Open Access
2022 Reconstruction of line-embeddings of graphons
Jeannette Janssen, Aaron Smith
Author Affiliations +
Electron. J. Statist. 16(1): 331-407 (2022). DOI: 10.1214/21-EJS1940


Consider a random graph process with n vertices corresponding to points vii.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 σSn 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 max1in|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].

Funding Statement

The authors acknowledge support from the National Science and Engineering Research Council through their Discovery Grant program.


Download Citation

Jeannette Janssen. Aaron Smith. "Reconstruction of line-embeddings of graphons." Electron. J. Statist. 16 (1) 331 - 407, 2022.


Received: 1 September 2020; Published: 2022
First available in Project Euclid: 10 January 2022

MathSciNet: MR4361745
zbMATH: 1490.60024
Digital Object Identifier: 10.1214/21-EJS1940

Primary: 60-04
Secondary: 60B20

Keywords: graphon , probabilistic algorithm , random graph , R-matrix

Vol.16 • No. 1 • 2022
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