  We propose a consistent polynomial-time method for the unseeded node matching
problem for networks with smooth underlying structures. Despite widely
conjectured by the research community that the structured graph matching
problem to be significantly easier than its worst case counterpart, well-known
to be NP-hard, the statistical version of the problem has stood a challenge
that resisted any solution both provable and polynomial-time. The closest
existing work requires quasi-polynomial time. Our method is based on the latest
advances in graphon estimation techniques and analysis on the concentration of
empirical Wasserstein distances. Its core is a simple yet unconventional
sampling-and-matching scheme that reduces the problem from unseeded to seeded.
Our method allows flexible efficiencies, is convenient to analyze and
potentially can be extended to more general settings. Our work enables a rich
variety of subsequent estimations and inferences.
