Source
Vilone and Polizzi (2024), PLOS ONE 19(1): e0296075, LIT-939, read in full in NOTE-810.
What was actually shown
Two agent-based models on scale-free networks (N = 5000, λ = 2.2, and λ = 1.5 as a check). Private beliefs b_i = ±1 are fixed. A hard-core minority (b = −1, always expressed) sits on every node of degree K_c, and the majority’s belief strengths σ_i are drawn uniformly from [0, 1].
- Model 1: no voluntary silence. An agent expresses its belief with
probability σ_i and otherwise its neighbours’ majority.
- Expressed opinion falls below belief for every K_c.
- The local climate r, what agents see around them, differs from the global expressed average by “at most around 15%” (K_c = 10).
- At K_c = 3, agents see slightly under 0.2 against a mean belief of about 0.67.
- Model 2: silence allowed. Agents choose ω_i ∈ {−1, 0, +1} to
maximise
G(ω_i) = −| (σ_i b_i + Σ_j A_i^j ω_j)/(|Γ_i| + 1) − ω_i |,
with A_i^j = deg(j)/deg(i).
- About 55–60% of agents end silent, “regardless of K_c” (Fig 3).
- Expressed opinion plateaus well below belief even as the minority falls under 0.1% (K_c = 20), where each hard-core agent silences about 150 others, far beyond its own neighbours (Fig 4).
- Here the locally perceived opinion is closer to the true beliefs than the global average is: the topological effect is “besides small, also non-universal”.
What could have come out otherwise: the silent fraction could have scaled with the minority’s size or centrality, and the topological term r − ⟨ω⟩ could have dominated, as the “majority illusion” account expects. Neither happened.
What this does not say
- Not that the community comes to believe the minority view. Beliefs are fixed by construction. The result is about what is expressed and perceived. Whether silence would in time change beliefs is outside the model, and the authors warn that fixed beliefs may exaggerate the dynamics.
- Not that structure, rather than members’ choices, sets the outcome. The paper’s point is close to the opposite. The direct effect of network geometry is small, and the bias is “mainly due to agents’ behavior”. Structure matters through degree-weighted influence and the cascade, and those are effects of the decision rule as much as of the graph.
- Not that a few charismatic agents are needed. A large, poorly connected minority produces the same silent fraction. Number and centrality trade off.
- Not a reversal. Expressed majority and minority swap places only when the minority exceeds about 30%. Below that the minority looks bigger, but does not look like the majority.
- Not shown in any real community. No empirical validation. The 55–60% figure agrees with one other model (Ross et al.’s social-bot model), which is agreement between simulations, not evidence.
- “Charisma” is degree, by stipulation.
Connections
- QUESTION-047. A community-level mechanism by which what a
community appears to hold, which is all any member can observe,
comes apart from what every member believes. It answers only the
question’s second clause (which mechanisms), and only for appearance.
No
answers:relation is declared, because the model has no belief that the community settles on. - THEORY-146. That account biases a community without corrupting any individual, through funding and recruitment. This one distorts a community’s public face without changing any individual’s belief, through expression. Both locate the effect above the members.
- THEORY-143. That account finds communication structure decisive for where an inquiring group settles. This one finds the network’s direct effect on perception small. The settings differ (learning against fixed belief), so there is no conflict, but the two should not be cited together as “structure matters”.
Note of 2026-10-22: the majority illusion
Lerman et al. (LIT-1390) measure something different and find a large effect: when the attribute is pushed onto hubs, “as many as 60%–70% of nodes have more than half of their neighbors in the active state, even though only 20% of the nodes are active” (Results, Fig 4). This theory’s topological effect is the gap between average local climate and global opinion, with the minority on nodes of one degree, and its source found it “besides small, also non-universal”. The two do not conflict. The title already confines “the network’s direct distortion of what members see is small” to the model that shows it; Lerman et al. show that the confinement matters, since a correlation between degree and attribute makes the distortion large.