Contribution
It gives the DeGroot averaging model (LIT-1527) a behavioural reading: agents update as Bayesians would on new information but fail to discount repetition, so the averaging is a bias with a name and a benchmark. From it come two results: influence that depends on network position apart from accuracy, with an exact test (Theorem 2) for when it does not, and long-run disagreement that is one-dimensional across any number of issues (Theorem 4).
Key insight
If each person keeps a running tally and folds in what neighbours say as though it were new, the neighbours’ reports, which already contain one’s own and each other’s earlier reports, are counted again each round. Whoever is listened to by those who are listened to is counted most. Influence is then the stationary distribution of a Markov chain on the listening graph, and the slowest-decaying disagreement is its second eigenvector, the same vector on every issue.
Assumptions
- Signals. Agent i starts with x⁰ᵢ = θ + εᵢ, θ ∈ ℝᴸ, errors normal, independent across agents and issues; i assigns j a precision π⁰ᵢⱼ, possibly subjective. A distribution-free version with counts of observations gives the same linear rule.
- Truthful reports. No strategic communication; §V says this is unreasonable for propaganda and campaigns.
- Persuasion bias. x^{t+1} = T_t x^t with T_t = (1 − λ_t)I + λ_t T, T_ij = q_ij π⁰ᵢⱼ / Σ_k q_ik π⁰ᵢₖ, the same relative weights every round; λ_t ∈ (0, 1] lets self-weight grow.
- Fixed, exogenous listening structure, with i ∈ S(i).
- Strong connectedness and Σ λ_t = ∞ (Assumption 1) for consensus; for unidimensionality, a real second eigenvalue and no ties, true for generic T.
Key results
- Theorem 1. Beliefs converge to the consensus w x⁰, where w is the unique solution of wT = w, all wᵢ > 0. Corollary 1: wᵢ = Σⱼ Tⱼᵢ wⱼ, so one is influential if listened to by the influential.
- Theorem 2. With correct precisions, the consensus is the precision-weighted optimum iff Σⱼ qⱼᵢ Tⱼⱼ = 1 for every i: the self-weights of those who listen to each agent sum to one. Typical networks fail it; a regular network with equal precisions meets it.
- Theorem 3. Fully rational agents who know the network converge to the correct beliefs after at most N² rounds; the proof is a dimension count, not constructive.
- Theorem 4. For generic T with α₂ real, the relative difference of opinion d^∞ᵢₗ = φᵢ Dₗ, φᵢ = a + b V₂ᶜᵢ, D = V₂ʳ x⁰: each agent’s position is the same on every issue, the line set by the second row eigenvector and initial beliefs.
- Theorem 5. Consensus converges at the rate of α₂; unidimensionality at the rate of α₃/α₂.
- Theorems 6–7 (bilateral listening). With equal precisions wᵢ/wⱼ = s(i)/#S(j): influence is the number of one’s neighbours, not their influence, since an influential neighbour also spreads his attention thinner. T is reversible, its eigenvalues real.
- Theorem 8 (agents on a line). The long-run order of opinion is the geographical order whatever the initial beliefs; agents are densest at the extremes, “a form of polarization”; α₂ ≈ 1 − π²/12M², α₃ ≈ 1 − π²/3M², so doubling the line slows convergence fourfold and opinions become one-dimensional three times faster than they agree.
- Appendix. Influence weights count spanning trees rooted at each agent (Theorem 9). Isolated groups converge to their own beliefs, and everyone else lies in the convex hull of theirs (Theorem 10).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Under persuasion bias, consensus weights are the listening matrix’s stationary distribution | strong | Theorem 1; checked on Example 1 |
| C2 | Consensus is correct iff the network is balanced | strong | Theorem 2; proof followed |
| C3 | Rational agents knowing the network reach correct beliefs in ≤ N² rounds | strong | Theorem 3; proof followed |
| C4 | Long-run disagreement is one-dimensional across issues | strong, for generic T with α₂ real | Theorem 4; proof skimmed; checked on Example 1 |
| C5 | Political left–right structure, propaganda, air-time and cult conformity are explained by persuasion bias | weak | §V, informal, with anecdotes (New Square’s 1400–12 vote) |
| C6 | Issues with more media attention should be more aligned with party lines | weak | a prediction, untested here |
Concepts
- Persuasion bias: failing to adjust for repetition in the information one receives, whether from one source over time or from sources that share sources.
- Listening structure: the directed graph S; T its weighted matrix.
- Social influence: wᵢ, agent i’s weight in the consensus.
- Self-importance: Tⱼⱼ, the weight an agent gives his own belief.
- Balanced: Σⱼ qⱼᵢ Tⱼⱼ = 1 for all i.
- Unidimensional opinions: long-run relative positions that coincide across issues.
Connections
The dynamics are DeGroot’s (1974, LIT-1527) and French’s and Harary’s social power; the paper’s addition is the reading as a bias against a rational benchmark, the balance test, and the second-eigenvector results. Golub and Jackson’s naïve learning (LIT-999, THEORY-321) takes this model to large networks and asks when the bias washes out. Jia, MirTabatabaei, Friedkin and Bullo’s reflected appraisal (LIT-1146) and Friedkin’s social power (LIT-1482) let the weights move across issues, which this paper holds fixed. The conclusion points to a companion paper, DeMarzo, Vayanos and Zwiebel (2001), “Social Networks and Financial Markets”, in which agents choose whom to listen to by trading profits and stable structures have everyone listening to the same experts, who “need not be the ones with the best information, and consequently, influence can be self-perpetuating independently of accuracy”. That paper is not registered.
Bearing on the record
- QUESTION-117. The question lists this as a candidate predecessor and quotes A15: “Endorsement-mediated reinforcement could sometimes amount to repeated counting of correlated evidence.” On this reading the fixed-weight model already gives one of A15’s suite, capture without fidelity: by Theorem 2, an unbalanced network’s consensus departs from the precision-weighted optimum with no change in anyone’s accuracy. It does not give the others. Its weights never move, so it has no hysteresis and no feedback from outcome to authority; operator turnover would only change T. Those need either reflected appraisal (LIT-1146) or the companion paper’s endogenous listening, which this paper describes but does not contain. So for QUESTION-117 the paper sets the floor: whatever the coupled meaning–authority model claims beyond position-weighted consensus has to come from moving weights.
- CLAIM-297. The claim is that authority updated against the interpretation it helped make dominant can amplify an initial advantage without gain in fidelity. This paper shows that even without that loop, fixed listening weights give an agent influence out of proportion to accuracy. The claim’s loop is an addition to that, not the only route to it. The companion paper’s result quoted above is the closest published statement of the loop the record has met; it should be registered and read before the claim’s mechanism is called new.
- Unidimensionality. Theorem 4 is a mechanism by which many issues collapse onto one axis from network structure alone. Nothing in the record found on 2026-10-29 by a search for “unidimensional” and “left-right” cites it.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Fixed weights. Agents never learn who is accurate or how redundant a source is; the persuasion assumption is exactly the constancy of T.
- Truthful reporting, which §V concedes fails in its own headline applications.
- No data. The political-science and marketing sections are conjecture and anecdote; the authors ask for experiments.
- Genericity. Theorem 4 needs α₂ real and no ties; one-way cycles give complex α₂ and cycling positions, a result “available upon request”.
- Errors and slips. None of substance found in what was checked. Extraction garbles fractions throughout, and Example 1’s eigenvector (3/2, −1/2, −3/2, −3/2) is correct only up to scale, as eigenvectors are. The step in Theorem 3’s proof from “sufficient statistic for every agent’s current statistic” to “for all past statistics” is asserted, not argued.
Open questions
- Does the coupled meaning–authority model (CLAIM-297) reduce, for some parameters, to the companion paper’s endogenous-listening model?
- Does Theorem 4’s unidimensionality survive weights that move with agreement, as in reflected appraisal?