logic of the two-node network proof can be followed almost exactly byreplacing x ( t ) and x ( t ) with x ( t ) and (cid:80) kj =1 x j ( t ) /k , respectively.The first set of results up to (II A) follow precisely given the substitution of terms above. We use this to establishonce again that for x ( t ) to switch indefinitely (cid:80) kj =1 x j ( t ) /k must oscillate between 1 / (2(1 − q )) and (1 − q ) / (2(1 − q )),i.e., between the upper and lower hemispheres of the outer region O . Furthermore, whenever the central node switchesorientation, the branch nodes’ average signal mix must also change from above to below 1 / t n , x ( t n ) > /
2, and (cid:80) kj =1 x j ( t n ) /k < /
2. At t n + 1, this orientation switches so that (cid:80) kj =1 x j ( t n + 1) /k > x ( t n + 1). Adapting Eqs. (7)and (8) to the present case, we have1 k k (cid:88) j =1 x j ( t n + 1) − x ( t n + 1) = k k (cid:88) j =1 ( x j ( t n ) + (1 − y )(1 − q ) x ( t n ) + (1 − y ) qg j ( t n )) − yx ( t n ) + (1 − y )(1 − q ) 1 k k (cid:88) j =1 x j ( t n ) + (1 − y ) q , where we have introduced a new indicator variable such that g j ( t ) = 1 is node j is positively oriented at time t ,and g j ( t ) = 0 otherwise. Let g ( t ) = (cid:80) kj =1 g j ( t ) /k be the fraction of positively oriented branch nodes. We can thensimplify the above: Strictly speaking, the signal diffusion mechanism would need to be modified for non-regular graphs to allow for signal diffusion to beequivalent to node averaging. More complex regular structures can also be shown to converge, but a star graph permits us to show howconvergence holds even with a strikingly different topology. We proceed with the star graph for the purpose of illustration. k k (cid:88) j =1 x j ( t n + 1) − x ( t n + 1) = [(1 − y )(1 − q ) − y ] x ( t n ) − k k (cid:88) j =1 x j ( t n ) − (1 − y ) q (1 − g ( t n )) < [(1 − y )(1 − q ) − y ] x ( t n ) − k k (cid:88) j =1 x j ( t n ) , where we used the fact that (1 − y ) q (1 − g ( t n )) >
0. This can be rewritten as1 k k (cid:88) j =1 x j ( t n + 1) − x ( t + 1) < δ x ( t n ) − k k (cid:88) j =1 x j ( t n ) , where δ = [(1 − y )(1 − q ) − y ] <
1, which re-establishes the result of (11): if the central node flips, it must convergestrictly closer to the branch nodes. Next, we establish that in the time steps where the central node does not switch
FIG. 8: A network structure consisting of a central node 0 and k = 6 branch nodes. All nodes are biased agents for the purposesof the toy example. orientation, the centre and branches still converge as long as the branch average is within O . The reasoning followsthe one of the previous section exactly given the appropriate substitutions, and we can replace the condition in (12)with: x ( t n ) − k k (cid:88) j =1 x j ( t ) > q (1 − g ( t n ))2 − q − µ − y . (14)Recall that x ( t n ) > / (cid:80) kj =1 x j ( t ) /k < (1 − q )(2(1 − q )) ∈ O l , therefore: x ( t n ) − k k (cid:88) j =1 x j ( t ) > − − q − q ) > q − q − µ − y ≥ q (1 − g ( t ))2 − q − µ − y for an arbitrarily small µ >
0. Hence, even if a switch does not occur, then the nodes will converge strictly closer.The final steps of the proof mirror those that follow (13) of the previous section, except a factor of g ( t n ) dampensthe ability of the branch nodes to escape the inner region even further. As such, we establish that even on a starnetwork structure, the biased agents cannot switch their orientation endlessly, and must eventually converge.8 C. Simulated dynamics and convergence criteria.
As has been established in the previous sections, settling is guaranteed under some simple network topologies chosenspecifically to hinder convergence. We round out the argument by noting that settling also occurs in simulations forthe k -regular network employed throughout the paper and in the following proofs.In what follows, we establish criteria for the case of a fixed q . Analogous criteria can be easily established for thecase of stochastic convergent weights w i ( t ) instead, although without much adding much insight. Furthermore, inpractice the stochasticity rapidly settles in numerical simulations, meaning that convergence can be safely studiedusing the asymptotic fixed q assumption.In order to guarantee that a network has in fact settled over the course of a simulation, we identify a “settling” rulefor the signal mix ˆ x ( t ). As we demonstrate in the following section, if one assumes that the biased agents at time t no longer switch orientations, one can calculate the steady state that would arise from this configuration of biasedagents. Call this ˆ x ∗ ( ˆ x ( t )). We can show that if the signal mix ˆ x ( t ) is sufficiently close to its corresponding steadystate ˆ x ∗ ( ˆ x ( t )) it will converge uniformly to that steady state without any further changes to any agent’s orientation.Define the difference between a signal mix and its steady state: (cid:15) ( t ) = ˆ x ( t ) − ˆ x ∗ . Recalling that the model’s dynamics is such that ˆ x ( t ) = ˆ A ˆ x ( t − ˆ x ∗ + (cid:15) ( t ) = ˆ A ˆ x ∗ + ˆ A(cid:15) ( t −
1) = ˆ x ∗ + ˆ A(cid:15) ( t − , and therefore: (cid:15) ( t ) = ˆ A(cid:15) ( t − . Finally, define (cid:15) ∗ ( t ) = max i ( | (cid:15) i ( t ) | ) = || (cid:15) ( t ) || ∞ . Then for any arbitrary biased node: (cid:15) i ( t + 1) = ˆ a ii (cid:15) i ( t ) + (1 − q ) (cid:88) j ˆ a ij (cid:15) j ( t ) , where ˆ a ij is the weight between node i and j in matrix ˆ A , and we use the fact that the ghost nodes are always attheir exact steady state, so their (cid:15) G = 0. Then taking the absolute distance and using the triangular inequality: | (cid:15) i ( t + 1) | ≤ ˆ a ii | (cid:15) i ( t ) | + (1 − q ) (cid:88) j ˆ a ij | (cid:15) j ( t ) |≤ ˆ a ii | (cid:15) ∗ ( t ) | + (1 − q ) (cid:88) j ˆ a ij | (cid:15) ∗ ( t ) | = (1 − q (1 − ˆ a ii )) | (cid:15) ∗ ( t ) | < | (cid:15) ∗ ( t ) | , and similarly for an unbiased agent, we can show: | (cid:15) i ( t + 1) | ≤ | (cid:15) ∗ ( t ) | . In short, for each steady state once the current signal mixes are within some (cid:15) -cube of the steady state, they mustremain within that (cid:15) -cube. Furthermore, biased agents at each time step must converge strictly closer to the steadystate. A larger q or smaller self-weight (ˆ a ii ) will cause faster convergence.Finally, we can also note because the network is strongly connected, there are some r ≥ all nodes must converge strictlycloser to the steady state than the maximum threshold of the (cid:15) -cube.Given all the above, we can now explicitly outline a “stable” region. Denote: (cid:15) s = min i (cid:18) | x ∗ i − | (cid:19) . (cid:15) s -ballof the steady state, then there can be no crossing the 0 . k -regularnetworks. We tested the condition on 1000 iterations each of the following parameter sets: n = 10 , p = 0 . k = { , , , , , } , f = { . , . , . , . , } , q = { . , . , . , . , . . } . The settling criteria was successfullyreached for every single run of the model, establishing extremely high confidence that the k -regular biased informationaggregation model always settles. III. SECTION S3: CONVERGENCE OF SIGNAL MIXES
So far, we have established that the random update matrix A ( t ) converges almost surely to a fixed update matrix A . Furthermore, we have demonstrated with extremely high confidence that biased agents settle in their orientationafter some finite time. As such, for biased k -regular networks, assume that there exists some time t ∗ after whichbiased agents cease switching their orientation. Define ˆ y ∗B as the steady state fraction of positively oriented biasedagents. Then the following holds.(1) The signal mix vector ˆ x ( t ) converges to some ˆ x ∗ = ˆ A ∗ ˆ x (0) for both biased and unbiased networks, where ˆ A ∗ isa steady-state matrix of influence weights which can be computed explicitly.(2) Unbiased networks achieve consensus, and converge to influence weights of a ∗ ij = 1 /n for all pairs ( i, j ). Thisensures that, for all i ∈ V , x ∗ i = x ∗ V = ¯ x (0), where ¯ x (0) = (cid:80) ni =1 s i is the intial average signal mix.(3) Biased networks where ˆ y ∗B = 0 , a ∗ ij = 0 for all pairs( i, j ) ∈ V , ˆ a ∗ i + = ˆ y ∗B and ˆ a ∗ i − = 1 − ˆ y ∗B for all i ∈ V .(4) Biased networks where 0 < ˆ y ∗B < a ∗ ij = 0 for all( i, j ) ∈ V , and ˆ a ∗ i + + ˆ a ∗ i − = 1 for all i ∈ V .We note that results regarding unbiased networks (part of (1) and all of (2)) are already well established results(see, for example, [36]) and are listed purely for comparison with biased networks. We focus on proving the remainderof the results.The results follow from the structure of ˆ x ∗ = ˆ A ∗ ˆ x (0) = lim t →∞ (cid:81) tτ =0 ˆ A ( τ ) ˆ x (0). We proceed by demonstrating thatdespite the stochasticity in the random update mechanism ˆ A ( τ ), the steady state converges to a fixed vector ˆ x ∗ .First note that for τ > t ∗ the biased agents will have ceased switching their orientation, and the random updatematrix ˆ A ( τ ) will have a fixed underlying structure ˆ A = E ( ˆ A ( τ )). The proof will follow by demonstrating thatlim t →∞ (cid:81) tτ =0 ˆ A ( τ ) = lim t →∞ ˆ A t . That is, the products of random matrices converges to the products of their expectation.Firstly recall the block structure of ˆ A ( τ ): ˆ A ( τ ) = (cid:20) Q ( τ ) R ( τ )0 I (cid:21) , with dimensions (clockwise from top-left): ( n × n ), ( n × × × Q ( τ ) = Q + (cid:15) Q ( τ ) a.s. −−→ QR ( τ ) = R + (cid:15) R ( τ ) a.s. −−→ R .
The properties above indicate that the blocks converge to their deterministic counterparts almost surely. This allowsus to state that for any (cid:15) and matrix norm || . || , there is guaranteed some t (cid:48) ≥ t ∗ such that for all τ > t (cid:48) , || Q − Q ( τ ) || = || (cid:15) Q ( τ ) || < (cid:15) . Also, the matrix Q is such that (cid:88) j Q ij < ∀ i ∈ B (cid:88) i Q ij < ∀ j ∈ ∂ B . Q is both row and column sub-stochastic. Row sub-stochasticity follows from the outgoing edges from the set of biased agents ( B ). For the k -regular graphs that arethe focus of our analysis this can be specifically shown to be (1 − q ) k +1 k +1 . Column sub-stochasticity follows from theneighbours of the biased agents ( ∂ B ) having incoming connections necessarily less than 1.We now define the product of the random matrices ˆ A ( τ ) as: t (cid:89) τ =0 ˆ A ( τ ) = ˜ A ( t,
0) = (cid:20) ˜ Q ( t,
0) ˜ R ( t, I (cid:21) , where ˜ Q ( t, R ( t,
0) are placeholder terms for the the random block matrix products which arise through productsof the random matrices ˆ A ( τ ). Consider also the deterministic analog to this expression: t (cid:89) τ =0 ˆ A = ˆ A t = ˙ A ( t,
0) = (cid:20) ˙ Q ( t,
0) ˙ R ( t, I (cid:21) . This formulation defines a random and analogous deterministic sequence for each of the blocks, denoted by ˜ A ( t, A ( t,
0) respectively.Firstly, we demonstrate that lim t →∞ ˙ Q ( t,
0) = lim t →∞ Q t = . Consider the 2-norm || . || . We consider first the deter-ministic matrix Q . Recall that since Q is doubly sub-stochastic, Q T Q is necessarily sub-stochastic and therefore: || Q || = (1 − δ ) < , for some δ ∈ (0 , t →∞ || Q t || ≤ lim t →∞ || Q || t = lim t →∞ (1 − δ ) t = 0 , therefore lim t →∞ Q t = (making use of the fact that || X || = 0 ⇐⇒ X = ). Now consider the term of interest ˜ Q ( t, || ˜ Q ( t, || = || t (cid:89) Q ( τ ) || ≤ t (cid:89) || Q ( τ ) || . Note that || Q ( τ ) || ≤ τ . However we can show that almost all || Q ( τ ) || < || Q ( τ ) || = || Q + (cid:15) Q ( τ ) || ≤ || Q || + || (cid:15) Q ( τ ) || = (1 − δ ) + || (cid:15) Q ( τ ) || We select some t (cid:48) such that for all τ > t (cid:48) , || (cid:15) Q ( τ ) || = µ < δ for some µ >
0. Therefore: || Q ( τ ) || ≤ (1 − δ + µ ) = (1 − δ ∗ ) < , (15)where δ ∗ = δ + µ . We can now conclude:lim t →∞ || ˜ Q ( t, || ≤ lim t →∞ t (cid:89) t ∗ (1 − δ ∗ ) = lim t →∞ (1 − δ ∗ ) t = 0 . We now show that lim t →∞ ˜ R ( t,
0) = lim t →∞ ˙ R ( t,
0) = ( I − Q ) − R . Consider firstly the deterministic sequence, which canbe defined through the following iterative relationship:˙ R ( t,
0) = Q ˙ R ( t − ,
0) +
R . (16)1Note again that ˙ R ( t,
0) refers to the t -th term in a deterministic sequence whereas Q and R are specific block matrices.The expression (16) can be straightforwardly solved in the limit:lim t →∞ ˙ R ( t,
0) = ( I − Q ) − R .
Consider now the random sequence, which can be defined analogously:˜ R ( t,
0) = Q ( t ) ˜ R ( t − ,
0) + R ( t ) . (17)Note here ˜ R ( t,
0) refers to the t -th term in a random sequence and Q ( t ) and R ( t ) are random block matrices thatoccur at time τ = t . In order to proceed we define:˜ R ( t,
0) = ˙ R ( t,
0) + E ( t ) , (18)where here E ( t ) is an error term capturing the difference between the terms of the deterministic and random sequencesat time τ = t . We substitute (18) into (17):˜ R ( t,
0) = Q ( t )( ˙ R ( t − ,
0) + E ( t − R ( t ) . (19)We now substitute the definition of Q ( τ ) and R ( τ ) into (19):˜ R ( t,
0) = Q ˙ R ( t − ,
0) + R + QE ( t −
1) + (cid:15) Q ( t )( ˙ R ( t − ,
0) + E ( t − (cid:15) R ( t ) . Note that we can substitute (16) for the two leading terms on the RHS:˜ R ( t, − R ( t,
0) = QE ( t −
1) + (cid:15) Q ( t )( ˙ R ( t − ,
0) + E ( t − (cid:15) R ( t ) . We can now take the 2-norm || . || : || E ( t ) || = || ˜ R ( t, − ˙ R ( t, || = || QE ( t −
1) + (cid:15) Q ( t )( ˙ R ( t − ,
0) + E ( t − (cid:15) R ( t ) ||≤ || Q || || E ( t − || + || (cid:15) Q ( t ) || || ˜ R ( t − || + || (cid:15) R ( t ) || . We can now substitute in (15) and once again make use of the fact that for any 0 < (cid:15) we can define t (cid:48) such that || (cid:15) Q ( t ) || , || (cid:15) R ( t ) || < (cid:15) . We also note that || ˜ R ( t − || < n (where n is the size of the network), therefore || E ( t ) || ≤ (1 − δ ) || E ( t − || + (cid:15) ( n + 1) , and therefore for a sufficiently small (cid:15) , there is a corresponding t (cid:48) such that for t > t (cid:48) : || E ( t ) || ≤ (1 − δ ∗ ) || E ( t − || < || E ( t − || . Finally we get: lim t →∞ || ˜ R ( t, − ˙ R ( t, || = lim t →∞ || E ( t ) || = 0 , which allows us to conclude that lim t →∞ ˜ R ( t,
0) = lim t →∞ ˙ R ( t,
0) = ( I − Q ) − R .We can combine these results to conclude:ˆ A ∗ = lim t →∞ t (cid:89) τ =0 ˆ A ( τ ) = lim t →∞ ˜ A ( t,
0) = (cid:34) lim t →∞ ˜ Q ( t,
0) lim t →∞ ˜ R ( t, I (cid:35) = (cid:20) I − Q ) − R I (cid:21) . ˆ x ∗ = ˆ A ∗ ˆ x (0) we can conclude Result (1) - that the signal mixes do converge. In particular: ˆ x ∗ = ˆ A ∗ ˆ x (0) = I − Q ) − R + ( I − Q ) − R − ˆ x (0)10 = ( I − Q ) − R + , that is, the steady state signal mixes of the agents not a function of the initial signals ˆ x (0). The signal mixes areinstead entirely a function of the steady state orientations of the biased agents, encoded by the vector R + , the edgesfrom the positive biased agents to the positive ghost nodes.Our remaining conclusions follow summarily from this. If all biased agents are negative (ˆ y ∗B = 0) R + is 0 and x ∗ is0 for all agents. Inversely if all biased agents are positive (ˆ y ∗B = 1), x ∗ is 1 for all agents. For any other configurationof biased agents, the steady state is determined by the closed form ( I − Q ) − R + . In this scenario z ∗ V > z ∗ R ≥
0. That is, unbiased agentsare no longer guaranteed to converge despite having no bias mechanism themselves. We investigate this and otherproperties of the unbiased agents in more detail in the next section.
IV. SECTION S4: STEADY STATE SIGNAL MIX DISTRIBUTION
We now seek to approximate the distribution of signal mixes of the agents once the steady state is reached. We willfirst approximate the average steady state signal mix of each sub-population in the network, followed by the steadystate signal mix variance, and finally the full distribution itself. We will do this for the k -regular network case usedin the body of the paper, and show via numerical simulations that it also captures the model’s dynamics on moreheterogeneous networks. Let us note that the results given in the following are the empirical distribution of the signalmixes for a given run of the model, as opposed to an ensemble over all possible runs of the model. A. Steady state expected signal mix.
As detailed in I, the model converges to a steady state ˆ x ∗ which is entirely contingent on the settled orientation ofthe biased agents in the network. In what follows, we will calculate an approximation for the model’s steady stateexpected signal mix conditional on a given fraction of positively oriented biased agents f + ( t ) = ˆ y B ( t ) f . We willthen show how under some reasonable assumptions the “settled” value of f + ( t ) can be approximated from the initialorientation f + (0).Consider an agent i picked uniformly at random at time t from the unbiased, positively oriented biased, negativelyoriented biased sub-populations. Let us denote the signal mixes of agents belonging to such sub-populations as ˆ x i U ( t ),ˆ x i B + ( t ) and ˆ x i B− ( t ), respectively. We are interested in establishing the expected steady state values for each of thesequantities, denoted as ˆ x ∗U = lim t →∞ E [ˆ x i U ( t )] = lim t →∞ ˆ x U ( t ) , with analogous definitions for ˆ x ∗B + and ˆ x ∗B − .We begin by considering the sub-population of unbiased agents at some finite t . We note the following:ˆ x U ( t + 1) = E [ˆ x i U ( t + 1)] = E (cid:20) ˆ x i U ( t ) + (cid:80) j ∈ ∂ i ˆ x j ( t ) k + 1 (cid:21) = ˆ x U ( t ) + k E [ˆ x j ( t )] k + 1 , (20)where E [ˆ x j ( t )] refers to the expected signal mix of a randomly picked agent j from the entire population, which ofcourse consists of the three aforementioned sub-populations. Therefore, we have E [ˆ x j ( t )] = (1 − f ) E [ˆ x j ( t ) | j ∈ U ] + f + ( t ) E [ˆ x j ( t ) | j ∈ B + ( t )] + f − ( t ) E [ˆ x j ( t ) | j ∈ B − ( t )] (21)= (1 − f )ˆ x U ( t ) + f + ( t )ˆ x B + ( t ) + f − ( t )ˆ x B − ( t ) . Plugging the above in (20) we getˆ x U ( t + 1) = 1 k + 1 (cid:2) (1 + k (1 − f ))ˆ x U ( t ) + kf + ˆ x B + ( t ) + kf − ˆ x B − ( t ) (cid:3) . x B + ( t + 1) = E [ˆ x i B + ( t + 1)] = E (cid:20) ˆ x i B + ( t ) + (cid:80) j ∈ ∂ i ((1 − w i ( t ))ˆ x j ( t ) + w i ( t )) k + 1 (cid:21) = ˆ x B + ( t ) + (1 − q ) k E [ˆ x j ( t )] + kqk + 1 . where we have explicitly referenced the random variable w i ( t ) representing the fraction of successfully distortednegative signals (see I), and made use of the fact that E [ w i ( t )] = q . We can use (21) again and writeˆ x B + ( t + 1) = 1 k + 1 (cid:2) k (1 − f )(1 − q )ˆ x U ( t ) + ((1 − q ) kf + ( t ) + 1)ˆ x B + ( t ) + (1 − q ) kf − ( t )ˆ x B − ( t ) + kq (cid:3) , and similarly for negatively oriented biased agents:ˆ x B − ( t + 1) = 1 k + 1 (cid:2) k (1 − f )(1 − q )ˆ x U ( t ) + (1 − q ) kf + ( t )ˆ x B + ( t ) + ((1 − q ) kf − + 1)( t )ˆ x B − ( t ) (cid:3) . We have therefore established the update rule for the expected signal mix of the three sub-populations at any time t . We collate this update rule into a matrix form for convenience: ξ ( t + 1) = 1 k + 1 ( F ( t ) + I ) ξ ( t ) + b , (22)where ξ ( t ) = [ˆ x U ( t ) , ˆ x B + ( t ) , ˆ x B − ( t )] T , b = (cid:20) , kqk + 1 , (cid:21) T and F ( t ) = k (1 − f ) f + ( t ) f − ( t )(1 − q )(1 − f ) (1 − q ) f + ( t ) (1 − q ) f − ( t )(1 − q )(1 − f ) (1 − q ) f + ( t ) (1 − q ) f − ( t ) . If we further simplify notation by defining ˆ F ( t ) = ( F ( t ) + I ) / ( k + 1), we get to the following compact expressionfor (22): ξ ( t + 1) = ˆ F ( t ) ξ ( t + 1) + b . The long-run evolution of the signal mixes can be determined from the above equation if the evolution of f + ( t )(and, consequently, of f − ( t )) in the matrix ˆ F ( t ) is known. Assume for the moment we are at some time t ∗ at whichthe system has settled, i.e., biased agents will keep their orientations intact and therefore will not cause the value of f + ( t ) to change for t > t ∗ . In this case we can write:lim t →∞ ξ ( t ) = lim t →∞ ˆ F ( t ∗ ) t ξ ( t ∗ ) + ( ˆ F ( t ∗ ) − I ) − b . It can be shown easily that, due to its double substochasticity, we have lim t →∞ ˆ F ( t ∗ ) t = 0, and thereforelim t →∞ ξ ( t ) = ( ˆ F ( t ∗ ) − I ) − b . The above limit allows to calculate the steady state signal mixes for all sub-populations explicitly:lim t →∞ ξ ( t ) = ˆ x ∗U ˆ x ∗B + ˆ x ∗B − = f + ( t ∗ ) /f (1 − q ) f + ( t ∗ ) /f + q (1 − q ) f + ( t ∗ ) /f = ˆ y B ( t ∗ )(1 − q )ˆ y B ( t ∗ ) + q (1 − q )ˆ y B ( t ∗ ) . (23)In the next Section we will approximate this result to the case where biased agents have not settled their orientationyet.4 B. Predicting the trajectory of biased agents’ orientations
Biased agents change their orientation when they receive a stream of incongruent signals that overcome their abilityto distort them using confirmation bias. There are two points in the evolution of the model where this is possible.Firstly, this may happen in the early stages of the evolution, where the information sets held by the agents arerelatively small and the stochasticity of the model can induce changes in orientation. Secondly, this may happen inthe long run, where sustained changes in orientation can be brought along when one of the two camps of biased agentsbecomes able to systematically bias the available information. This leads to the composition of signals experiencedby each node to change consistently in one direction, which can cause large scale switches in orientation, which inturn triggers a domino effect, as newly switched nodes will accelerate the rate at which signals are distorted.Let us capture this notion more formally. Consider the expected long term signal mix of each sub-populationassuming the biased agents have settled ((23)). Suppose the positively oriented biased agents have an expectedsteady state signal mix ˆ x ∗B + < /
2. If such steady state value is to be reached, then some positively oriented biasedagents’ signal mixes must fall below 1 /
2, thereby switching orientation to negative. If this happens, then ˆ y B ( t ) fallsand the steady state signal mix for all agents strictly decreases . This, in turn, means more positively orientedbiased agents switch orientation to reach their steady state, and so forth until all such agents switch to a negativeorientation, yielding ˆ y B = 0. A corresponding outcome can be determined for negatively oriented biased agents allbeing converted. We can therefore determine, for any given t ∗ , the approximate conditions under which we expectall positively oriented biased agents to switch their orientations to negative in the eventual steady state. Settingˆ x ∗B + < / − q )ˆ y B ( t ∗ ) + q <
12 = ⇒ ˆ y B ( t ∗ ) < − q , (24)and, correspondingly, for all negatively oriented biased agents to be tipped to positive we have the following condition:ˆ y B ( t ∗ ) > − q − q . (25)Let us now consider the case t = 0, which means we are approximating the expected trajectory of the entiresystem given a starting fraction of positively oriented biased agents f + ∗ (0) /f = ˆ y B (0). We then have the followingapproximate result for the steady state signal mix of the unbiased sub-population:ˆ x ∗U = ˆ y ∗B ≈ ˆ y B (0) for − q ) ≤ ˆ y B (0) ≤ − q − q ) y B (0) > − q − q ) y B (0) < − q ) , where the latter two conditions derive from Eqs. (24) and (25), while the first condition is the same reported in (23)adapted for the case t = 0. C. Steady state signal mix variance.
In the previous Section we have provided approximations for the first moment of the steady state signal mixesof the unbiased agents, as well as those of the two biased agent sub-populations. We have also approximated thelong term “settled” fractions of positively and negatively oriented biased agents. We noted that for ˆ y B (0) > − q − q ) (ˆ y B (0) < − q ) ), the steady state signal mix is likely to asymptotically reach 1 (0). Under these conditions, all agentseventually trivially possess the same signal +1 ( − − q ) ≤ ˆ y B (0) ≤ − q − q ) , and as such use (23) to approximate: This can be proven rigorously with the results from the previous Section: the steady state mix is ( I − Q ) − R + . R + is a vector with 0for each negative biased agent, and ( I − Q ) − is element-wise >
0. A biased agent switching to positive turns a previous zero elementof R + to positive, and adds another strictly positive vector to the steady state signal mix. The same argument is made in reverse for apositive to negative switch t →∞ ξ ( t ) = ˆ y B ( t ∗ )(1 − q )ˆ y B ( t ∗ ) + q (1 − q )ˆ y B ( t ∗ ) ≈ ˆ y B (0)(1 − q )ˆ y B (0) + q (1 − q )ˆ y B (0) Correspondingly, we note f + ∗ = ˆ y B ( t ∗ ) = ˆ y B (0). We would now like to characterise the distribution of signal mixesfor each sub-population at the steady state beyond its first moment. We begin with an approximation of the variance,under the asymptotic limit of large populations n → ∞ .For convenience we define the steady state signal mix variance of any sub-population G as σ G . In the following, wewill provide approximate expressions for the steady state signal mix variances σ U , σ B + , and σ B − , and for the overallvariance σ V .Consider an agent i picked uniformly at random from the entire population. The variance of such an agent’s steadystate signal mix Var[ˆ x ∗ i ] represents the variance across the entire population σ V . From the law of total variance, thiscan be broken down as follows σ V = Var[ˆ x ∗ i ] = E (cid:2) Var[ˆ x ∗ i ] | i ∈ {U , B + , B − } (cid:3) + Var (cid:2) E (cid:2) ˆ x ∗ i | i ∈ { U, B + , B − } (cid:3) (cid:3) , (26)where E (cid:2) Var[ˆ x ∗ i ] | i ∈ {U , B + , B − } (cid:3) = (1 − f ) σ U + f + ∗ σ B + + f −∗ σ B − , (27)and Var (cid:2) E (cid:2) ˆ x ∗ i | i ∈ {U , B + , B − } (cid:3) (cid:3) = (1 − f ) (cid:26) f + ∗ f − E (cid:104) E [ˆ x ∗ i | i ∈ {U , B + , B − } ] (cid:105)(cid:27) (28)+ f + ∗ (cid:26) (1 − q ) f + ∗ f + q − E (cid:104) E [ˆ x ∗ i | i ∈ {U , B + , B − } ] (cid:105)(cid:27) + f −∗ (cid:26) (1 − q ) f + ∗ f − E (cid:104) E [ˆ x ∗ i | i ∈ {U , B + , B − } ] (cid:105)(cid:27) . Noting that E (cid:104) E [ˆ x ∗ i | i ∈ {U , B + , B − } ] (cid:105) = (1 − f ) f + ∗ f + f + ∗ (cid:18) (1 − q ) f + ∗ f + q (cid:19) + f −∗ (cid:18) (1 − q ) f + ∗ f (cid:19) = f + ∗ f we can considerably simplify (28):Var (cid:2) E (cid:2) ˆ x ∗ i | i ∈ {U , B + , B − } (cid:3) (cid:3) = q f + ∗ f −∗ f = f q ˆ x ∗U (1 − ˆ x ∗U ) , where in the last step we have used the fact that ˆ x ∗U = f + ∗ /f , as per (23).(28) provides a compact expression for the second contribution for the overall variance σ V in (26). We now turn tothe first term ((27)). In order to be able to calculate it, we must compute the variance of each sub-population. Letus begin with the unbiased agent sub-population: σ U = Var[ˆ x ∗U ] = Var k (cid:88) j ∈ ∂ i ˆ x ∗ j = σ V k + 2 k (cid:88) ( j,(cid:96) ) ∈ ∂ i Cov[ˆ x ∗ j , ˆ x ∗ (cid:96) ] = σ V k + O (cid:18) k (cid:19) , (29)where in the last term we have assumed the covariance term to decay as k − , which will be proved in the next Section.In analogy with the above results, for the biased agent sub-population (either positively or negatively oriented) wehave: σ B = Var[ˆ x ∗ i B ] = (1 − q ) σ V k + O (cid:18) k (cid:19) . (30)Substituting the two expressions above in (27), and combining the result with the one obtained in (28), we finallyobtain the following result for the overall variance in (26): σ V = 1 − f q (2 − q ) k σ V + f q ˆ x ∗U (1 − ˆ x ∗U ) + O (cid:18) k (cid:19) . σ V we get σ V = kf q ˆ x ∗U (1 − ˆ x ∗U ) k + f q (2 − q ) − kk + f q (2 − q ) − O (cid:18) k (cid:19) ≈ f q ˆ x ∗U (1 − ˆ x ∗U ) , where we have used the fact that f q (2 − q ) − ∈ [ − , f q (2 − q ) − (cid:28) k even for moderate connectivity.Finally, we can specialize the above result to the three sub-populations via Eqs. (29) and (30): σ U ≈ f q ˆ x ∗U (1 − ˆ x ∗U ) k (31) σ B ± ≈ f ( q (1 − q )) ˆ x ∗U (1 − ˆ x ∗U ) k . (32) D. Explicit neighbourhood covariance expressions.
In this Section we establish that the covariance term appearing in (29) can indeed be assumed to be of order k − .The first thing to note is that the term Cov[ˆ x ∗ j , ˆ x ∗ (cid:96) ] for two generic agents can be bounded above by the covarianceCov[ˆ x ∗ j , ˆ x ∗ (cid:96) | j, (cid:96) ∈ U ] between unbiased agents’ steady state signal mixes. To see this, suppose j, (cid:96) ∈ B + :Cov[ˆ x ∗ j , ˆ x ∗ (cid:96) | j, (cid:96) ∈ B + ] = Cov (cid:34) (1 − q ) (cid:80) h ∈ ∂ j ˆ x ∗ h + qkk , (1 − q ) (cid:80) m ∈ ∂ (cid:96) ˆ x ∗ m + qkk (cid:35) = (1 − q ) k Cov (cid:88) h ∈ ∂ j ˆ x ∗ h , (cid:88) m ∈ ∂ (cid:96) ˆ x ∗ m < k Cov (cid:88) h ∈ ∂ j ˆ x ∗ h , (cid:88) m ∈ ∂ (cid:96) ˆ x ∗ m = Cov[ˆ x ∗ j , ˆ x ∗ (cid:96) | j, (cid:96) ∈ U ] . Let Cov(ˆ x ∗ j , ˆ x ∗ l | ( j, l ) ∈ U ) = σ , denote the least upper bound for the covariance between two nodes of distance 2 FIG. 9: The covariance of i ’s neighbours, j and k (grey), can be decomposed into the covariance between its neighbours (blackand white). This consists of the covariance between neighbours that are two steps apart (black to white) as well as those thathave distance four steps apart (white to white). apart. Similarly, let σ d be the same for nodes of distance d apart. In the remainder of this section we are seeking toestablish a relationship between these upper bounds and in doing so recursively determine the upper bound at d = 2.It is worth reiterating that we are approximating the variance of the steady state signal mixes at the asymptoticlimit n → ∞ . Given this assumption, a k -regular tree will approximate a Cayley tree. A useful consequence of thisassumption is that the network is globally tree-like, and loops vanish in the limit. As such, only a single path existsbetween any two nodes. Therefore, in (29) we have: σ U = σ V k + 2 k (cid:88) ( j,(cid:96) ) ∈ ∂ i Cov[ˆ x ∗ j , ˆ x ∗ (cid:96) ] ≤ σ V k + 2 k (cid:88) ( j,(cid:96) ) ∈ ∂ i σ = σ V k + σ k ( k − . σ = O ( k − ), the whole expression will be of order O ( k − ). To do this note:Cov[ˆ x ∗ j , ˆ x ∗ (cid:96) ] = 1 k Cov ˆ x ∗ i + (cid:88) m ∈ ∂ j /i ˆ x ∗ m , ˆ x ∗ i + (cid:88) n ∈ ∂ (cid:96) /i ˆ x ∗ n = 1 k Cov[ˆ x ∗ i , ˆ x ∗ i ] + (cid:88) ( m,n ) ∈ [ ∂ j × ∂ (cid:96) ] / ( i,i ) Cov[ˆ x ∗ m , ˆ x ∗ n ] = 1 k Cov[ˆ x ∗ i , ˆ x ∗ i ] + (cid:88) m ∈ ∂ j /i Cov[ˆ x ∗ i , ˆ x ∗ m ] + (cid:88) n ∈ ∂ (cid:96) /i Cov[ˆ x ∗ i , ˆ x ∗ n ] + (cid:88) ( m,n ) ∈ [ ∂ j /i × ∂ (cid:96) /i ] Cov[ˆ x ∗ m , ˆ x ∗ n ] . In this last step, we explicitly break down the covariance sum into the covariance between the neighbours of j and l ,which exist at various distances to one another. This is illustrated in 9.We can group these covariance pairs by their distance and bound them using our defined bounds σ d :Cov[ˆ x ∗ j , ˆ x ∗ (cid:96) ] ≤ k (cid:0) σ U + 2( k − σ + ( k − σ (cid:1) . This allows us to recursively define σ = 1 k (cid:0) σ U + 2( k − σ + ( k − σ (cid:1) . Re-arranging this expression we get: σ = 1( k − + 1 σ U + ( k − ( k − − σ = 1( k − + 1 σ + ( k − ( k − − σ . In the final step, we have replaced σ U with σ - which emphasizes that this term is merely the covariance of a nodewith a node at distance 0 (i.e. its own variance), and σ U is the largest possible variance expression amongst the biasedand unbiased nodes. We can easily (though quite tediously) repeat this process for the covariance of nodes at anydistance d to establish: σ d = 1( k − + 1 σ d − + ( k − ( k − − σ d +2 . This linear recurrence relation can be solved with the boundary conditions that σ = σ U and lim d →∞ σ d = 0 . Thisestablishes: σ d = σ U ( k − d , and therefore: σ = σ U ( k − = O ( k − ) . Therefore, to finalise (29): σ U = 1 k σ V + O (cid:18) k ( k − k ) σ (cid:19) = 1 k σ V + O ( k − ) . E. Steady state signal mix normality.
We finally proceed to demonstrate that the distribution of signal mixes is approximately normal when k and n arelarge. Assume firstly that n → ∞ , which ensures that the model’s k -regular network becomes a Cayley tree with In other words, nodes at infinitely long distances have a covariance that decays to zero k = (cid:15)n for some (cid:15) >
0, ensuring k also grows arbitrarily large, but still can bearbitrarily smaller than n .As we have already established in the previous section, the covariance between the steady state signal mixes ofunbiased agents at distance 2 decays as k − , which implies that such signals mixes become asymptotically independentin the aforementioned limits. Therefore, the steady state signal mix of an unbiased agent ˆ x ∗ i U = (cid:80) j ∈ ∂ i ˆ x ∗ j /k becomesthe sum of an infinitely large set of independent random variables. Furthermore, the variables will follow one of threedistributions, depending on which sub-population the agent’s neighbors belong to:ˆ x ∗ i U = 1 k (cid:88) j ∈ ∂ i ˆ x ∗ j = 1 k (cid:88) j ∈ ∂ i ∩ U ˆ x ∗ j + (cid:88) j ∈ ∂ i ∩B + ˆ x ∗ j + (cid:88) j ∈ ∂ i ∩B − ˆ x ∗ j . Each of the above contributions is a sum of an infinitely large set of independent and identically distributed variables,which implies that each of them is normally distributed. This, in turn, implies that the steady state signal mixes ofthe unbiased agents (and, by generalisation, of the biased agents) is asymptotically normal. From the results obtainedfor the first two moments of the signal mix distributions in the previous sections (see Eqs. (23) and (31)), we canconclude that when n → ∞ and k = (cid:15)n we haveˆ x U d → N (cid:18) ˆ x ∗U , f q ˆ x ∗U (1 − ˆ x ∗U ) k (cid:19) ˆ x B + d → N (cid:32) (1 − q )ˆ x ∗U + q, f ( q (1 − q )) ˆ x ∗U (1 − ˆ x ∗U ) k (cid:33) ˆ x B − d → N (cid:32) (1 − q )ˆ x ∗U , f ( q (1 − q )) ˆ x ∗U (1 − ˆ x ∗U ) k (cid:33) . The normality of the distribution for the unbiased agents is demonstrated in Figure 3.
V. SECTION S5: ACCURACY
We can now aggregate our results in order to approximate the accuracy of a social network. As described in themain body of the paper, the accuracy of a network A ( G ) is the expected fraction of accurate unbiased agents in thesteady state, i.e. accuracy quantifies the probability Prob( y i U = +1) that a randomly picked unbiased agent in arandom realisation of the model will correctly learn the ground truth .This is of course a complex outcome determined by a dynamic series of processes worth recapping. First, the modelwill generate initial signals for all agents, both biased and unbiased. All agents will share their signals, but biasedagents will selectively sample incoming signals based on their current orientation. Over time, biased agents are able toinfluence the set of signals in the system, and the system converges towards a steady state where each agent possessesan equilibrium mix of signals. Accurate agents are those whose equilibrium signal mix is contains more positive thannegative signals, i.e. x ∗ i > / A ( G ) = 12 (cid:90) dx ∗U P ( x ∗U ) erfc (cid:18) − x ∗U √ σ U (cid:19) , (33)where P ( x ∗U ) is the distribution of the average signal mix across unbiased agents as determined by (IV B) (see Eq. (3)of the main paper), while the complementary error function quantifies the fraction of unbiased agents whose steadystate signal mix is above 1 /
2, and are therefore accurate, under the normal approximation outlined in the previoussection. The definition of accuracy could very easily be extended to all agents instead of just unbiased agents, but we retain discussion tounbiased agents for simplicity k -regular and Erd˝os-R´enyi networks. As can be seen, the average accuracyobtained across independent numerical simulations of the model closely matches the expected value obtained with(33), even for relatively low network size and average degree (the results reported were obtained for n = 10 and k = 8). The wider error bars for lower f reflect the expected outcome that most runs of the model will result in totalconsensus on either X = ± f grows, the agents arehighly polarised and the fraction of accurate and inaccurate agents will be relatively constant. -2 -1 FIG. 10: Non-monotonic changes in expected accuracy as f increases. The model’s prediction are compared to numericalresults obtained with simulations on both k -regular (light blue) and Erd˝os-R´enyi (purple) networks. The parameters used inthe simulations were n = 10 , p = 0 . k = 8, q = 1. VI. SECTION S6: REGRESSION RESULTSA. Theory and model interpretation.
Our model is stylized, and therefore largely agnostic as to a particular interpretation of its parameters. Nevertheless,it is quite well suited to provide an initial exploration on a number of issue. In this Section, we shall test the model’sability to shed light on the impact that Internet access has on shaping popular opinion on specific issues (globalwarming in this case). In order to do this, we first specify how we are going to relate our model’s parameters toreal-world measurable quantities.There are two convenient (and pragmatically equivalent) interpretations of the model in the context of Internet use.Consider the agent-specific ghost node interpretation, where each ghost node attached to a biased agent represents anaggregation of the “filter bubble” (passive algorithmic affects) and “selective exposure” (actively selecting informationin a biased way) effects. An increase in Internet access therefore translates to an increase in access to these self-confirmatory effects, and corresponds to changing unbiased agents into biased agents (an increase in f ). Alternatively,one could consider a scenario where the fraction of biased agents is fixed, in which case an increase in Internet wouldimprove their ability to obtain self-confirmatory information (an increase in q ). For the purposes of this exploration,however, the two effects are equivalent, and for convenience we only retain the interpretation where f increases.As far as the interpretation of the degree variable k is concerned, the important distinction to make here is that weare not interested in “social networks” as a catch-all term for the number of family and friends one has. Rather, giventhe model, we are interested in the degree to which individuals actively exchange information with their underlyingsocial network with regards to the topic of interest. Therefore, for k we wish to measure the volume of active socialinformation diffusion in a given population.As per (31), one of our model’s main results is that f and k work in opposite directions when it comes to po-0larisation - an increase in confirmatory behaviours increases polarisation and is equivalent to a reduction in socialinformation. Furthermore, if the majority of the population accurately learns the ground truth ( x ∗U > / B. Data sources and measuring variables
In order to test the model’s predictions in the aforementioned context, we gathered data from the Yale Programmeon Climate Change Communication 2016 Opinion Maps [49], which provides state and county level survey data onopinions on global warming, as well as behaviours such as the propensity to discuss climate change with friends andfamily. We combined this with FCC 2016 county level data on residential high speed Internet access [50]. Finally, wealso used a supplemental source in the data aggregated by the Joint Economic Council’s Social Capital Project [51],a government initiative aiming to measure social capital at a county level by aggregating a combination of state andcounty level data from sources such as the American Community Survey, the Current Population Survey, and theIRS.In this context, we measured accuracy as the estimated fraction of the population believing that “global warming ishappening”. We refer to this as “GW Accuracy”. In other words, we are attempting to examine the degree to whichsocial information and access to confirmatory bias mechanisms affect the ability of individuals to accurately learn anobjective, measurable and uncontroversial ground truth (that global temperatures are rising).Internet access is measured by the FCC’s data on county-level high speed broadband penetration amongst residen-tials (in [39], the authors utilise another instrumental variable approach to argue that increased broadband penetrationdoes in fact increase Internet use). In Table I we demonstrate preliminary ordinary least squares regression results byregressing GW Accuracy on Internet access, accounting for a range of covariates such as median age, median income,county population size and the fraction of adults with college degrees. The results indicate that even after controllingfor relevant covariates, the net effect of Internet access on accuracy is positive (and by interpretation, the effect ofpolarisation on this particular ground truth is negative).However, this alone is insufficient as research indicates Internet access is likely to improve the degree to whichindividuals can communicate information to friends and family, which in our model is precisely the variable k . TheYale Climate Change data includes a measure estimating the fraction of the county population that discusses globalwarming regularly with family and friends (“Social Discussion”). To sense check this, Table I (column 2) demonstratesthat increased Internet access does indeed improve the ability to discuss matters with friends and family, even aftercontrolling for relevant covariates, which is consistent with a broad set of empirical research on the topic (see [52] fora review).Therefore, this allows us to construct our final model in Table 1(3) where we regress GW Accuracy on both SocialDiscussion and Internet access (and the covariates). We can now interpret the coefficient on Internet access as theresidual effect of Internet access after controlling for the effect it has on Social Discussion. One way of thinking aboutthis is to consider all causal pathways from Internet access to belief formation - some fraction of them will be viaimproved access to social and discussion networks (communication platforms, online social networks, and forums),and the remaining fraction will be non-social (algorithmic effects, filter bubbles, online news media, selective exposure, Strictly speaking the result refers to the variance in information sets, but we exploit the monotonic relationship between informationvariance σ x ∗U and polarization z ∗U for the remainder of this section One may note that the impact of median income on this regression, and all subsequent results. is negative. We have verified this resultthrough a number of additional checks. It appears that the inclusion of college education heavily affects this coefficient, implying thatthe effect of income on global warming beliefs is heavily mediated by access to education. We also performed some further checks byincluding dummy variables for political partisanship using county level voting results for the 2016 presidential elections. While politicalpartisanship provides additional explanatory power over and above the current set of variables, the coefficient for income when includingit is still negative. Unpacking the exact nature of this relationship would require a broader range of economic and political factors,which is clearly outside the scope this initial analysis, so we exclude partisanship and continue with the original model, allowing thecoefficients to be taken at face value. TABLE I: Initial Regression Results
Dependent variable:
GW Accuracy Social Discussion GW Accuracy(1) (2) (3)Social Discussion 1.057 ∗∗∗ (0.019)Internet Access 1.550 ∗∗ ∗∗∗ -2.400 ∗∗∗ (0.664) (0.447) (0.471)Median Age -0.044 ∗∗∗ -0.020 ∗ -0.023 ∗ (0.017) (0.011) (0.012)log(Median Household Income) -6.691 ∗∗∗ -1.659 ∗∗∗ -4.938 ∗∗∗ (0.464) (0.313) (0.327)log(Total Pop) 0.690 ∗∗∗ -0.399 ∗∗∗ ∗∗∗ (0.071) (0.048) (0.050)College Education 0.335 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ (4.846) (3.267) (3.501)Observations 2,933 2,933 2,933 R R ∗∗∗ ∗∗∗ ∗∗∗ (df = 5; 2927) (df = 5; 2927) (df = 6; 2926) Note: ∗ p < ∗∗ p < ∗∗∗ p < etc). By accounting for the former effects by observing the discussion network size in Social Discussion, the residualeffect of internet access will aggregate all these other effects. This lines up with the interpretation of f in our model -Internet users will have access to these effects (“biased agents”) and non-Internet users will not. The results confirmour hypothesis - Social Discussion ( k ) and residual Internet Access ( f ) act in opposite directions when it comes tolearning the ground truth, even after conditioning on a range of covariates.It is worth unpacking these results in detail. The direct effect of a 1 percentage point increase in Internet access onglobal warming accuracy is negative ( − . (3 . of 1 . × . ≈ .
95. The net effect, of course, is positive(3 . − .
40 = 1 . . However, breaking down the causal mechanisminto its constituent elements - direct internet use effects vs socially mediated internet effects - allows us the capturethe nuance of what is actually happening. C. Accounting for simultaneous causality.
A clear shortcoming of the above analysis is the fact that the variable “Social Discussion” is likely to have a reversecausal relationship with the outcome variable of “GW Accuracy”. That is, the more likely individuals are to believeglobal warming is happening, the more likely they are to discuss this topic with friends and family.In order to account for this, we will take an instrumental variable approach. That is, we need some instrumentthat can account for independent variation in discussion with family and friends, which is otherwise unlikely to affectthe belief in global warming. We note as before that k can be interpreted as the fraction of the “underlying socialnetwork” that is activated to transmit social information related to the topic of global warming. We are thereforeinterested in a variable that can measure the pre-existing strength of these underlying social networks. To do so, we Table 1, Column 3, Row 2. Table 1, Column 2, Row 2. Table 1, Column 3, Row 1. Table 1, Column 1, Row 2. ) refers broadly to something “related to social relationships, social networks, and civilsociety”. More specifically, it is measured with an intention to reflect communities with “an abundance of close,supportive relationships” [51].The index itself measures a spectrum of factors, and in particular a “Community Health” subindex. The subindexis calculated as the leading principal component across a variety of state and county-level measures of communityengagement (where people ostensibly meet and socialise with friends and family), including religious congregations,non-religious non-profit activities, public meeting attendance, working with neighbours to fix things, attending ameeting where politics was discussed, etc. This index is then validated by examining bivariate correlations with abattery of county level benchmarks and measures of social dysfunction.The strength of this instrument is established in Table II (column 1), where a first stage least squares regressionis run to show that improvements in Community Health do translate to improved discussion with friends and family(controlling for covariates).The validity is established through a series of additional checks. Factors such as religious attendance, publicmeetings, etc. are unlikely to have a causal effect on people’s beliefs about global warming independent of them beinga medium to allow for social discussion of these topics. The only other reasonable and plausibly significant causalchannel is if these factors are caused by or cause an increase membership in social groups (for instance, politicalparties) that are strongly associated with reduced belief in global warming. In particular, it is well-established thatmembers of the Republican Party have a reduced belief in the existence of Global Warming [54]. To check this, weexamined the bivariate correlation between Community Health and the percentage of GOP votes cast in the 2016presidential election. The results were weak, with a correlation of only 0 .
14, meaning only 1 .
8% of the variation inthe measures were explained by the relationship.Having established the strength and validity of the instrument, we demonstrate the results from the two stageleast squares regression results in Table II (column 2). We can see the qualitative results of the simpler model havebeen preserved, with the effects predictably attenuated. However, the results are still significant, and corroborate ourtheory. After separating out the social and confirmatory effects of Internet access, we can see the impact on Accuracy(and Polarisation) both occur in the direction that we predict.Once again, let us unpack the results. The direct effect of a 1 percentage point increase in internet access on globalwarming accuracy is negative ( − . (3 . of 0 . × . ≈ .
74. The net effect, of course, is positive(2 . − .
71 = 1 . f , or q ). However, it leads to a significant positiveimpact on social discussion ( k ), and the net result of this is positive. This result remains robust even after controllingfor a battery of relevant covariates.It should be emphasized that this result is merely an initial exploration of how our model can provide some testablepredictions to empirical data, as opposed to a detailed effort to understand the effect of Internet access on globalwarming beliefs. Having said that, the initial results are encouraging, and we hope the clarity of the analytic resultsof our model pave the way for testing variations of the idea of biased information aggregation in a range of outcomesand settings. D. Making sense of broader empirical results.
We have seen so far that our model can help us decompose the effect of internet access on learning in the specificcase of global warming facts. We now see if the model can help us better understand the seemingly conflicting findingswe have found in existing research as indicated above. It should be said that the following interpretations are meantonly to be indicative of how our model can help shape our theoretical understanding of empirical phenomena, ratherthan a detailed exploration of the specific empirical questions these papers explore. i.e. Putnam [53] (1995, p.19), “...social capital refers to connections among individuals’ social networks and the norms of reciprocityand trustworthiness that arise from them”. Table 2, Column 2, Row 3. Table 2, Column 1, Row 2. Table 2, Column 2, Row 2. TABLE II: IV Regression Results
Dependent variable:
Social Discussion GW Accuracy
OLS instrumental(First Stage LS) variable (2SLS) (1) (2)Community Health Index 1.501 ∗∗∗ (0.081)Social Discussion 0.872 ∗∗∗ (0.060)Internet Access 3.143 ∗∗∗ -1.712 ∗∗∗ (0.424) (0.523)log(Median Household Income) -1.814 ∗∗∗ -5.281 ∗∗∗ (0.297) (0.346)log(Total Pop) 0.300 ∗∗∗ ∗∗∗ (0.059) (0.056)Median Age -0.081 ∗∗∗ -0.028 ∗∗ (0.011) (0.012)College Education 0.245 ∗∗∗ ∗∗∗ (0.009) (0.021)Constant 42.621 ∗∗∗ ∗∗∗ (3.095) (4.356)Observations 2,932 2,932 R R ∗∗∗ (df = 6; 2925) Note: ∗ p < ∗∗ p < ∗∗∗ p < In [41], the authors argue that internet access has not had an effect on political polarisation because the demographicwith the lowest increase in internet use - the elderly - has had the highest increase in political polarisation. However,it is also well established that older people have smaller network sizes than younger people [55] and growing evidenceof demographic shifts suggest that older people are increasingly living alone [56]. This translates to a direct fall in k for such populations, and without a corresponding increase in k provided by internet access, we would in fact expectto see higher polarisation in such a group.In [39], the authors argue that an increase in internet access leads to an increase in political polarisation. Firstly,it is worth noting that the overall effect size is very small - increasing the number of broadband providers by 10%increases political polarisation by 0 .
003 points (on a scale between 0 and 1). This is consistent with notion thatsocial connectivity will dampen the direct effect of biased media, and it is possible one could uncouple the effectof the internet on social connectivity as opposed to enabling confirmation bias with some proxy measure for socialconnectivity. What is also noteworthy is that the researchers included the level of “political interest” per county as amediating variable in parts of the analysis. So for example, if we allow f to represent the fraction of respondents ineach county with such strong partisan interest, then q could represent the level of bias these agents can display due toaccess to partisan media on the internet. Under this interpretation we can make sense of the interaction terms in theregression results - the effect of internet access on polarisation was considerably higher for counties where politicalinterest is higher, which is exactly what we predict from the product ( f q ) in (31).In [40], the author argues that internet access leads to a decrease in political polarisation. This study looks solelyat Twitter networks over time (but shows how they relate to political polarisation data offline). The author findsthat more diverse Twitter networks lead to reduced polarisation over time. Again, our model predicts the following -since everyone is already on Twitter in this scenario, the fractions f and q are untouched. However, the author notesthat more diverse networks are directly correlated with larger networks - a larger k . It follows therefore that theseusers with reduced polarisation experienced an increased k without a corresponding change in f or q , and the resultsfollow.4All in all, our biased learning model has proven to provide useful insight into a long-standing debate about animportant empirical topic. We show that it allows us compress a large and complex set of causal mechanisms in theliterature down to the effect of three terms of interest - the prevalence of biased agents ( f ), degree of bias ( q ), andsocial connectivity ( k ). In doing so, we were able to shed insights on the mechanisms at play when it came to internetaccess, and provide the beginnings of a more uniform understanding of what previously conflicting data has suggestedto date. [1] A. M. McCright and R. E. Dunlap, The Sociological Quarterly , 155 (2011).[2] Z. Horne, D. Powell, J. E. Hummel, and K. J. Holyoak, Proceedings of the National Academy of Sciences , 10321(2015).[3] D. Levy, N. Newman, R. Fletcher, A. Kalogeropoulos, and R. K. Nielsen, Report of the Reuters Institute for the Study ofJournalism. Available online: http://reutersinstitute. politics. ox. ac. uk/publication/digital-news-report-2014 (2014).[4] J. Gottfried and E. Shearer, News Use Across Social Medial Platforms 2016 (Pew Research Center, 2016).[5] A. Mitchell and R. Weisel,
Political polarization & media habits: From fox news to facebook, how liberals, and conservativeskeep up with politics. pew research center (2014).[6] D. Nikolov, D. F. Oliveira, A. Flammini, and F. Menczer, PeerJ Computer Science , e38 (2015).[7] M. D. Conover, J. Ratkiewicz, M. Francisco, B. Gon¸calves, F. Menczer, and A. Flammini, in Fifth international AAAIconference on weblogs and social media (2011).[8] A. L. Schmidt, F. Zollo, M. Del Vicario, A. Bessi, A. Scala, G. Caldarelli, H. E. Stanley, and W. Quattrociocchi, Proceedingsof the National Academy of Sciences , 3035 (2017).[9] A. Hannak, P. Sapiezynski, A. Molavi Kakhki, B. Krishnamurthy, D. Lazer, A. Mislove, and C. Wilson, in
Proceedings ofthe 22nd international conference on World Wide Web (ACM, 2013), pp. 527–538.[10] E. Jonas, S. Schulz-Hardt, D. Frey, and N. Thelen, Journal of personality and social psychology , 557 (2001).[11] R. S. Nickerson, Review of general psychology , 175 (1998).[12] M. Del Vicario, A. Bessi, F. Zollo, F. Petroni, A. Scala, G. Caldarelli, H. E. Stanley, and W. Quattrociocchi, Proceedingsof the National Academy of Sciences , 554 (2016).[13] J. An, D. Quercia, M. Cha, K. Gummadi, and J. Crowcroft, EPJ Data Science , 12 (2014).[14] S. Messing and S. J. Westwood, Communication research , 1042 (2014).[15] E. Bakshy, S. Messing, and L. A. Adamic, Science , 1130 (2015).[16] S. Flaxman, S. Goel, and J. M. Rao, Public opinion quarterly , 298 (2016).[17] S. Goel, W. Mason, and D. J. Watts, Journal of personality and social psychology , 611 (2010).[18] R. Hegselmann, U. Krause, et al., Journal of artificial societies and social simulation (2002).[19] M. Del Vicario, A. Scala, G. Caldarelli, H. E. Stanley, and W. Quattrociocchi, Scientific reports , 40391 (2017).[20] W. Quattrociocchi, G. Caldarelli, and A. Scala, Scientific reports , 4938 (2014).[21] J. Lorenz, International Journal of Modern Physics C , 1819 (2007).[22] P. M. DeMarzo, D. Vayanos, and J. Zwiebel, The Quarterly journal of economics , 909 (2003).[23] B. Golub and M. O. Jackson, American Economic Journal: Microeconomics , 112 (2010).[24] D. Acemoglu and A. Ozdaglar, Dynamic Games and Applications , 3 (2011).[25] M. Mobius and T. Rosenblat, Annu. Rev. Econ. , 827 (2014).[26] B. Golub and E. Sadler (2017).[27] D. Acemo˘glu, G. Como, F. Fagnani, and A. Ozdaglar, Mathematics of Operations Research , 1 (2013).[28] D. Acemoglu, A. Ozdaglar, and A. ParandehGheibi, Games and Economic Behavior , 194 (2010).[29] M. Mobilia, A. Peterson, and S. Redner, Journal of Statistical Mechanics: Theory and Experiment (2007).[30] Z. Kunda, Psychological bulletin , 480 (1990).[31] M. Rabin and J. L. Schrag, The Quarterly Journal of Economics , 37 (1999).[32] D. P. Redlawsk, The Journal of Politics , 1021 (2002).[33] B. Nyhan and J. Reifler, Political Behavior , 303 (2010).[34] F. Zollo, A. Bessi, M. Del Vicario, A. Scala, G. Caldarelli, L. Shekhtman, S. Havlin, and W. Quattrociocchi, PloS one ,e0181821 (2017).[35] A. Bessi, F. Petroni, M. Del Vicario, F. Zollo, A. Anagnostopoulos, A. Scala, G. Caldarelli, and W. Quattrociocchi, in WWW (Companion Volume) (2015), pp. 355–356.[36] M. H. DeGroot, Journal of the American Statistical Association , 118 (1974).[37] G. Livan and M. Marsili, Entropy , 3031 (2013).[38] K. Christensen, Tech. Rep. (2002).[39] Y. Lelkes, G. Sood, and S. Iyengar, American Journal of Political Science , 5 (2017).[40] P. Barber´a, Job Market Paper, New York University (2014).[41] L. Boxell, M. Gentzkow, and J. M. Shapiro, Proceedings of the National Academy of Sciences , 10612 (2017).[42] P. D. Howe, M. Mildenberger, J. R. Marlon, and A. Leiserowitz, Nature Climate Change , 596 (2015).[43] S. Blancke, F. Van Breusegem, G. De Jaeger, J. Braeckman, and M. Van Montagu, Trends in plant science , 414 (2015).[44] D. Centola, J. Becker, D. Brackbill, and A. Baronchelli, Science , 1116 (2018). [45] F. Newport, Gallup (2015).[46] H. C, Information Communication and Society , 1148 (2010), https://doi.org/10.1177/0002764209356247,URL https://doi.org/10.1177/0002764209356247 .[53] R. Putnam and P. Putnam, Bowling Alone: The Collapse and Revival of American Community , A Touchstone book(Simon & Schuster, 2000), ISBN 9780743203043, URL https://books.google.co.uk/books?id=rd2ibodep7UC