The Annals of Probability

An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence

Christian Borgs, Jennifer T. Chayes, Henry Cohn, and Yufei Zhao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We extend the $L^{p}$ theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of $W$-random graphs.

Article information

Source
Ann. Probab., Volume 46, Number 1 (2018), 337-396.

Dates
Received: February 2015
Revised: March 2017
First available in Project Euclid: 5 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1517821225

Digital Object Identifier
doi:10.1214/17-AOP1187

Mathematical Reviews number (MathSciNet)
MR3758733

Zentralblatt MATH identifier
06865125

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 82B99: None of the above, but in this section

Keywords
Sparse graph limit graphon graph convergence graph quotient

Citation

Borgs, Christian; Chayes, Jennifer T.; Cohn, Henry; Zhao, Yufei. An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence. Ann. Probab. 46 (2018), no. 1, 337--396. doi:10.1214/17-AOP1187. https://projecteuclid.org/euclid.aop/1517821225


Export citation

References

  • [1] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 23.
  • [2] Bollobás, B. and Riordan, O. (2009). Metrics for sparse graphs. In Surveys in Combinatorics 2009. London Mathematical Society Lecture Note Series 365 211–287. Cambridge Univ. Press, Cambridge.
  • [3] Borgs, C., Chayes, J. T., Cohn, H. and Zhao, Y. (2014). An $L^{p}$ theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions. Available at arXiv:1401.2906.
  • [4] Borgs, C., Chayes, J. and Gamarnik, D. (2017). Convergent sequences of sparse graphs: A large deviations approach. Random Structures Algorithms 51 52–89.
  • [5] Borgs, C., Chayes, J., Kahn, J. and Lovász, L. (2013). Left and right convergence of graphs with bounded degree. Random Structures Algorithms 42 1–28.
  • [6] Borgs, C., Chayes, J. and Lovász, L. (2010). Moments of two-variable functions and the uniqueness of graph limits. Geom. Funct. Anal. 19 1597–1619.
  • [7] Borgs, C., Chayes, J., Lovász, L., Sós, V. T. and Vesztergombi, K. (2006). Counting graph homomorphisms. In Topics in Discrete Mathematics. Algorithms Combin. 26 315–371. Springer, Berlin.
  • [8] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008). Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.
  • [9] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2012). Convergent sequences of dense graphs II. Multiway cuts and statistical physics. Ann. of Math. (2) 176 151–219.
  • [10] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin. Corrected reprint of the second (1998) edition.
  • [11] Frieze, A. and Kannan, R. (1999). Quick approximation to matrices and applications. Combinatorica 19 175–220.
  • [12] Lovász, L. and Szegedy, B. (2007). Szemerédi’s lemma for the analyst. Geom. Funct. Anal. 17 252–270.
  • [13] Price, G. B. (1940). On the completeness of a certain metric space with an application to Blaschke’s selection theorem. Bull. Amer. Math. Soc. 46 278–280.