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Cited by 427Opinion Dynamics and Social InfluenceSocial Media and PoliticsComplex Network Analysis Techniques

Modeling echo chambers and polarization dynamics in social networks

Open access
Baumann, Fabian, Lorenz-Spreen, Philipp, Sokolov, Igor M., Starnini, Michele·2020·Physical Review Letters
DOI 10.1103/physrevlett.124.048301
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Abstract

Echo chambers and opinion polarization recently quantified in several sociopolitical contexts and across different social media raise concerns on their potential impact on the spread of misinformation and on the openness of debates. Despite increasing efforts, the dynamics leading to the emergence of these phenomena remain unclear. We propose a model that introduces the dynamics of radicalization as a reinforcing mechanism driving the evolution to extreme opinions from moderate initial conditions. Inspired by empirical findings on social interaction dynamics, we consider agents characterized by heterogeneous activities and homophily. We show that the transition between a global consensus and emerging radicalized states is mostly governed by social influence and by the controversialness of the topic discussed. Compared with empirical data of polarized debates on Twitter, the model qualitatively reproduces the observed relation between users' engagement and opinions, as well as opinion segregation in the interaction network. Our findings shed light on the mechanisms that may lie at the core of the emergence of echo chambers and polarization in social media.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proposes an activity-driven network model of opinion dynamics with radicalization, heterogeneous activity, and homophily, showing transitions between neutral consensus, one-sided radicalization, and metastable polarization. It qualitatively reproduces empirical Twitter patterns linking user engagement to opinion extremity and opinion segregation in interaction networks.

Key findings

  • The model yields three regimes: neutral consensus for small social interaction strength \(K\) and controversialness \(\alpha\); radicalization for larger \(K\) and/or \(\alpha\); and, with homophily \(\beta>0\), a metastable polarized state with bimodal opinions that eventually turns into a one-sided radicalized state. The lifetime of the polarized state increases at least exponentially with \(\beta\).
  • For \(\beta=0\), the transition from consensus to radicalization is captured analytically by a critical controversialness \(\alpha_c\) depending on \(K\), reciprocity \(r\), and average activity \(\langle a\rangle\); the dashed line still captures the transition for moderate \(\beta\).
  • The model qualitatively reproduces empirical Twitter debates on abortion, Obamacare, and guncontrol: bimodal opinion distributions, a U-shaped relation between user activity and opinion extremity, and echo chambers in the \((x,\langle x\rangle_{\mathrm{NN}})\) plane. The empirical datasets had \(N_u=4130\), \(r=0.69\) for abortion; \(N_u=4828\), \(r=0.62\) for Obamacare; and \(N_u=1838\), \(r=0.61\) for guncontrol.
  • Results are robust to reciprocity \(r\) and asymmetric initial conditions; echo chambers persist even for very small reciprocity, e.g. \(r=0.1\).

Methods

  • Agent-based model of \(N\) agents with continuous opinions \(x_i\), coupled by an ODE with social interaction strength \(K\) and sigmoidal influence \(\tanh(\alpha x)\), where \(\alpha\) is interpreted as topic controversialness. Interactions occur on a directed activity-driven temporal network with heterogeneous activities \(a_i\), power-law activity distribution \(F(a)\sim a^{-\gamma}\), \(m\) contacts, reciprocity \(r\), and homophily controlled by \(\beta\).
  • Numerical integration used \(dt=0.01\), network updates each step, \(N=1000\), \(m=10\), \(\epsilon=10^{-2}\), \(\gamma=2.1\), and \(r=0.5\) unless otherwise stated; initial opinions were uniformly spaced on \([-1,1]\). For comparison with Twitter data, \(r=0.65\) was used. A mean-field approximation was derived for \(\beta=0\).
  • Empirical comparison used three Twitter datasets from Garimella et al. on abortion, Obamacare, and guncontrol. Users’ political leaning scores were inferred from links to news organizations with known political leaning, and the directed follower network among users was reconstructed; activity was measured as the fraction of tweets containing links to news organizations of known political leaning.

Limitations

  • Stated in the paper: the model is based on a minimal number of assumptions and does not account for features such as targeted advertising or different credibility of users.
  • Stated in the paper: the polarized state is metastable and, for moderate \(\beta\), eventually turns into a one-sided radicalized state; its lifetime increases with homophily.
  • Evident from the text: empirical validation is qualitative rather than quantitative, limited to three Twitter topics and specific event windows, with opinions inferred from linked news organizations; the model assumes a clear time-scale separation between fast network dynamics and slow opinion change.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevlett.124.048301

Cites

48 of 48 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 40 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § Modeling echo chambers and polarization dynamics in social networks · p. 4

    For all three Twitter topics (guncontrol, Obamacare, abortion), user engagement rises towards the extremes of opinion space.

    “Fig. 3(a) shows the average engagement, or activity a , of users as a function of their opinions x . For all three topics under consideration, the engagement rises towards the extremes of opinion space.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [2] K. Garimella, G. De Francisci Morales, A. Gionis, and M. Mathioudakis, in Proceedings of the 2018 World Wide Web Conference , WWW'18 (International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, Switzerland, 2018) pp. 913-922. source

      Not checked in the free run.

  • Findingown data
    § Modeling echo chambers and polarization dynamics in social networks · p. 3

    For small values of K and α, the model reaches a neutral consensus in which all opinions converge to zero.

    “For small values of K and α , a neutral consensus is reached, in which the opinions of all agents converge towards zero, Fig. 1(a).”
    • Also stated in: § Modeling echo chambers and polarization dynamics in social networks · p. 3 · plain
      “In the dark region, the system approaches a neutral consensus”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Opinion dynamics is solely driven by interactions among agents, with no intrinsic individual preferences.
    • The authors state this as an assumption: The social influence function is odd, increases monotonically with an agent's conviction, and caps the influence of extreme opinions.
  • Findingown data
    § Modeling echo chambers and polarization dynamics in social networks · p. 4

    The model qualitatively reproduces opinion segregation in the interaction network (echo chambers) observed in empirical data.

    “Both our model (Fig. 4(a)) and empirical data (Figs. 4(b) - (d)) clearly show two bright areas characterized by a high density of users with likely-minded neighbors, identifying two echo chambers corresponding to opposite opinion groups.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [2] K. Garimella, G. De Francisci Morales, A. Gionis, and M. Mathioudakis, in Proceedings of the 2018 World Wide Web Conference , WWW'18 (International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, Switzerland, 2018) pp. 913-922. source

      Not checked in the free run.

    • The authors state this as an assumption: Interactions are ruled by homophily.
  • Findingcitation
    § Modeling echo chambers and polarization dynamics in social networks · p. 4

    The empirical Twitter data sets show a bimodal shape of expressed opinions across all three considered data sets.

    “The data on x i yield the distributions of expressed opinions, P ( x ), which show a bimodal shape across all three considered data sets (see Fig. S1(a) in Ref. [47]).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [2] K. Garimella, G. De Francisci Morales, A. Gionis, and M. Mathioudakis, in Proceedings of the 2018 World Wide Web Conference , WWW'18 (International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, Switzerland, 2018) pp. 913-922. source

      Not checked in the free run.

  • Findingreasoning
    § Modeling echo chambers and polarization dynamics in social networks · p. 1

    The model qualitatively reproduces the observed relation between users' engagement and opinions in polarized Twitter debates: more active users tend to show more extreme opinions.

    “Compared with empirical data of polarized debates on Twitter, the model qualitatively reproduces the observed relation between users' engagement and opinions”
    • Also stated in: § Modeling echo chambers and polarization dynamics in social networks · p. 4 · boosted
      “This characteristic U-shaped relation is well reproduced by our model, see Fig. 3(b)”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [2] K. Garimella, G. De Francisci Morales, A. Gionis, and M. Mathioudakis, in Proceedings of the 2018 World Wide Web Conference , WWW'18 (International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, Switzerland, 2018) pp. 913-922. source

      Not checked in the free run.

    • The authors state this as an assumption: Opinion dynamics is solely driven by interactions among agents, with no intrinsic individual preferences.
    • The authors state this as an assumption: Interactions are ruled by homophily.
    • The authors state this as an assumption: Agents sharing similar opinions mutually reinforce each other and move towards more extreme views, describing a radicalization dynamics.
  • May combine two claims. 18
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 8

  • May combine two claims. The quote only partly states this claim.

    Findingreasoning
    § Modeling echo chambers and polarization dynamics in social networks · p. 3

    The introduction of homophily (β > 0) leads agents to reinforce their opinions and segregate into two groups on opposite sides of the neutral consensus, producing a polarized state with a bimodal distribution of opinions.

    “In this scenario, a polarized state characterized by a bimodal distribution of opinions emerges”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Opinion dynamics is solely driven by interactions among agents, with no intrinsic individual preferences.
    • The authors state this as an assumption: Interactions are ruled by homophily.
    • The authors state this as an assumption: Agents sharing similar opinions mutually reinforce each other and move towards more extreme views, describing a radicalization dynamics.
    • The authors state this as an assumption: The social influence function is odd, increases monotonically with an agent's conviction, and caps the influence of extreme opinions.

  • May combine two claims.

    Findingnone given
    § Modeling echo chambers and polarization dynamics in social networks · p. 3

    Larger values of α and/or K destabilize the neutral consensus and give rise to radicalization, in which opinions do not converge, are widely spread, and may reach values far outside the initial opinion interval.

    “Larger values of α and/or K destabilize this consensus and give rise to radicalization. These are situations in which agents' opinions do not converge, are widely spread, and may reach values far outside of the initial opinion interval.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Opinion dynamics is solely driven by interactions among agents, with no intrinsic individual preferences.
    • The authors state this as an assumption: Agents sharing similar opinions mutually reinforce each other and move towards more extreme views, describing a radicalization dynamics.
    • The authors state this as an assumption: The social influence function is odd, increases monotonically with an agent's conviction, and caps the influence of extreme opinions.
    • The authors state this as an assumption: The parameter α is interpreted as the controversialness of the issue.

  • May combine two claims.

    Findingnone given
    § Modeling echo chambers and polarization dynamics in social networks · p. 3

    The polarized state in the model is metastable and, for moderate values of β, eventually turns to a one-sided radicalized state.

    “The polarized state in our model is metastable and (for moderate values of β ) eventually turns to a one-sided radicalized state.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Opinion dynamics is solely driven by interactions among agents, with no intrinsic individual preferences.
    • The authors state this as an assumption: Interactions are ruled by homophily.
    • The authors state this as an assumption: Agents sharing similar opinions mutually reinforce each other and move towards more extreme views, describing a radicalization dynamics.
    • The authors state this as an assumption: The social influence function is odd, increases monotonically with an agent's conviction, and caps the influence of extreme opinions.

  • May combine two claims.

    Findingown data
    § POLARIZED OPINION DISTRIBUTIONS · p. 7

    For K=3, α=3, β=1, r=0.65, the model enters a polarized state characterized by bimodal opinion distributions, which are in qualitative agreement with the investigated Twitter data.

    “For sufficiently high values of the parameters K , α and β (here K = 3, α = 3, β = 1, r = 0 . 65) the model enters a polarized state characterized by bimodal opinion distributions, which are in qualitative agreement with the investigated Twitter data.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data
    § EMPIRICAL DATA SETS · p. 10

    For the abortion dataset, the final number of users is 4130 and measured reciprocity is 0.69.

    “The final numbers of users ( N u ) and measured reciprocities ( r ) for each data set are: abortion : N u = 4130, r = 0 . 69, obamacare : N u = 4828, r = 0 . 62, guncontrol : N u = 1838, r = 0 . 61.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § ROBUSTNESS WITH RESPECT TO ASYMMETRIC INITIAL CONDITIONS AND THE EFFECT OF FLUCTUATIONS · p. 14

    The consensus behavior does not depend on the value of δ and results in the same final state even with all opinions initialized on one side (δ < -1 or δ > 1).

    “Note, that this behavior does not depend on the value of δ and results in the same final state even with all opinions initialized on one side, i.e. for δ &lt; -1 or δ &gt; 1.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § ROBUSTNESS WITH RESPECT TO ASYMMETRIC INITIAL CONDITIONS AND THE EFFECT OF FLUCTUATIONS · p. 14

    The model does not critically depend on the choice of uniformly spaced initial opinions, and the emergence of consensus and opinion polarization is recovered also for highly asymmetric cases.

    “Here, we demonstrate that the model does not critically depend on this choice and the emergence of consensus and opinion polarization is recovered also for highly asymmetric cases.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § ROBUSTNESS WITH RESPECT TO ASYMMETRIC INITIAL CONDITIONS AND THE EFFECT OF FLUCTUATIONS · p. 14

    For symmetric initial conditions, the system is purely fluctuation driven and both final states σ(x_i) = ±1 are realized with equal probabilities.

    “While for symmetric initial conditions the system is purely fluctuation driven and both final states, σ ( x i ) ± = 1, are realized with equal probabilities, initial conditions with δ = 0 favor final states with σ ( x i ) = σ ( δ ).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § ROBUSTNESS WITH RESPECT TO RECIPROCITY · p. 12

    The emergence of polarized states does not rely on high values of r; even for r=0.1 the opinions clearly split in two groups on opposite sides of the neutral consensus.

    “First, and most importantly, we find that the emergence of polarized states does not rely on high values of r . Even in the case of r = 0 . 1 the opinions clearly split in two groups on opposite sides of the neutral consensus.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § LIFETIME OF POLARIZED STATES · p. 8

    The mean lifetimes of polarized states show a strong dependence on β, even exceeding exponential growth for higher values of homophily.

    “Note the logscale on the y -axis, i.e. the strong dependence of the mean lifetimes on β , which even exceed an exponential growth for higher values of homophily.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § RELATION BETWEEN USER OPINIONS AND ACTIVITIES · p. 11

    The U-shaped relation between opinions x and activities a is a generic feature of the radicalization dynamics and occurs as soon as the system is in a polarized state.

    “The U-shaped relation between opinions x and activities a is a generic feature of the radicalization dynamics and occurs as soon as the system is in a polarized state.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § APPROXIMATION OF THE CRITICAL CONTROVERSIALNESS α c · p. 9

    For λ > 0 the neutral consensus destabilizes, hence λ = 0 defines the critical value of controversialness α_c.

    “For ˜ λ &gt; 0 the neutral consensus destabilizes, hence, ˜ λ = 0 defines the critical value of controversialness α c , i.e.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The temporal adjacency matrix can be approximated by its time average because network dynamics are fast.
    • The authors state this as an assumption: Homophily can be neglected (β=0) in the mean-field approximation.
    • The authors state this as an assumption: Activity values follow a power law distribution F(a) = ... normalized on [ε,1].

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data
    § EMPIRICAL DATA SETS · p. 10

    For the guncontrol dataset, the final number of users is 1838 and measured reciprocity is 0.61.

    “The final numbers of users ( N u ) and measured reciprocities ( r ) for each data set are: abortion : N u = 4130, r = 0 . 69, obamacare : N u = 4828, r = 0 . 62, guncontrol : N u = 1838, r = 0 . 61.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data
    § EMPIRICAL DATA SETS · p. 10

    For the obamacare dataset, the final number of users is 4828 and measured reciprocity is 0.62.

    “The final numbers of users ( N u ) and measured reciprocities ( r ) for each data set are: abortion : N u = 4130, r = 0 . 69, obamacare : N u = 4828, r = 0 . 62, guncontrol : N u = 1838, r = 0 . 61.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductreasoning
    § Modeling echo chambers and polarization dynamics in social networks · p. 5

    The model does not take into account some features of empirical social networks that might additionally drive polarization, such as targeted advertising or different credibility of users.

    “It is important to remark that our model is based on a minimal number of assumptions. Thus it does not take into account some features of empirical social networks which might additionally drive polarization phenomena, such as targeted advertising or different credibility of users.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Study design and conductcitation
    § Modeling echo chambers and polarization dynamics in social networks · p. 2

    The model's results are robust with respect to the reciprocity parameter r and asymmetric initial conditions.

    “In [47] we show that the obtained results are robust with respect to r , and asymmetric initial conditions.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [47] source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § Modeling echo chambers and polarization dynamics in social networks · p. 1

    Traditional models based on constructive opinion dynamics predict that unrestricted interaction would eventually lead to consensus, even on controversial issues.

    “Within the traditional models based on constructive opinion dynamics [5], such unrestricted modes of interaction would eventually lead to a consensus, even on controversial issues.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [5] source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § Modeling echo chambers and polarization dynamics in social networks · p. 2

    Informational influence theory for group polarization holds that agents with moderate opinions may become extreme while interacting in a group.

    “This mechanism builds on the idea of informational influence theory [38] for the phenomenon of group polarization, where agents with moderate opinions may become extreme while interacting in a group [31].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [38] source

      Not checked in the free run.

    • cites [31] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § Modeling echo chambers and polarization dynamics in social networks · p. 5

    In the case of controversial issues, a reinforcement mechanism leads to radicalization dynamics and may drive groups of agents away from the global consensus.

    “In the case of controversial issues, a reinforcement mechanism leads to radicalization dynamics and may drive groups of agents away from the global consensus.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Opinion dynamics is solely driven by interactions among agents, with no intrinsic individual preferences.
    • The authors state this as an assumption: Agents sharing similar opinions mutually reinforce each other and move towards more extreme views, describing a radicalization dynamics.
    • The authors state this as an assumption: The social influence function is odd, increases monotonically with an agent's conviction, and caps the influence of extreme opinions.
    • The authors state this as an assumption: The parameter α is interpreted as the controversialness of the issue.

  • May combine two claims.

    Premisecitation
    § Modeling echo chambers and polarization dynamics in social networks · p. 1

    Echo chambers have been quantified in several controversial debates on different social media platforms and may be related to the spread of misinformation.

    “Echo chambers have been quantified in several controversial debates on different social media platforms [2, 16, 17] and may be related to the spread of misinformation [18].”
    Quote in the paperNot a substantial claim0 of 4 sources checked
    • cites [2] K. Garimella, G. De Francisci Morales, A. Gionis, and M. Mathioudakis, in Proceedings of the 2018 World Wide Web Conference , WWW'18 (International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, Switzerland, 2018) pp. 913-922. source

      Not checked in the free run.

    • cites [16] source

      Not checked in the free run.

    • cites [17] source

      Not checked in the free run.

    • cites [18]

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § EMPIRICAL DATA SETS · p. 10

    In Ref. [2], for each dataset, the directed follower network among users has been reconstructed: a directed link from node u to node v indicates that user u follows user v.

    “In Ref. [2], for each dataset, the directed follower network among users has been reconstructed: a directed link from node u to node v indicates that user u follows user v .”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [2] K. Garimella, G. De Francisci Morales, A. Gionis, and M. Mathioudakis, in Proceedings of the 2018 World Wide Web Conference , WWW'18 (International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, Switzerland, 2018) pp. 913-922. source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § EMPIRICAL DATA SETS · p. 10

    For each user, a political leaning score is inferred on the basis of the content s/he produces, by using a ground truth of political leaning scores of various news organizations obtained from Bakshy et al. [4].

    “For each user, a political leaning score is inferred on the basis of the content s/he produces, by using a ground truth of political leaning scores of various news organizations (e.g., nytimes.com) obtained from Bakshy et al. [4].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [4] K. Garimella, G. D. F. Morales, A. Gionis, and M. Mathioudakis, Trans. Soc. Comput. 1 , 3:1 (2018). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § APPROXIMATION OF THE CRITICAL CONTROVERSIALNESS α c · p. 9

    Neglecting homophily (β=0), the overall probability that agent i is influenced by j has two contributions: j contacts i, or i contacts j and the link is reciprocal.

    “Neglecting homophily ( β = 0) the overall probability that agent i is influenced by j has two different contributions. Either, j contacts i , or i contacts j and the link is reciprocal.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Homophily can be neglected (β=0) in the mean-field approximation.
23 of 23

  • The authors state this as an assumption.

    Opinion dynamics is solely driven by interactions among agents, with no intrinsic individual preferences.

    “We assume that the opinion dynamics is solely driven by the interactions among agents”
    “we model opinion dynamics as a purely collective, self-organized process without any intrinsic individual preferences.”

  • The authors state this as an assumption.

    Interactions are ruled by homophily.

    “we assume here that interactions are ruled by homophily [22, 23].”

  • The authors state this as an assumption.

    Social interactions evolve much faster than opinions, yielding a clear time-scale separation between network and opinion dynamics.

    “Here, we focus on a regime in which social interactions evolve much faster than opinions, like it is reasonable to assume for online social media.”
    “This yields a clear time-scale separation between the network's and opinion dynamics.”

  • The authors state this as an assumption.

    Agents sharing similar opinions mutually reinforce each other and move towards more extreme views, describing a radicalization dynamics.

    “The model introduces a mechanism by which agents sharing similar opinions can mutually reinforce each other and move towards more extreme views, thus describing a radicalization dynamics , also known as group polarization in social psychology [31, 32].”

  • The authors state this as an assumption.

    The social influence function is odd, increases monotonically with an agent's conviction, and caps the influence of extreme opinions.

    “Its odd non-linear shape guarantees that an agent j influences others in the direction of its own opinion's sign, σ ( x j ), that this influence increases monotonically with the agent's conviction | x j | , and that the social influence of extreme opinions is capped, as suggested by experimental findings [35].”

  • The authors state this as an assumption.

    The parameter α is interpreted as the controversialness of the issue.

    “Therefore, the parameter α is interpreted as the controversialness of the issue.”

  • The authors state this as an assumption.

    The parameter β may include various homophilic effects in interactions, both endogenous and exogenous.

    “Note that the parameter β may include various homophilic effects in interactions, both endogeneous (due to the intrinsic behavior of agents) and exogeneous (i.e. due to the algorithms of social media platforms [27]).”

  • The authors state this as an assumption.

    The temporal adjacency matrix can be approximated by its time average because network dynamics are fast.

    “Fast network dynamics allow the adjacency matrix A ij ( t ) (cf. Eq. (1) of the main text) to be approximated by its time average, yielding”

  • The authors state this as an assumption.

    Homophily can be neglected (β=0) in the mean-field approximation.

    “Neglecting homophily ( β = 0) the overall probability that agent i is influenced by j has two different contributions.”

  • The authors state this as an assumption.

    Activity values follow a power law distribution F(a) = ... normalized on [ε,1].

    “For a power law activity distribution F ( a ) = 1 -γ 1 -ϵ 1 -γ a -γ , normalized on the interval a ∈ [ ϵ, 1], we have”
10 of 10

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • There is substantial evidence that segregated online communities emerge around politically contentious topics.

    “Although recent evidence suggests that general information consumption on social media might not be an echo-chamber for many users (21, 22), there is nevertheless substantial evidence supporting the argument that segregated online communities emerge around politically contentious topics (23, 19, 24, 25, 26, 27, 20).”

    From Troll and divide: the language of online polarization (2022) · cites this paper as [23]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Conflicting reciprocity values for Figure 6

    Minor

    The text says Figure 6 compares r = 1 with r = 0.65, but its caption says the plotted cases use r = 1 and r = 0.5.

  • Supplement reverses the Figure 1 panel labels

    Minor

    The supplementary text points to Figures 1(b) and 1(c) for polarized and one-sided radicalized states, respectively. The Figure 1 caption assigns those states to the opposite panels.

2 of 2

Cites

Flagged first, then by citations

Research works printed but not linked: 7

  1. M. D. Conover, 2011, International AAAI Conference on Web and Social Media, North America
  2. A. Hanna, 2013, Proceedings of the 2nd Workshop on Politics, Elections and Data, PLEAD ’13
  3. J. Borge-Holthoefer, 2015, Proceedings of the 18th ACM Conference on Computer Supported Cooperative Work & Social Computing, CSCW 2015, Vancouver, Canada, 2015
  4. K. Garimella, 2018, Proceedings of the 2018 World Wide Web Conference, WWW ’18
  5. R. Hegselmann, 2002, J. Artif. Soc. Simul., 5
  6. K. Garimella, 2018, T. Soc. Comput. Simul., 1, 3
  7. M. D. Conover, 2012, Eur. Phys. J. Data Sci., 1, 6

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Authors
Baumann, F., Lorenz-Spreen, P., Sokolov, I. M., & Starnini, M.
Affiliations
  • Humboldt-Universität zu Berlin (DE)
  • Max Planck Institute for Human Development (DE)
  • Institute for Scientific Interchange (IT)
ORCID
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
2020-01-27
Volume
124
Issue
4
Pages
048301-048301
ISSN
  • 0031-9007 (ISSN-L)
  • 1079-7114
DOI
10.1103/physrevlett.124.048301
OpenAlex
W2955488027
Semantic Scholar
1ad1c78329fc97906d15cb131d57df8a51a82c62
Cited by
427
References
48
Citations by year
2026: 56 · 2025: 69 · 2024: 75 · 2023: 81 · 2022: 54 · 2021: 63 · 2020: 28 · 2019: 1
FWCI
23.90
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • MPG.PuRe (Max Planck Society) (submittedVersion)
  • arXiv (Cornell University) (cc-by)
Funders
  • Deutsche Forschungsgemeinschaft
  • Fundação de Amparo à Pesquisa do Estado de São Paulo
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Opinion Dynamics and Social Influence
  • Social Sciences › Communication › Social Media and Politics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
Keywords
echo chambers, opinion polarization, radicalization, social influence, homophily, agent-based modeling, social media, misinformation, user engagement, Twitter
MeSH terms
Communication, Humans, Models, Theoretical, Information Dissemination, Social Networking

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-09.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-09.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

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