Consistency of Spectral Seriation

12/08/2021

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by   Amine Natik, et al.

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Consider a random graph G of size N constructed according to a graphon w : [0,1]^2↦ [0,1] as follows. First embed N vertices V = {v_1, v_2, …, v_N} into the interval [0,1], then for each i < j add an edge between v_i, v_j with probability w(v_i, v_j). Given only the adjacency matrix of the graph, we might expect to be able to approximately reconstruct the permutation σ for which v_σ(1) < … < v_σ(N) if w satisfies the following linear embedding property introduced in [Janssen 2019]: for each x, w(x,y) decreases as y moves away from x. For a large and non-parametric family of graphons, we show that (i) the popular spectral seriation algorithm [Atkins 1998] provides a consistent estimator σ̂ of σ, and (ii) a small amount of post-processing results in an estimate σ̃ that converges to σ at a nearly-optimal rate, both as N →∞.