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

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Summary

Introduction

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.

High-level algorithm descriptions
Basic notation
A simple assumption
Main results
Running time
Main contributions and related work
Consequences for efficiency of downstream tasks
Seriation
Testing graphons
Estimating graphons
Paper guide
Some further notation
Algorithms for Theorem 1
Algorithms for Theorem 2
Proof of strengthened version of Theorem 1
A stronger result
Construction of probability spaces
Unit interval graphs and the LexBFS Algorithm
Properties of latent variables
Properties of subsampled graphs
Proof sketch
Analysis of Algorithm 3
Analysis of Algorithm 4
Analysis of Algorithm 2
Analysis of Algorithms 5 and 1 and proof of Theorem 3
Iterative error reduction
Other Initial Sketches
Algorithm 7: basic concepts
Small Noise
Algorithm 7: correctness
Proof of Theorem 2
Finding α
Large error of embeddings
Bad events for latent variables
Bad events for subsampled graphs

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