Source
Golub & Jackson (2010), LIT-999, §§II–V and Appendix A–C, as read in NOTE-840.
What was actually shown
Setting. Each agent starts with an independent signal of mean μ and variance at least σ², and repeatedly replaces its belief by a fixed weighted average of others’ beliefs, p(t) = T p(t−1). All agents use the same rule; only the weights differ.
The consensus. For a convergent T the limit belief of every agent in a closed group is s·p(0), where s is the unit left eigenvector of T (Proposition 1, Theorem 3). Influence is recursive: s_i = Σ_j T_ji s_j.
The characterization. A sequence of growing networks is wise, all limit beliefs converging in probability to μ, if and only if the largest influence weight tends to zero (Lemma 1, Proposition 2; proof checked). This holds for every admissible signal distribution or for none, so “the network structure alone determines wisdom”.
Structure. With reciprocal, equal weights, influence is proportional to degree and wisdom holds iff max degree / total degree → 0 (Corollary 1). A finite group that everyone attends to within t steps with weight bounded below prevents wisdom (Proposition 3). Influence can be concentrated by indirect structure invisible a few steps out (Example 4). Balance together with minimal out- or in-dispersion suffices (Theorems 1 and 4).
Speed. Whether consensus is reached quickly is independent of whether it is right; all four combinations occur (§V).
What could have failed: the theorem is an exact if-and-only-if. A network with vanishing maximal influence whose consensus stayed off the truth would contradict it, and Lemma 1’s variance bound rules that out.
What this does not say
- Not that the members are sound. They are naïve by design: they do not discount repeated information. A rational community might aggregate correctly where this one fails.
- Not about real groups. No data are analysed. Whether real communities update like DeGroot agents is not tested here.
- Not about finite societies. The criterion is asymptotic; it does not grade a group of fixed size.
- Not about correlated errors, strategic agents or changing trust. Signals are independent and weights fixed.
- Not that sparse or slow networks are wiser. Speed and wisdom are independent; the variable is concentration of influence, not density.
Connections
- QUESTION-047 asks whether a community can settle on a belief through community-level mechanisms with its members’ reasoning held fixed. In this model it does: the rule is fixed and identical, and the settled belief is the truth or a leader’s error according to the influence structure. That answers the question in part, in a model, with naïve rather than sound members.
- THEORY-143 and THEORY-148 make network structure decide the outcome in bandit models of inquiry, through density and transient diversity. This adds a third structural variable, the concentration of influence, in a model with no exploration.
Note of 2026-10-19: Lemma 1 and Proposition 2, re-derived for a strongly connected society
Take beliefs p_i(0) in [0, 1], independent, with mean μ and variance at least σ² > 0, and a stochastic weight matrix T(n) that is strongly connected and aperiodic.
- By Proposition 1, every agent’s limit belief is s·p(0), where s is the unit left eigenvector of T, with s ≥ 0 and Σ s_i = 1.
- Its mean is μ Σ s_i = μ, and its variance is Σ s_i² Var p_i(0).
- If the largest weight s_1 tends to 0: each variance is at most 1/4 on [0, 1], so the variance of s·p(0) is at most ¼ Σ s_i² ≤ ¼ s_1 Σ s_i = s_1/4, which tends to 0. By Chebyshev, the consensus converges in probability to μ.
- If s_1 does not tend to 0, then along a subsequence s_1 ≥ c > 0 and the variance is at least σ² s_1² ≥ σ²c². Since |s·p(0) − μ| ≤ 1, convergence in probability would force the variance to 0, so the society is not wise.
Steps 3 and 4 use only the bounds on the variances, so the criterion holds for every admissible signal distribution or for none, which is the sense of “network structure alone”. With several closed groups the limit beliefs are several consensus values, and the maximum over them is not covered by 3. That case, and whether real groups behave so, are QUESTION-184.
Note of 2026-10-25: influence concentration in diffusion form
Used without citation at A36 and A37 of the exchange of 2026-10-24, continued. In a linear network of accumulators the influence-weighted support has variance σ²Σℓᵢ² for independent, equal noises, and A36’s “effective population” 1/Σℓᵢ² is this theorem’s influence concentration in diffusion form. A36 called it “an immediate mathematical connection between network influence and collective statistical efficiency”. Since max ℓᵢ² ≤ Σℓᵢ² ≤ max ℓᵢ, the collective noise vanishes exactly when the largest weight does, this theorem’s condition (CASE-160). That ℓ is DeGroot’s consensus weight π is his Theorem 3 (THEORY-960). The reading is unchanged.
Note of 2026-10-25: DeGroot’s original conditions
DeGroot’s 1974 paper (LIT-1527, read in NOTE-1157 from a copy the owner supplied; THEORY-960) gives a consensus from every start exactly when the weight matrix has a single closed class of communicating members and that class is aperiodic. The consensus is Σπᵢ Fᵢ with π the unique stationary vector, which gives members outside the closed class no weight. He names the two failures: two closed classes, and a periodic closed class, of which he says “a consensus is not reached” (§5). He gives no proofs; the record re-derived them.
So this theorem’s “strongly connected” narrowing, made in the note of 2026-10-19, is stronger than consensus needs: a society with one closed class and transient members reaches a consensus, and its influence weights are the closed class’s. The re-derivation above carries over with s zero off that class. The several-closed-groups question in QUESTION-184 is Golub and Jackson’s question of wisdom, which DeGroot does not address. The reading is unchanged.