Ordered graph limits and their applications

Omri Ben-Eliezer, Eldar Fischer, Amit Levi, Yuichi Yoshida

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 vertex-ordered 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 locality-preserving 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 well-known 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].

Original languageEnglish
Title of host publication12th Innovations in Theoretical Computer Science Conference, ITCS 2021
EditorsJames R. Lee
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771771
StatePublished - 1 Feb 2021
Externally publishedYes
Event12th Innovations in Theoretical Computer Science Conference, ITCS 2021 - Virtual, Online
Duration: 6 Jan 20218 Jan 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference12th Innovations in Theoretical Computer Science Conference, ITCS 2021
CityVirtual, Online

Bibliographical note

Publisher Copyright:
© Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Yuichi Yoshida.


  • Cut distance
  • Graph limits
  • Graphon
  • Ordered graph
  • Removal lemma

ASJC Scopus subject areas

  • Software


Dive into the research topics of 'Ordered graph limits and their applications'. Together they form a unique fingerprint.

Cite this