Abstract
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in analytic language. We extend the theory of graph limits to the ordered setting, presenting a limit object for dense vertexordered graphs, which we call an orderon. As a special case, this yields limit objects for matrices whose rows and columns are ordered, and for dynamic graphs that expand (via vertex insertions) over time. Along the way, we devise an ordered localitypreserving variant of the cut distance between ordered graphs, showing that two graphs are close with respect to this distance if and only if they are similar in terms of their ordered subgraph frequencies. We show that the space of orderons is compact with respect to this distance notion, which is key to a successful analysis of combinatorial objects through their limits. For the proof we combine techniques used in the unordered setting with several new techniques specifically designed to overcome the challenges arising in the ordered setting.
We derive several applications of the ordered limit theory in extremal combinatorics, sampling, and property testing in ordered graphs. In particular, we prove a new ordered analogue of the wellknown result by Alon and Stav [RS&A'08] on the furthest graph from a hereditary property; this is the first known result of this type in the ordered setting. Unlike the unordered regime, here the Erdős–Rényi random graph 𝐆(n, p) with an ordering over the vertices is not always asymptotically the furthest from the property for some p. However, using our ordered limit theory, we show that random graphs generated by a stochastic block model, where the blocks are consecutive in the vertex ordering, are (approximately) the furthest. Additionally, we describe an alternative analytic proof of the ordered graph removal lemma [Alon et al., FOCS'17].
BibTeX  Entry
@InProceedings{beneliezer_et_al:LIPIcs.ITCS.2021.42,
author = {Omri BenEliezer and Eldar Fischer and Amit Levi and Yuichi Yoshida},
title = {{Ordered Graph Limits and Their Applications}},
booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages = {42:142:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771771},
ISSN = {18688969},
year = {2021},
volume = {185},
editor = {James R. Lee},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13581},
URN = {urn:nbn:de:0030drops135815},
doi = {10.4230/LIPIcs.ITCS.2021.42},
annote = {Keywords: graph limits, ordered graph, graphon, cut distance, removal lemma}
}
Keywords: 

graph limits, ordered graph, graphon, cut distance, removal lemma 
Collection: 

12th Innovations in Theoretical Computer Science Conference (ITCS 2021) 
Issue Date: 

2021 
Date of publication: 

04.02.2021 