Abstract
Consider a random graph process with n vertices corresponding to points vi∼i.i.d.Unif[0,1] embedded randomly in the interval, and where edges are inserted between vi,vj independently with probability given by the graphon w(vi,vj)∈[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 σ∈Sn an ordering of these vertices if vσ(i)<vσ(j) for all i<j, and ask: how can we accurately estimate σ from an observed graph? We present a randomized algorithm with output σˆ that, for a large class of graphons, achieves error max1≤i≤n|σ(i)−σˆ(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˜(nϵ) 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 [20, 21] and testing whether a graphon is diagonally increasing [11].
Highlights
In this paper, we propose and analyze new algorithms for estimating latent vertex labellings given an observed random graph
We present a of graphons, randomized algorithm with output σachieves error max1≤i≤n |σ(i) − σ(i)|
The basic seriation problem considers a Robinsonian similarity matrix, i.e. a symmetric n by n matrix A = [ai,j]1≤i,j≤n with the property: there exists a permutation σ ∈ Sn so that every row of the permuted matrix Aσ = [aσ(i),σ(j)]1≤i,j≤n is unimodal with the maximum occurring at the diagonal
Summary
We propose and analyze new algorithms for estimating latent vertex labellings given an observed random graph. An approximate ordering with an error rate similar to that presented in this paper has been obtained Generally such results only apply to a narrow class of random graphs, and the restriction of the problem to this specific class allows for the use of highly specialized methods. Our paper introduces and analyzes two new algorithms for the noisy graph seriation problem, which approximate the line-embedded permutation for graphs sampled from a large general class of diagonally increasing graphons. Many latent-position models are analyzed by algorithms that first estimate all latent positions and plug this estimator into a formula for a quantity of interest (in seriation this quantity of interest is the ordering, but see [26] for applications of the same approach to other problems) This approach seems sensible under the condition that the first step is not much less accurate than the second step.
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