Sharp and interpretable conditions for wisdom in large networks

Identify sharp, interpretable necessary and sufficient conditions under which a sequence of irreducible row-stochastic matrices (M(n)) yields wisdom—i.e., the associated eigenvector centralities c_i(n) uniformly converge to 0 as n → ∞—ensuring that consensus under independent, unbiased initial opinions converges to the true mean.

Background

Golub and Jackson (2010) introduce wisdom—vanishing centralities in large societies—linking it to convergence of consensus to the true mean with independent initial opinions.

The paper gives a simple obstruction (bounded-size prominent sets) and interpretable sufficient conditions, but the author notes that sharp and interpretable conditions have not been established, leaving a gap between necessary obstructions and sufficient criteria.