False-merge growth law in multi-cluster entity resolution

Prove that the mean damage g(s) caused by one false merge between two clusters of size s under uniformly random query ordering satisfies g(s)=Θ(s^{2/3}), and consequently establish the stated first-order risk scaling for K clusters of size s.

Background

For entity resolution with transitivity pruning, the quantity g(s) is the expected number of wrong final pair decisions caused by one incorrectly accepted cross-cluster “same” answer when the two affected clusters each have size s. The paper derives bounds, exact values for small s, and a first-order decomposition showing that a single false merge is localized to the two involved entities.

The authors conjecture that g(s) grows as s^{2/3}. This would imply that, for K clusters of equal size s, the within-cluster contribution scales as Θ(ε_w K s^{7/3}) and the cross-cluster contribution as Θ(ε_x K^2 s^{8/3}) at first order. Simulations support the exponent but do not determine the asymptotic constant; the paper states that a proof would require a scaling limit for the constrained merging process near the critical window of the within-cluster Erdős–Rényi graphs.