Contribution
DeGroot’s averaging model was known to reach consensus; this paper asks when the consensus is right. It gives an exact answer in terms of influence weights, a network-level characterization in a special case, a structural obstruction, and structural sufficient conditions for large societies of naïve updaters to aggregate dispersed information correctly.
Key insight
The consensus is a weighted average of the initial signals, with weights given by the network’s left unit eigenvector. A law of large numbers applies to that average exactly when no single weight stays large. So correctness of the group belief is a property of where influence concentrates in the network, not of how each agent updates.
Assumptions
- Updating. p(t) = T p(t−1), T an n × n row-stochastic matrix, fixed over time, possibly asymmetric (§I). Agents do not correct for repeated or correlated information (“persuasion bias”); only the first round is Bayesian, under normal signals with T_ij ∝ the precision i assigns to j.
- Signals. p_i(0)(n) ∈ [0, 1], mean μ, independent within each n, variance at least σ² > 0 (needed only for the “only if” of Lemma 1).
- Convergence. Each T(n) is convergent (strongly aperiodic, Theorem 2); strong connectivity is not required.
- Large societies. Wisdom is a double limit: t → ∞ for each network, then n → ∞ along a sequence of networks.
Key results
- Proposition 1. For strongly connected T: convergent ⇔ aperiodic ⇔ there is a unique left eigenvector s of eigenvalue 1, entries summing to 1, with lim (T^t p)_i = s·p for every i. s_i is i’s influence: s_i = Σ_j T_ji s_j.
- Theorems 2–3 (App. A). T is convergent iff strongly aperiodic (aperiodic on every closed group). Each minimal closed group reaches its own consensus s_B p_B; agents outside closed groups converge to mixtures of these and have influence 0.
- Equal weights, undirected (§II.C). T_ij = G_ij/d_i gives s_i = d_i / Σ_j d_j.
- Lemma 1 (law of large numbers). plim s(n)p(0)(n) = μ iff s_1(n) → 0 (s_1 the largest weight). Proof checked: var ≤ Σ s_i² ≤ s_1 → 0 one way; var ≥ s_1²σ² the other.
- Proposition 2 (characterization). A sequence of convergent stochastic matrices is wise, plim max_i |p_i(∞)(n) − μ| = 0, iff s_1(n) → 0. Holds when: the signal assumptions above, for every such signal distribution or none.
- Corollary 1. Reciprocal equal-weight networks are wise iff max_i d_i / Σ_j d_j → 0: “disproportionate popularity … is the only obstacle”.
- Proposition 3 with Lemma 5. If a finite family B_n is uniformly prominent, min_{i∉B}(T^t)_{i,B} ≥ α > 0 for some t, the society is not wise; s_B ≥ κπ_B/(1 + π_B). Checked.
- Examples 3–4. An unbalanced star keeps the centre’s influence at (1−ε)/(1−ε+δ) for any n; a one-way line gives agent 1 influence near 1 for small δ although agent 1 is prominent only in n − 1 steps. So direct weight, even indirect weight a few steps out, can mislead.
- Theorem 1 (sufficiency). Balance (no family of size ≤ j(n) → ∞ gets unboundedly more weight from outside than it gives out) plus minimal out-dispersion (any large enough finite family gives at least r to any family of nearly all of society) imply wisdom. Neither alone suffices (Examples 3–4). Proof checked.
- Theorem 4 (App. C). Balance plus minimal in-dispersion also suffice.
- §V, speed. Convergence time scales with −1/log|λ₂(T)|. Four examples show speed and wisdom vary independently: both, neither, or either one.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | The DeGroot consensus is the left-eigenvector-weighted average of initial beliefs | strong | Proposition 1, Theorem 3 (known results, proved) |
| C2 | A growing society is wise iff the maximal influence vanishes | strong | Lemma 1, Proposition 2, checked |
| C3 | With reciprocal equal weights, wisdom ⇔ max degree / total degree → 0 | strong | Corollary 1 |
| C4 | A finite uniformly prominent group prevents wisdom | strong | Proposition 3, Lemma 5, checked |
| C5 | Balance and dispersion together suffice for wisdom | strong | Theorems 1 and 4 |
| C6 | Speed of convergence and wisdom are independent | moderate | four examples, §V |
| C7 | Influence cannot be read off direct or few-step attention | strong | Examples 3–4, exact formulas |
Concepts
- Wise. Limiting beliefs converge jointly in probability to μ as n grows (Definition 3).
- Influence weight s_i. Agent i’s weight in the consensus; the unit left eigenvector entry.
- Prominent group. One from which every outsider receives weight within t steps (Definition 4); uniformly prominent if bounded below along the sequence (Definition 5).
- Balance; minimal out-/in-dispersion. Properties 1–3, conditions on weight between small families and the rest.
Connections
The model is DeGroot (1974); the Bayesian motivation and persuasion bias are DeMarzo, Vayanos and Zwiebel (2003), whose knife-edge condition for optimal aggregation in finite societies this replaces by asymptotic conditions. It contrasts with observational learning (Bala and Goyal 1998; Acemoglu et al. 2008), where wisdom needs some stubborn or arbitrarily informed agent, and with eigenvector centrality (Katz 1953; Bonacich 1987; PageRank). Speed is left to the authors’ homophily paper (arXiv:0811.4013) and to conductance and Cheeger bounds.
Bearing on the record
- QUESTION-047. The paper is a model in which every member uses the same updating rule and the community’s settled belief is fixed by network structure: change the weights, hold the rule fixed, and the consensus changes from the truth to the leader’s signal. That is the form QUESTION-047 asks for, in a model, not a real community. The consensus s·p(0) is reached by every agent, but no agent computes or aims at it.
- A correction to the filing. LIT-999’s body says “Every member updates soundly”. The paper’s agents are explicitly naïve: they do not adjust for repeated and cross-contaminated information, and only the first round is Bayesian. So the case is of naïve, uniform members and a structure-determined group belief, not of sound members. QUESTION-047’s “individually sound inquirers” fits it only if a fixed reasonable rule counts as sound. The status note’s “a structural condition on the network, not the soundness of its members, decides” is right.
- CLAIM-030 and CLAIM-060. The collective outcome here is entirely derived from members’ rules and weights by linear algebra; it is “systemic” only in that no member’s rule mentions it. Whether that counts for CLAIM-030 is a question for that claim; the paper gives no mechanism beyond the weights.
- THEORY-143 and THEORY-148 (Zollman-type bandit models) concern sparse networks and transient diversity. This model has no exploration and no action, so it neither supports nor conflicts with them; it adds a different structural variable, concentration of influence.
- THEORY-321 is filed from this reading.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Fixed weights; agents never learn whom to trust.
- Independent signals with a common mean; no correlated errors and no strategic agents.
- Asymptotic in n; finite societies are not graded.
- The sufficient conditions can be hard to check (§VI); they are not necessary.
Open questions
- Which network-formation processes yield balance and dispersion.
- Whether agents who reweight over time, ignoring insular groups, restore wisdom.
- Whether a small admixture of more rational agents changes the outcome.