Ordered Graph Limits and Their Applications

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



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2021.42.pdf
  • Filesize: 0.58 MB
  • 20 pages

Document Identifiers

Author Details

Omri Ben-Eliezer
  • Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, USA
Eldar Fischer
  • Faculty of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel
Amit Levi
  • Cheriton School of Computer Science, University of Waterloo, Canada
Yuichi Yoshida
  • Principles of Informatics Research Division, National Institute of Informatics (NII), Tokyo, Japan

Cite AsGet BibTex

Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Yuichi Yoshida. Ordered Graph Limits and Their Applications. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.42

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 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].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Functional analysis
  • Mathematics of computing → Nonparametric representations
  • Mathematics of computing → Extremal graph theory
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • graph limits
  • ordered graph
  • graphon
  • cut distance
  • removal lemma

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Emmanuel Abbe. Community detection and stochastic block models: Recent developments. Journal of Machine Learning Research, 18(177):1-86, 2018. Google Scholar
  2. Noga Alon, Omri Ben-Eliezer, and Eldar Fischer. Testing hereditary properties of ordered graphs and matrices. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 848-858, 2017. Google Scholar
  3. Noga Alon, Eldar Fischer, Michael Krivelevich, and Mario Szegedy. Efficient testing of large graphs. Combinatorica, 20(4):451-476, 2000. Google Scholar
  4. Noga Alon, Eldar Fischer, Ilan Newman, and Asaf Shapira. A combinatorial characterization of the testable graph properties: It’s all about regularity. SIAM Journal on Computing, 39(1):143-167, 2009. Google Scholar
  5. Noga Alon and Asaf Shapira. A characterization of the (natural) graph properties testable with one-sided error. SIAM Journal on Computing, 37(6):1703-1727, 2008. Google Scholar
  6. Noga Alon and Uri Stav. What is the furthest graph from a hereditary property? Random Structures & Algorithms, 33(1):87-104, 2008. Google Scholar
  7. Maria Axenovich and Ryan R. Martin. Multicolor and directed edit distance. Journal of Combinatorics, 2(4), 2011. Google Scholar
  8. Omri Ben-Eliezer and Eldar Fischer. Earthmover resilience and testing in ordered structures. In Proceedings of the 33rd Conference on Computational Complexity (CCC), pages 18:1-18:35, 2018. Google Scholar
  9. Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Yuichi Yoshida. Limits of ordered graphs and their applications. Full version of this work. URL: http://arxiv.org/abs/1811.02023.
  10. Christian Borgs, Jennifer Chayes, and David Gamarnik. Convergent sequences of sparse graphs: A large deviations approach. Random Structures & Algorithms, 51(1):52-89, 2017. Google Scholar
  11. Christian Borgs, Jennifer Chayes, and László Lovász. Moments of two-variable functions and the uniqueness of graph limits. Geometric and Functional Analysis, 19(6):1597-1619, 2010. Google Scholar
  12. Christian Borgs, Jennifer Chayes, László Lovász, Vera T Sós, Balázs Szegedy, and Katalin Vesztergombi. Graph limits and parameter testing. In Proceedings of the 38th ACM Symposium on the Theory of Computing (STOC), pages 261-270, 2006. Google Scholar
  13. Christian Borgs, Jennifer Chayes, László Lovász, Vera T. Sós, and Katalin Vesztergombi. Counting graph homomorphisms. In Topics in Discrete Mathematics, pages 315-371. Springer Berlin Heidelberg, 2006. Google Scholar
  14. Christian Borgs, Jennifer Chayes, László Lovász, Vera T. Sós, and Katalin Vesztergombi. Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Advances in Mathematics, 219(6):1801-1851, 2008. Google Scholar
  15. Christian Borgs, Jennifer Chayes, László Lovász, Vera T. Sós, and Katalin Vesztergombi. Convergent sequences of dense graphs II. multiway cuts and statistical physics. Annals of Mathematics, 176(1):151-219, 2012. Google Scholar
  16. Christian Borgs and Jennifer T. Chayes. Graphons: A nonparametric method to model, estimate, and design algorithms for massive networks. CoRR, abs/1706.01143, 2017. URL: http://arxiv.org/abs/1706.01143.
  17. Persi Diaconis and Svante Janson. Graph limits and exchangeable random graphs. Rendiconti di Matematica e delle sue Applicazioni. Serie VII, 28(1):33-61, 2008. Google Scholar
  18. Gábor Elek and Balázs Szegedy. A measure-theoretic approach to the theory of dense hypergraphs. Advances in Mathematics, 231(3):1731-1772, 2012. Google Scholar
  19. Eldar Fischer and Ilan Newman. Testing versus estimation of graph properties. SIAM Journal on Computing, 37(2):482-501, 2007. Google Scholar
  20. John M. Franks. A (terse) introduction to Lebesgue integration, volume 48 of Student Mathematical Library. American Mathematical Society, 2009. Google Scholar
  21. Michael Freedman, László Lovász, and Alexander Schrijver. Reflection positivity, rank connectivity, and homomorphism of graphs. Journal of the American Mathematical Society, 20:37-51, 2007. Google Scholar
  22. Alan Frieze and Ravi Kannan. The regularity lemma and approximation schemes for dense problems. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 12-20, 1996. Google Scholar
  23. Alan Frieze and Ravi Kannan. Quick approximation to matrices and applications. Combinatorica, 19(2):175-220, 1999. Google Scholar
  24. Frederik Garbe, Robert Hancock, Jan Hladký, and Maryam Sharifzadeh. Limits of latin squares. arXiv preprint, 2020. Extended abstract appeared in Eurocomb 2019 under the name Theory of limits of sequences of Latin squares. URL: http://arxiv.org/abs/2010.07854.
  25. Carlos Hoppen, Yoshiharu Kohayakawa, Carlos Gustavo Moreira, Balázs Ráth, and Rudini Menezes Sampaio. Limits of permutation sequences. Journal of Combinatorial Theory, Series B, 103(1):93-113, 2013. Google Scholar
  26. Svante Janson. Poset limits and exchangeable random posets. Combinatorica, 31(5):529-563, 2011. Google Scholar
  27. László Lovász. Large networks and graph limits, volume 60. American Mathematical Society, 2012. Google Scholar
  28. László Lovász and Balázs Szegedy. Limits of dense graph sequences. Journal of Combinatorial Theory, Series B, 96(6):933-957, 2006. Google Scholar
  29. László Lovász and Balázs Szegedy. Szemerédi’s lemma for the analyst. Geometric And Functional Analysis, 17(1):252-270, 2007. Google Scholar
  30. László Lovász and Balázs Szegedy. Testing properties of graphs and functions. Israel Journal of Mathematics, 178(1):113-156, 2010. Google Scholar
  31. Ryan R. Martin. The edit distance in graphs: Methods, results, and generalizations. In Andrew Beveridge, Jerrold R. Griggs, Leslie Hogben, Gregg Musiker, and Prasad Tetali, editors, Recent Trends in Combinatorics, pages 31-62. Springer International Publishing, 2016. Google Scholar
  32. Ryan R. Martin and Maria Axenovich. Avoiding patterns in matrices via a small number of changes. SIAM Journal of Discrete Mathematics, 20(1):49-54, 2006. Google Scholar
  33. Peter Orbanz and Daniel M. Roy. Bayesian models of graphs, arrays and other exchangeable random structures. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(2):437-461, 2015. Google Scholar
  34. Endre Szemerédi. Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes. Colloq. Internat. CNRS, volume 260, pages 399-401, 1976. Google Scholar
  35. Yuichi Yoshida. Gowers norm, function limits, and parameter estimation. In Proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1391-1406, 2016. Google Scholar
  36. Yufei Zhao. Hypergraph limits: A regularity approach. Random Structures & Algorithms, 47(2):205-226, 2015. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail