Reconstruction of line-embeddings of graphons

Abstract

Consider a random graph process with n vertices corresponding to points $v_i\stackrel{i.i.d.}{\sim }\mathrm{Unif}[0,1]$ embedded randomly in the interval, and where edges are inserted between $v_i,v_j$ independently with probability given by the graphon $w(v_i,v_j)\in [0,1]$. Following [11], we call a graphon w diagonally increasing if, for each x, $w(x,y)$ decreases as y moves away from x. We call a permutation $\mathit{\sigma }\in S_n$ an ordering of these vertices if $v_{\mathit{\sigma }(i)}<v_{\mathit{\sigma }(j)}$ for all $i<j$, and ask: how can we accurately estimate σ from an observed graph? We present a randomized algorithm with output $\hat{\mathit{\sigma }}$ that, for a large class of graphons, achieves error ${max}_{1\le i\le n}|\mathit{\sigma }(i)-\hat{\mathit{\sigma }}(i)|=\tilde{O}(\sqrt{n})$ 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 $\sqrt{n}$ and obtain the vastly better rate $\tilde{O}(n^{\mathit{ϵ}})$ for any $\mathit{ϵ}>0$. 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 [11].