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arxiv: 2605.20418 · v1 · pith:URWS6SY4new · submitted 2026-05-19 · ⚛️ physics.soc-ph · cs.SI

A Bounded-Confidence Model of Opinion Dynamics with Adaptive Interaction Probabilities

Leila Thompsky , Yuexuan (Yolanda) Wu , Mason A. Porter , Jiajie Luo This is my paper

Pith reviewed 2026-05-21 06:45 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cs.SI
keywords bounded confidence modelopinion dynamicsadaptive networksDeffuant-Weisbuch modelconvergence timenetwork densityinteraction probabilitieseffective graph
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The pith

Adaptive edge weights in a bounded-confidence model decrease convergence time for dense networks but increase it for sparse networks when confidence bounds are small.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper modifies the Deffuant-Weisbuch model by introducing adaptive edge weights that raise interaction probabilities after agents compromise and lower them otherwise. This change encodes the tendency for people to interact more with those they have agreed with before. The authors provide proofs for convergence of opinions, stabilization of weights, and behavior of the effective graph of receptive pairs. Their simulations demonstrate that these adaptive weights make the system reach agreement faster on densely connected networks but slower on sparsely connected ones for small confidence bounds.

Core claim

By making edge weights adaptive in the Deffuant-Weisbuch bounded-confidence model, the opinions of agents converge to clusters, the edge weights evolve to reflect the history of compromises, and the effective graph updates to show current receptive interactions. For small confidence bounds, adding this adaptivity reduces the number of steps needed to reach consensus on dense networks while increasing that number on sparse networks.

What carries the argument

Adaptive edge weights that modify interaction probabilities based on whether prior interactions resulted in compromise.

If this is right

  • The adaptive model still converges to opinion clusters for any initial conditions.
  • Edge weights reach a steady state determined by the frequency of compromising interactions.
  • The effective graph is the time-varying subgraph consisting of pairs with sufficiently close opinions.
  • For small confidence bounds, convergence time decreases with adaptivity on dense networks and increases on sparse networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Social networks with high connectivity may form opinion consensus more rapidly if positive interactions increase future contact rates.
  • Testing the weight update rule against data from social media platforms could validate or refine how reinforcement affects interaction probabilities.
  • Similar adaptive mechanisms might be added to other opinion dynamics models to study their effects on cluster formation.

Load-bearing premise

The specific rule for increasing or decreasing edge weights after each interaction based on whether agents compromised accurately reflects real social reinforcement.

What would settle it

Running simulations of the adaptive model versus the non-adaptive model on a complete graph and on a cycle graph, then checking if convergence time decreases on the complete graph and increases on the cycle graph for a small fixed confidence bound.

Figures

Figures reproduced from arXiv: 2605.20418 by Jiajie Luo, Leila Thompsky, Mason A. Porter, Yuexuan (Yolanda) Wu.

Figure 1
Figure 1. Figure 1: FIG. 1. The numbers of major opinion clusters in simulations of our adaptive edge-weighted DW model versus the confidence [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The logarithm of the convergence times of simulations of our adaptive edge-weighted DW model versus the confidence [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The logarithm of the convergence times of simulations of our adaptive edge-weighted DW model versus the confidence [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The logarithm of the convergence times of simulations of our adaptive edge-weighted DW model versus the confidence [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The number of major opinion clusters in simulations of our adaptive edge-weighted DW model versus the confidence [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The logarithm of the convergence times in simulations of our adaptive edge-weighted DW model versus the confidence [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The numbers of major opinion clusters in simulations of our adaptive edge-weighted DW model versus the confidence [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The Shannon entropies at the convergence time [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The logarithm of the convergence times in simulations of our adaptive edge-weighted DW model versus the confidence [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The number of major opinion clusters in simulations of our adaptive edge-weighted DW model versus the confidence [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
read the original abstract

Models of opinion dynamics aim to capture how individuals' opinions change when they interact with each other. One well-known model of opinion dynamics is the Deffuant--Weisbuch (DW) model, which is a type of bounded-confidence model (BCM). In the DW model, agents have pairwise interactions, and they are receptive to other agents' opinions when their opinions are sufficiently close to each other. In this paper, we extend the DW model by studying it on networks with heterogeneous and adaptive edge weights between pairs of agents. These edge weights govern the interaction probabilities between the agents and thereby encode the idea that people are more likely to communicate with individuals with whom they have previously compromised or had other positive interactions. We prove theoretical guarantees of our adaptive edge-weighted DW model's convergence properties, the long-time dynamics of its edge weights, and the model's associated ``effective graph", which is a time-dependent subgraph that includes edges only between agents that are receptive to each other's opinions. We support our theoretical results with numerical simulations of our adaptive edge-weighted DW model on a variety of networks and find that including adaptive edge weights yields different qualitative dynamics for different types of networks. In particular, for small confidence bounds, we observe that incorporating adaptive edge weights decreases the convergence time for dense networks but increases the convergence time for sparse networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript extends the Deffuant-Weisbuch bounded-confidence model to networks with heterogeneous adaptive edge weights, where weights increase after a compromise and decrease otherwise. It establishes theoretical guarantees for opinion convergence, the long-time behavior of the edge weights, and the associated time-dependent effective graph (the subgraph of currently receptive pairs). Simulations on multiple network topologies then show that, for small confidence bounds, the adaptive rule shortens convergence time on dense networks while lengthening it on sparse networks.

Significance. The combination of an adaptive-interaction rule with rigorous convergence and effective-graph analysis is a useful addition to the bounded-confidence literature. The reported topology-dependent effect on convergence time, if robust, supplies a concrete, falsifiable prediction about how social reinforcement modulates consensus formation in different social structures. The proofs and effective-graph construction are the primary strengths; the simulations serve mainly to illustrate the qualitative distinction between dense and sparse regimes.

major comments (2)
  1. [§3] §3 (Convergence and effective-graph analysis): the proof that the effective graph eventually becomes static appears to rely on the edge weights converging to a limit; however, the argument does not explicitly rule out persistent oscillations between two or more effective graphs when the adaptation rate is comparable to the opinion-update rate. A short additional lemma or remark addressing this case would strengthen the claim that the long-time dynamics are fully characterized.
  2. [§4.3] §4.3 (Sparse-network simulations): the reported increase in convergence time for sparse networks is load-bearing for the central qualitative claim, yet the manuscript presents only mean convergence times without reporting the number of independent runs or standard deviations. Adding these statistics (or a table of raw values) is necessary to confirm that the observed lengthening is not an artifact of a single atypical realization.
minor comments (3)
  1. [§2] The definition of the adaptation rule (increase after compromise, decrease otherwise) is stated clearly, but the precise functional form (linear, multiplicative, etc.) and the normalization that keeps weights in [0,1] should be written as an explicit equation in §2 for reproducibility.
  2. [Figure 3] Figure 3 (or whichever panel shows the dense vs. sparse comparison) would benefit from a legend that explicitly labels the adaptive and non-adaptive curves; the current caption is slightly ambiguous about which line corresponds to which variant.
  3. [Introduction] A brief remark on the relation to existing adaptive-interaction models (e.g., those that adapt based on opinion similarity rather than compromise outcome) would help readers situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the specific suggestions that help strengthen the presentation. We respond to each major comment below and have revised the manuscript to incorporate the requested clarifications and statistical details.

read point-by-point responses

  1. Referee: [§3] §3 (Convergence and effective-graph analysis): the proof that the effective graph eventually becomes static appears to rely on the edge weights converging to a limit; however, the argument does not explicitly rule out persistent oscillations between two or more effective graphs when the adaptation rate is comparable to the opinion-update rate. A short additional lemma or remark addressing this case would strengthen the claim that the long-time dynamics are fully characterized.

    Authors: We agree that an explicit treatment of comparable timescales strengthens the result. Our existing proof already shows that opinions converge to a finite number of clusters and that each edge weight is updated by a rule that is monotonic in the long run (increasing only upon successful compromise and otherwise decreasing toward a lower bound). Because there are only finitely many possible effective graphs on a finite vertex set, and the weight dynamics cannot sustain indefinite cycling once opinions have clustered, the effective graph must stabilize. We have added a short remark in §3 making this argument explicit and confirming that the long-time characterization holds independently of the relative rates. revision: yes

  2. Referee: [§4.3] §4.3 (Sparse-network simulations): the reported increase in convergence time for sparse networks is load-bearing for the central qualitative claim, yet the manuscript presents only mean convergence times without reporting the number of independent runs or standard deviations. Adding these statistics (or a table of raw values) is necessary to confirm that the observed lengthening is not an artifact of a single atypical realization.

    Authors: We concur that statistical reporting is required for the simulation claims. In the revised §4.3 we now state that every mean convergence time is computed from 100 independent realizations with distinct random initial opinions and seeds. Standard deviations are reported alongside the means (both in the text and via error bars in the figures) for the sparse-network cases, confirming that the lengthening effect is statistically consistent and not driven by outliers. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines an explicit update rule for adaptive edge weights based on whether agents compromise during interactions, proves convergence properties and long-time edge-weight behavior directly from the model equations, constructs an effective graph as a time-dependent subgraph of receptive pairs, and reports qualitative simulation results on convergence times for dense versus sparse networks under small confidence bounds. These elements are internally consistent and do not reduce by construction to fitted parameters, self-referential definitions, or load-bearing self-citations; the central claim is a direct numerical observation from the independently specified dynamics.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on the standard bounded-confidence assumption plus a new adaptive weight update rule whose functional form is not derived from data but postulated to encode positive-interaction reinforcement.

free parameters (2)
  • confidence bound
    Standard DW parameter controlling when agents interact; value chosen by user and affects the reported dense/sparse difference.
  • adaptation rate or update strength for edge weights
    Controls how much weights change after each interaction; appears as a tunable parameter in the extension.
axioms (2)
  • domain assumption Agents only interact when their opinions differ by less than the confidence bound.
    Core premise of all bounded-confidence models invoked throughout.
  • ad hoc to paper Edge weights increase after compromise and decrease otherwise, encoding positive reinforcement.
    The adaptive rule is introduced by the authors to capture the stated social idea.

pith-pipeline@v0.9.0 · 5780 in / 1311 out tokens · 32584 ms · 2026-05-21T06:45:46.428170+00:00 · methodology

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Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages · 1 internal anchor

  1. [1]

    positive

    studied a BCM with a neighborhood-based network adaptivity, in which agents rewire preferentially to other agents whose neighbors’ opinions are close to their own. Agostinelli et al. [30] examined a BCM on an adaptive network with polyadic interactions to analyze the im- portance of group interactions in opinion dynamics. Re- searchers have also examined ...

    work page 2005

  2. [2]

    M. O. Jackson,Social and Economic Networks, Social and Economic Networks (Princeton University Press, Princeton, NJ, USA, 2008)

    work page 2008

  3. [3]

    Wasserman and K

    S. Wasserman and K. Faust,Social Network Analysis: Methods and Applications(Cambridge University Press, Cambridge, UK, 1994)

    work page 1994

  4. [4]

    E. C. Baek, R. Hyon, K. L´ opez, M. A. Porter, and C. Parkinson, Perceived community alignment increases information sharing, Nature Comm.16, 5864 (2025)

    work page 2025

  5. [5]

    Noorazar, K

    H. Noorazar, K. R. Vixie, A. Talebanpour, and Y. Hu, From classical to modern opinion dynamics, Int. J. Mod- ern Phys. C31, 2050101 (2020)

    work page 2020

  6. [6]

    Opinion dynamics: Statistical physics and beyond

    M. Starnini, F. Baumann, T. Galla, D. Garcia, G. I˜ niguez, M. Karsai, J. Lorenz, and K. Sznajd- Weron, Opinion dynamics: Statistical physics and be- yond, arXiv:2507.11521 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  7. [7]

    Caldarelli, O

    G. Caldarelli, O. Artime, G. Fischetti, A. N. Stefano Guarino, F. Saracco, P. Holme, and M. de Domenico, The physics of news, rumors, and opinions, arXiv:2510.15053 (2025)

    work page arXiv 2025

  8. [8]

    Newman,Networks, 2nd ed

    M. Newman,Networks, 2nd ed. (Oxford University Press, Oxford, UK, 2018)

    work page 2018

  9. [9]

    Chandler and R

    D. Chandler and R. Munday,A Dictionary of Media and Communication, 3rd ed. (Oxford University Press, Ox- ford, UK, 2020)

    work page 2020

  10. [10]

    Millon, M

    T. Millon, M. J. Lerner, and I. B. Weiner,Handbook of Psychology, Personality and Social Psychology(John Wi- ley & Sons, Inc., Hoboken, NJ, USA, 2003)

    work page 2003

  11. [11]

    Bernardo, C

    C. Bernardo, C. Altafini, A. Proskurnikov, and F. Vasca, Bounded confidence opinion dynamics: A survey, Auto- matica159, 111302 (2024)

    work page 2024

  12. [12]

    X. F. Meng, R. A. Van Gorder, and M. A. Porter, Opin- ion formation and distribution in a bounded-confidence model on various networks, Phys. Rev. E97, 022312 (2018)

    work page 2018

  13. [13]

    Hegselmann and U

    R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis and simulation, J. Artif. Soc. Soc. Simul.5(3), 2 (2002)

    work page 2002

  14. [14]

    Deffuant, D

    G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, Mixing beliefs among interacting agents, Adv. Cplx. Sys. 3, 87 (2000)

    work page 2000

  15. [15]

    Lorenz, Heterogeneous bounds of confidence: Meet, discuss and find consensus!, Complexity15, 43 (2010)

    J. Lorenz, Heterogeneous bounds of confidence: Meet, discuss and find consensus!, Complexity15, 43 (2010). 21 FIG. 10. The number of major opinion clusters in simulations of our adaptive edge-weighted DW model versus the confidence boundcfor different values of the edge-weight increase parameterγand the edge-weight decrease parameterδon the Smith College ...

    work page 2010

  16. [16]

    W. Su, G. Chen, and Y. Hong, Noise leads to quasi- consensus of Hegselmann–Krause opinion dynamics, Au- tomatica85, 448 (2017)

    work page 2017

  17. [17]

    Vaz Martins, M

    T. Vaz Martins, M. Pineda, and R. Toral, Mass media and repulsive interactions in continuous-opinion dynam- ics, Europhys. Lett.91, 48003 (2010)

    work page 2010

  18. [18]

    Kann and M

    C. Kann and M. Feng, Repulsive bounded-confidence model of opinion dynamics in polarized communities (2023), arXiv:2301.02210

    work page arXiv 2023

  19. [19]

    Hickok, Y

    A. Hickok, Y. Kureh, H. Z. Brooks, M. Feng, and M. A. Porter, A bounded-confidence model of opinion dynamics on hypergraphs, SIAM J. Appl. Dyn. Sys.21, 1 (2022)

    work page 2022

  20. [20]

    Schawe and L

    H. Schawe and L. Hern´ andez, Higher order interactions destroy phase transitions in deffuant opinion dynamics model, Comm. Phys.5, 32 (2022)

    work page 2022

  21. [21]

    G. J. Li and M. A. Porter, Bounded-confidence model of opinion dynamics with heterogeneous node-activity lev- els, Phys. Rev. Res.5, 023179 (2023)

    work page 2023

  22. [22]

    Sˆ ırbu, D

    A. Sˆ ırbu, D. Pedreschi, F. Giannotti, and J. Kert´ esz, Al- gorithmic bias amplifies opinion fragmentation and po- larization: A bounded confidence model, PLoS ONE14, e0213246 (2019)

    work page 2019

  23. [23]

    Berner, T

    R. Berner, T. Gross, C. Kuehn, J. Kurths, and S. Yanchuk, Adaptive dynamical networks, Phys. Rep. 1031, 1 (2023)

    work page 2023

  24. [24]

    Sugishita, M

    K. Sugishita, M. A. Porter, M. Beguerisse-D´ ıaz, and N. Masuda, Opinion dynamics on tie-decay networks, Phys. Rev. Res.3, 023249 (2021)

    work page 2021

  25. [25]

    Ahmad, M

    W. Ahmad, M. A. Porter, and M. Beguerisse-D´ ıaz, Tie- decay networks in continuous time and eigenvector-based centralities, IEEE Trans. Net. Sci. Eng.8, 1759 (2021)

    work page 2021

  26. [26]

    Kozma and A

    B. Kozma and A. Barrat, Consensus formation on adap- tive networks, Phys. Rev. E77, 016102 (2008)

    work page 2008

  27. [27]

    Kozma and A

    B. Kozma and A. Barrat, Consensus formation on coe- volving networks: Groups’ formation and structure, J. Phys. A. Math. Theor.41, 224020 (2008)

    work page 2008

  28. [28]

    U. Kan, M. Feng, and M. A. Porter, An adaptive bounded-confidence model of opinion dynamics on net- works, J. Cplx. Net.11, cnac055 (2023)

    work page 2023

  29. [29]

    Brede, How does active participation affect consen- sus: Adaptive network model of opinion dynamics and in- fluence maximizing rewiring, Complexity2019, 1486909 (2019)

    M. Brede, How does active participation affect consen- sus: Adaptive network model of opinion dynamics and in- fluence maximizing rewiring, Complexity2019, 1486909 (2019)

    work page 2019

  30. [30]

    Krishnagopal and M

    S. Krishnagopal and M. A. Porter, Bounded-confidence models of opinion dynamics with neighborhood effects, Phys. Rev. E112, 054317 (2025)

    work page 2025

  31. [31]

    Agostinelli, M

    C. Agostinelli, M. Mancastroppa, and A. Barrat, Group adaptation drives opinion dynamics in higher-order net- works, arXiv:2602.19684 (2026). 22

    work page arXiv 2026

  32. [32]

    G. J. Li, J. Luo, and M. A. Porter, Bounded-confidence models of opinion dynamics with adaptive confidence bounds, SIAM J. App. Dyn. Sys.24, 994 (2025)

    work page 2025

  33. [33]

    Holme and M

    P. Holme and M. E. J. Newman, Nonequilibrium phase transition in the coevolution of networks and opinions, Phys. Rev. E74, 056108 (2006)

    work page 2006

  34. [34]

    R. T. Durrett, J. P. Gleeson, A. L. Lloyd, P. J. Mucha, F. Shi, D. Sivakoff, J. E. S. Socolar, and C. Varghese, Graph fission in an evolving voter model, Proc. Natl. Acad. Sci. U.S.A.109, 3682 (2012)

    work page 2012

  35. [35]

    Y. H. Kureh and M. A. Porter, Fitting in and breaking up: A nonlinear version of coevolving voter models, Phys. Rev. E101, 062303 (2020)

    work page 2020

  36. [36]

    Golovin, J

    A. Golovin, J. M¨ olter, and C. Kuehn, Polyadic opinion formation: The adaptive voter model on a hypergraph, Ann. der Physik536, 2300342 (2024)

    work page 2024

  37. [37]

    A. M. Timpanaro, Emergence of echo chambers in a noisy adaptive voter model (2024), arXiv:2409.12933

    work page arXiv 2024

  38. [38]

    Oh and A

    P. Oh and A. Schauf, Self-organizing group structure through rewiring for collective decision-making in evolv- ing environments, Sci. Rep.15(2025)

    work page 2025

  39. [39]

    K. S. Cook, Exchange: Social, inInternational Encyclo- pedia of the Social & Behavioral Sciences, edited by J. D. Wright (Elsevier, Amsterdam, The Netherlands, 2015) 2nd ed., pp. 482–488

    work page 2015

  40. [40]

    Lorenz, A stabilization theorem for dynamics of con- tinuous opinions, Physica A355, 217 (2005)

    J. Lorenz, A stabilization theorem for dynamics of con- tinuous opinions, Physica A355, 217 (2005)

    work page 2005

  41. [41]

    M. E. J. Newman, Finding community structure in net- works using the eigenvectors of matrices, Phy. Rev. E74, 036104 (2006)

    work page 2006

  42. [42]

    A. L. Traud, P. J. Mucha, and M. A. Porter, Social struc- ture of Facebook networks, Physica A391, 4165 (2012)

    work page 2012

  43. [43]

    M. F. Laguna, G. Abramson, and D. H. Zanette, Minori- ties in a model for opinion formation, Complexity9, 31 (2004)

    work page 2004

  44. [44]

    W. Han, Y. Feng, X. Qian, Q. Yang, and C. Huang, Clusters and the entropy in opinion dynamics on complex networks, Physica A559, 125033 (2020)

    work page 2020

  45. [45]

    A bounded-confidence model of opinion dynamics with adaptive edge probabilities

    L. Thompsky and Y. Wu, Code and figure repository for “A bounded-confidence model of opinion dynamics with adaptive edge probabilities” (2026), available athttps: //github.com/Yolandayx929/adaptive-edge-BCM

    work page 2026

  46. [46]

    H. Z. Brooks and M. A. Porter, A model for the influ- ence of media on the ideology of content in online social networks, Phys. Rev. Res.2, 023041 (2020)

    work page 2020

  47. [47]

    Smith, How TikTok reads your mind (2021), The New York Times, available athttps://www.nytimes.com/ 2021/12/05/business/media/tiktok-algorithm.html; 5 December 2021

    B. Smith, How TikTok reads your mind (2021), The New York Times, available athttps://www.nytimes.com/ 2021/12/05/business/media/tiktok-algorithm.html; 5 December 2021

    work page 2021

  48. [48]

    Mosseri, Instagram ranking explained (2023), Insta- gram blog, available athttps://about.instagram.com/ blog/announcements/instagram-ranking-explained/; 31 May 2023

    A. Mosseri, Instagram ranking explained (2023), Insta- gram blog, available athttps://about.instagram.com/ blog/announcements/instagram-ranking-explained/; 31 May 2023

    work page 2023

  49. [49]

    Milli, M

    S. Milli, M. Carroll, Y. Wang, S. Pandey, S. Zhao, and A. D. Dragan, Engagement, user satisfaction, and the amplification of divisive content on social media, PNAS Nexus4, pgaf062 (2025)

    work page 2025