nucleation

THEORY-916: In the DeGroot–Friedkin model, where a group reaches DeGroot consensus on each issue and each member's self-weight on the next issue equals their control over the last, with fixed relative interpersonal weights that form a strongly connected network, self-weights converge from every non-autocratic start to one equilibrium ordered by the eigenvector centralities of those weights: power accumulates above a centrality threshold, is autocratic exactly when the network is a star and democratic exactly when it is doubly stochastic; and in the networks that are not strongly connected that the authors treat, members to whom some other member does not listen, even indirectly, end with none

3 min read

Properties6
StatusProposed
Version2
Tagsnetwork-science, social-science, mathematics
SourceLIT-1146 — Opinion Dynamics and the Evolution of Social Power in Influence Networks
Bears onQUESTION-348 — In groups that discuss a sequence of issues under a stable network of relative influence, measured independently of the members' self-weights, does each member's influence over outcomes come to order as the network's eigenvector centralities and concentrate above them, as the DeGroot–Friedkin model proves for its own dynamics?
CorroboratesCLAIM-448 — An ordinal potential describes only the winning-move relation it is computed on; where an organization's legibility regime selects which contests are admitted or recorded, an acyclic record, even a persistent potential, can be the product of selective contestation, so observed coherence is not evidence that the organization's perspectives are integrated, and an intervention that improves its self-governance may at first increase the disagreement observed

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Source

Jia, MirTabatabaei, Friedkin & Bullo, LIT-1146, read in NOTE-1135: Lemmas 2.2–3.2, Theorem 4.1, Proposition 4.2 and Section 5 (pp. 376–385), with proofs in Appendices B–G, skimmed.

What was actually shown

A proof, not an experiment. The model: on each issue, opinions follow DeGroot averaging with W(x) = diag(x) + (I − diag(x))C and reach consensus; then x(s+1) = w(x(s)), the dominant left eigenvector of W.

  • The only parameter is c, the eigenvector centrality of C (Lemma 2.2).
  • Interior equilibrium. Unless C is a star, every non-autocratic start converges to one x*, ordered as c (Theorem 4.1).
  • Accumulation. Above a threshold power exceeds centrality; below it, falls short (Proposition 4.2). “social power tends to accumulate at the top of the hierarchy” (p. 367).
  • The extremes. A star converges to autocracy; a doubly stochastic C to equal power, and only then (Lemmas 3.1, 3.2).
  • The unreached. With a reducible C, members not globally reachable end with zero power (Section 5); in Krackhardt’s advice network, the four managers nobody consults.

What this does not say

  • That groups behave this way. The authors: “an assessment of the validity of the model has just begun” (p. 386).
  • That it is robust. Anchored dynamics differ, and “At the present time, we cannot assert robustness” (p. 386).
  • That authority is chosen or maintained. C is fixed and exogenous; nobody decides who is heard.
  • That the equilibrium is autocratic in general. Only for a star. Short of that, the most central member gains disproportionately but does not take everything.

Connections

  • CLAIM-448. The baseline it names, now in exact form. Every issue ends in consensus, so the record is always coherent; its content is weighted toward the central, and the unreached contribute nothing. That is a proved instance of coherence without integration, produced by uptake rather than by a regime of admission.
  • THEORY-902. The experiment on the same mechanism under anchored dynamics.
  • THEORY-321. Golub & Jackson’s DeGroot result: a consensus is right in large societies only if no member’s influence weight stays large. Under reflected appraisal the most central member’s weight grows, which is the condition THEORY-321 says defeats the wisdom of crowds. That combination is the record’s, not the paper’s.

Note of 2026-10-29: the same rule under the Friedkin–Johnsen step

Friedkin’s reflected-appraisal theory, LIT-1404, read in NOTE-1374, uses the same appraisal rule with a different single-issue step: Friedkin–Johnsen anchoring on initial positions instead of DeGroot consensus. Its model differs, so it is not a counter-example to this account. But the same rule gives autocracy under that step. Re-run as printed on 300 random complete networks each of sizes 5, 10 and 20, every one ended with one self-weight above 0.99; the DeGroot–Friedkin update on 100 random complete networks of sizes 5 and 10 gave a median largest self-weight of 0.29 and 0.13, never above 0.40. The comparison is THEORY-1161, Proposed. This account is unchanged and stays Proposed.

Note of 2026-10-30: what holds it back

The proofs of Lemmas 3.1–3.2, Theorem 4.1, Proposition 4.2 and Section 5 were skimmed, not checked step by step (NOTE-1135), and the record holds no published independent proof. The title asserts the theorems, so it waits on that check (ADR-065 §5). Whether groups behave as the model does is QUESTION-348.

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