Reconstruction of Line-Embeddings of Graphons
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Consider a random graph process with n vertices corresponding to points v_i∼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) ∈ [0,1]. Following Chuangpishit et al. (2015), 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 σ∈ S_n an ordering of these vertices if v_σ(i) < v_σ(j) for all i < j, and ask: how can we accurately estimate σ from an observed graph? We present a randomized algorithm with output σ̂ that, for a large class of graphons, achieves error max_1 ≤ i ≤ n | σ(i) - σ̂(i)| = O^*(√(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 √(n) and obtain the vastly better rate O^*(n^ϵ) for any 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, and testing whether a graphon is diagonally increasing.
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