Optimal power of few in sparse majority dynamics

Determine whether the optimal initial Red advantage for achieving Red unanimity in majority dynamics on an Erdős–Rényi random graph G(n,p), under the fixed-advantage coloring scheme, is of order C_ε p^{-1/2} throughout the sparse-density regime, particularly for densities below the currently established dense regime.

Background

The paper considers majority dynamics on G(n,p) with a fixed initial advantage: the Red camp has more vertices than the Blue camp by a parameter α. The quantity αmin(n,p,ε) is the smallest initial advantage for which Red wins with probability at least 1−ε. Tran and Vu conjectured that the optimal power-of-few value is Cε p^{-1/2}, independent of n up to the dependence on p.

That conjecture had been confirmed in the dense regime p>log^{-1/16}n, but the paper states that the corresponding result for lower densities remained unresolved. The paper proves a different sufficient bound, of order p^{-3/2}n^{-1/2}log n, in a range of sparse densities, thereby making progress without resolving the conjectured optimal p^{-1/2} dependence in the full sparse regime.