The second reconstruction transition for random graphs from 2-neighborhoods

Abstract

We prove that, with high probability, the Erdős--Rényi random graph $G(n,p)$ is reconstructible up to isomorphism from the multiset of its rooted induced 2-neighborhoods whenever $p\geq n^{-3/4}(\log n)^3$. Together with the non-reconstructibility result of Johnston, Kronenberg, Roberts and Scott, our result locates the second reconstruction transition at the scale $p=n^{-3/4}$ up to polylogarithmic factors, resolving a question they posed. Our proof relies on a global rigidity argument exploiting 5-cycle constraints.