Reconstruction of Line-Embeddings of Graphons

Abstract

Consider a random graph process with n vertices corresponding to points v i i.i.d. ∼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 [CGH + 15], 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 σ that, for a large class of graphons, achieves error max 1≤i≤n |σ(i) − σ(i)| = O * ( √ n) with high probability; we also show that this is the bestpossible 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 ǫ ) for any ǫ > 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 [GLZ15, GJ20] and testing whether a graphon is diagonally increasing [CGH + 15].

Localization in 1D non-parametric latent space models from pairwise affinities

“…In a recent work, Janssen and Smith [Janssen and Smith, 2020] consider the related seriation problem for R-matrices (Example 3 in the introduction), in a geometric setting where F σ * i ,σ * j = g(|i − j|/n) for some unknown permutation σ * , and some unknown function g. Hence, in addition to be a pre-R matrix, F is also a Toeplitz matrix. Under additional assumptions on the squared matrix (F σ * i ,σ * j ) 2 , they establish that an algorithm based on a (thresholded version) of the square matrix of observations A 2 , achieves, with high probability, an error bound…”

“…As a byproduct of our analysis, we provide some more explicit recovery bounds in our specific setting with noisy observations. Closer to our contribution, Jannssen and Smith [Janssen and Smith, 2020] observe a noisy version of a pre-R matrix and, under some assumptions on the affinity function f , learn a permutation that satisfies max i∈…”