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arxiv: 2605.20418 · v2 · pith:URWS6SY4new · submitted 2026-05-19 · ⚛️ physics.soc-ph · cs.SI· cs.SY· eess.SY· math.DS· math.PR

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 2026-05-25 06:31 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cs.SIcs.SYeess.SYmath.DSmath.PR
keywords opinion dynamicsbounded confidenceadaptive networksDeffuant-Weisbuch modelconvergenceeffective graphinteraction probabilities
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The pith

An adaptive edge-weighted Deffuant-Weisbuch model converges while its interaction probabilities evolve to define a time-dependent effective graph of receptive agent pairs.

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

The paper extends the standard bounded-confidence model by letting edge weights adapt after each compromise, raising the chance of future interactions between agents who have already aligned. It proves that opinions still converge, that the weights settle into long-term behavior, and that the system can be reduced to an effective graph containing only currently receptive pairs. Simulations then show that this adaptation shortens convergence time on dense networks but lengthens it on sparse ones when the confidence bound is small. A sympathetic reader would care because the extension captures the realistic idea that successful conversations make future ones more likely, while still delivering rigorous guarantees on the resulting dynamics.

Core claim

The authors introduce an adaptive edge-weighted version of the Deffuant-Weisbuch model in which edge weights increase after compromise according to a positive-feedback rule. They prove that the opinions converge, that the edge weights reach a long-time limit, and that the system is equivalent to a time-dependent effective graph consisting solely of edges between agents whose opinions lie within the confidence bound. Numerical experiments on various networks confirm that the adaptive weights produce qualitatively different convergence times depending on network density.

What carries the argument

The adaptive edge-weight update rule that raises interaction probability after compromise and thereby generates the effective graph of receptive pairs.

If this is right

  • Opinions reach stable clusters whose sizes depend on the final effective graph.
  • The effective graph becomes fixed once all edge weights have converged.
  • Convergence time decreases with adaptive weights on dense networks.
  • Convergence time increases with adaptive weights on sparse networks for small confidence bounds.

Where Pith is reading between the lines

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

  • The mechanism suggests that real social networks with strong positive feedback from agreement may reach consensus faster than static models predict when connectivity is high.
  • One could replace the update rule with a version that also allows weight decreases after disagreement and check whether the convergence proofs still hold.
  • The effective-graph reduction might be used to analyze other bounded-confidence variants without resimulating every interaction.

Load-bearing premise

The model assumes that edge weights increase after every compromise according to one specific positive-feedback rule.

What would settle it

Running the model on a small complete graph with a fixed small confidence bound and observing that the edge weights fail to stabilize or that the set of active edges does not match the predicted effective graph after a long time.

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

0 major / 3 minor

Summary. The paper extends the Deffuant-Weisbuch bounded-confidence model to networks with heterogeneous, adaptive edge weights that increase interaction probability after compromise. It claims to prove convergence of opinions to clusters, the long-time dynamics of the edge weights under a positive-feedback update rule, and the structure of a time-dependent effective graph (the subgraph of receptive pairs). Simulations on multiple network classes are presented to show that adaptive weights decrease convergence time on dense networks but increase it on sparse networks for small confidence bounds.

Significance. If the proofs are correct, the work supplies a rigorous analytic framework for adaptive interaction probabilities in BCMs, with the effective-graph construction offering a useful reduction for long-time analysis. The topology-dependent simulation results are a concrete, falsifiable prediction that distinguishes the model from the static-weight DW case. Explicit model equations and proof strategy are supplied, which is a strength.

minor comments (3)
  1. The abstract states the adaptive rule only qualitatively; a one-sentence description of the positive-feedback update (e.g., the functional form of the weight increment) would improve readability without lengthening the abstract appreciably.
  2. Figure captions for the simulation panels should explicitly state the network size N, number of Monte-Carlo realizations, and the precise values of the confidence bound ε and the adaptation rate used in each panel.
  3. Notation for the effective graph G_eff(t) is introduced in the text but never appears in a displayed equation; adding a compact definition (e.g., Eq. (X)) would aid cross-referencing in the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines an explicit positive-feedback adaptive rule for edge weights as an independent modeling choice and then derives convergence properties, long-time edge-weight evolution, and effective-graph structure via direct proofs from the stated model equations. No fitted parameters are renamed as predictions, no self-citation chains bear the central claims, and no ansatz or uniqueness result is smuggled in. The theoretical guarantees are self-contained against the model definition; simulations are presented only as supporting evidence. This is the standard case of an internally consistent derivation with no reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract supplies no explicit free parameters, standard axioms, or invented entities beyond the adaptive weight rule itself; details of the weight-update function and any background lemmas are absent.

invented entities (1)
  • adaptive edge weights no independent evidence
    purpose: to encode that people are more likely to communicate with individuals with whom they have previously compromised or had other positive interactions
    Central modeling addition described in the abstract; no independent evidence supplied.

pith-pipeline@v0.9.0 · 5793 in / 1177 out tokens · 53202 ms · 2026-05-25T06:31:44.655113+00:00 · methodology

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Reference graph

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