Source
Lerman, Yan & Wu (2016), PLOS ONE 11(2): e0147617, arXiv:1506.03022, LIT-1390, read in full in NOTE-1085 from the publisher’s PDF. The Supporting Information (S1 File, S1 Fig) was not read.
What was actually shown
- Synthetic scale-free networks (N = 10,000, α = 2.1, 2.4, 3.1, 5% of nodes active). The share of nodes with more than half their neighbours active rises with the degree–attribute correlation ρ_kx, with disassortativity and with tail weight; at α = 2.1 it reaches 60–80% (Fig 2).
- Erdős–Rényi networks. The effect is much smaller but present, more so with 10–20% active (Fig 3).
- Six real topologies (co-authorship, protein interaction, Digg, Enron, Twitter, political blogs), with 5–30% of nodes activated and ρ_kx raised by swapping attributes. In the three disassortative ones, 60–70% of nodes see an active majority when 20% are active; less in the assortative ones (Fig 4).
- A model (Eqs 3–5) treating a node’s neighbours as independent draws, with the probability of an active neighbour set by the joint degree distribution and P(x|k), reproduces the simulated shares. A Gaussian shortcut fails for heavy tails and strong (dis)assortativity (Fig 5).
What could have come out otherwise: the share in the paradox regime could have stayed near the global prevalence once real topologies, with their clustering and assortativity, replaced configuration-model graphs. It did not, in the disassortative ones.
What this does not say
- Not that real attributes are distributed this way. On every network the authors choose the active nodes and push ρ_kx to the values they want. Whether opinions, behaviours or adoptions are in fact concentrated on hubs strongly enough is not measured.
- Not that anyone is misled. No person’s perception, estimate, belief or behaviour is measured. Each node’s view of its neighbours is, by construction, accurate; the illusion is only in generalizing from them. That people do generalize this way is the authors’ assumption.
- Not a loud minority. The mechanism is degree. A minority that speaks more often or more forcefully is a different mechanism (silence and expression, THEORY-242).
- Not pluralistic ignorance. Pluralistic ignorance misjudges what others hold; the majority illusion samples correctly from a skewed neighbourhood. The paper offers itself as a possible explanation of the first, citing Prentice & Miller (LIT-1370), without testing it.
- Not that contagion is accelerated. That rests on a 14-node illustration with a threshold model (Fig 1); no contagion is simulated.
Connections
- THEORY-242. Vilone and Polizzi set themselves against this paper and find the network’s direct, topological distortion small in their model. The two measure different things (an average local climate against a share of nodes with a local majority) under different placements of the minority (all nodes of one degree, against hubs at high ρ_kx), so both can hold. The two accounts are complementary routes to an apparent majority: one by who is seen, the other by who stays silent.
- CLAIM-415. The claim grounds itself on this paper for a perceived plurality with no real one behind it. This account supplies the structural half of that: a collective majority attributed from an accurate but skewed sample. It supplies nothing about how the attribution organizes a person, which is the claim’s own step.