Consider a random graph process with n vertices corresponding to points embedded randomly in the interval, and where edges are inserted between independently with probability given by the graphon . Following , we call a graphon w diagonally increasing if, for each x, decreases as y moves away from x. We call a permutation an ordering of these vertices if for all , and ask: how can we accurately estimate σ from an observed graph? We present a randomized algorithm with output that, for a large class of graphons, achieves error with high probability; we also show that this is the best-possible convergence rate for a large class of algorithms and proof strategies. Under an additional assumption that is satisfied by some popular graphon models, we break this “barrier” at and obtain the vastly better rate for any . These improved seriation bounds can be combined with previous work to give more efficient and accurate algorithms for related tasks, including estimating diagonally increasing graphons [20, 21] and testing whether a graphon is diagonally increasing .
The authors acknowledge support from the National Science and Engineering Research Council through their Discovery Grant program.
Jeannette Janssen. Aaron Smith. "Reconstruction of line-embeddings of graphons." Electron. J. Statist. 16 (1) 331 - 407, 2022. https://doi.org/10.1214/21-EJS1940