An 𝐿^{𝑝} theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions

Borgs, Christian; Chayes, Jennifer; Cohn, Henry; Zhao, Yufei

doi:10.1090/tran/7543

An $L^p$ theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions
HTML articles powered by AMS MathViewer

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

Trans. Amer. Math. Soc. 372 (2019), 3019-3062 Request permission

Abstract:

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper.

References

David Aldous and Russell Lyons, Processes on unimodular random networks, Electron. J. Probab. 12 (2007), no. 54, 1454–1508. MR 2354165, DOI 10.1214/EJP.v12-463
David Aldous and J. Michael Steele, The objective method: probabilistic combinatorial optimization and local weak convergence, Probability on discrete structures, Encyclopaedia Math. Sci., vol. 110, Springer, Berlin, 2004, pp. 1–72. MR 2023650, DOI 10.1007/978-3-662-09444-0_{1}
Noga Alon and Joel H. Spencer, The probabilistic method, 3rd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2008. With an appendix on the life and work of Paul Erdős. MR 2437651, DOI 10.1002/9780470277331
Itai Benjamini and Oded Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001), no. 23, 13. MR 1873300, DOI 10.1214/EJP.v6-96
P. J. Bickel and A. Chen, A nonparametric view of network models and Newman–Girvan and other modularities, Proc. Natl. Acad. Sci. USA 106 (2009), 21068–21073. 10.1073/pnas.0907096106.
Béla Bollobás and Oliver Riordan, Metrics for sparse graphs, Surveys in combinatorics 2009, London Math. Soc. Lecture Note Ser., vol. 365, Cambridge Univ. Press, Cambridge, 2009, pp. 211–287. MR 2588543
Christian Borgs, Jennifer T. Chayes, Henry Cohn, and Yufei Zhao, An $L^p$ theory of sparse graph convergence II: LD convergence, quotients and right convergence, Ann. Probab. 46 (2018), no. 1, 337–396. MR 3758733, DOI 10.1214/17-AOP1187
C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, and K. Vesztergombi, Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing, Adv. Math. 219 (2008), no. 6, 1801–1851. MR 2455626, DOI 10.1016/j.aim.2008.07.008
C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, and K. Vesztergombi, Convergent sequences of dense graphs II. Multiway cuts and statistical physics, Ann. of Math. (2) 176 (2012), no. 1, 151–219. MR 2925382, DOI 10.4007/annals.2012.176.1.2
Sourav Chatterjee and S. R. S. Varadhan, The large deviation principle for the Erdős-Rényi random graph, European J. Combin. 32 (2011), no. 7, 1000–1017. MR 2825532, DOI 10.1016/j.ejc.2011.03.014
Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman, Power-law distributions in empirical data, SIAM Rev. 51 (2009), no. 4, 661–703. MR 2563829, DOI 10.1137/070710111
Amin Coja-Oghlan, Colin Cooper, and Alan Frieze, An efficient sparse regularity concept, SIAM J. Discrete Math. 23 (2009/10), no. 4, 2000–2034. MR 2594968, DOI 10.1137/080730160
David Conlon and Jacob Fox, Bounds for graph regularity and removal lemmas, Geom. Funct. Anal. 22 (2012), no. 5, 1191–1256. MR 2989432, DOI 10.1007/s00039-012-0171-x
David Conlon, Jacob Fox, and Yufei Zhao, Extremal results in sparse pseudorandom graphs, Adv. Math. 256 (2014), 206–290. MR 3177293, DOI 10.1016/j.aim.2013.12.004
David Conlon, Jacob Fox, and Yufei Zhao, A relative Szemerédi theorem, Geom. Funct. Anal. 25 (2015), no. 3, 733–762. MR 3361771, DOI 10.1007/s00039-015-0324-9
Rick Durrett, Probability: theory and examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, Cambridge University Press, Cambridge, 2010. MR 2722836, DOI 10.1017/CBO9780511779398
Helmut Finner, A generalization of Hölder’s inequality and some probability inequalities, Ann. Probab. 20 (1992), no. 4, 1893–1901. MR 1188047
Péter E. Frenkel, Convergence of graphs with intermediate density, Trans. Amer. Math. Soc. 370 (2018), no. 5, 3363–3404. MR 3766852, DOI 10.1090/tran/7036
Alan Frieze and Ravi Kannan, Quick approximation to matrices and applications, Combinatorica 19 (1999), no. 2, 175–220. MR 1723039, DOI 10.1007/s004930050052
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481–547. MR 2415379, DOI 10.4007/annals.2008.167.481
W. Hoeffding, The strong law of large numbers for $U$-statistics, North Carolina State University, Institute of Statistics Mimeograph Series No. 302, 1961. http://repository.lib.ncsu.edu/dr/handle/1840.4/2128
Svante Janson, Graphons, cut norm and distance, couplings and rearrangements, New York Journal of Mathematics. NYJM Monographs, vol. 4, State University of New York, University at Albany, Albany, NY, 2013. MR 3043217
Y. Kohayakawa, Szemerédi’s regularity lemma for sparse graphs, Foundations of computational mathematics (Rio de Janeiro, 1997) Springer, Berlin, 1997, pp. 216–230. MR 1661982
Y. Kohayakawa and V. Rödl, Szemerédi’s regularity lemma and quasi-randomness, Recent advances in algorithms and combinatorics, CMS Books Math./Ouvrages Math. SMC, vol. 11, Springer, New York, 2003, pp. 289–351. MR 1952989, DOI 10.1007/0-387-22444-0_{9}
D. Kunszenti-Kovács, L. Lovász, and B. Szegedy, Multigraph limits, unbounded kernels, and Banach space decorated graphs, preprint, 2014. arXiv:1406.7846
László Lovász, Large networks and graph limits, American Mathematical Society Colloquium Publications, vol. 60, American Mathematical Society, Providence, RI, 2012. MR 3012035, DOI 10.1090/coll/060
László Lovász and Balázs Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B 96 (2006), no. 6, 933–957. MR 2274085, DOI 10.1016/j.jctb.2006.05.002
László Lovász and Balázs Szegedy, Szemerédi’s lemma for the analyst, Geom. Funct. Anal. 17 (2007), no. 1, 252–270. MR 2306658, DOI 10.1007/s00039-007-0599-6
Eyal Lubetzky and Yufei Zhao, On replica symmetry of large deviations in random graphs, Random Structures Algorithms 47 (2015), no. 1, 109–146. MR 3366814, DOI 10.1002/rsa.20536
Russell Lyons, Asymptotic enumeration of spanning trees, Combin. Probab. Comput. 14 (2005), no. 4, 491–522. MR 2160416, DOI 10.1017/S096354830500684X
Michael Mitzenmacher, A brief history of generative models for power law and lognormal distributions, Internet Math. 1 (2004), no. 2, 226–251. MR 2077227
Jaroslav Ne et il and Patrice Ossona de Mendez, Sparsity, Algorithms and Combinatorics, vol. 28, Springer, Heidelberg, 2012. Graphs, structures, and algorithms. MR 2920058, DOI 10.1007/978-3-642-27875-4
Jaroslav Ne et il and Patrice Ossona de Mendez, A model theory approach to structural limits, Comment. Math. Univ. Carolin. 53 (2012), no. 4, 581–603. MR 3016428
J. Nešetřil and P. Ossona de Mendez, A unified approach to structural limits and limits of graphs with bounded tree-depth, preprint, 2013. arXiv:1303.6471
Jaroslav Ne et il and Patrice Ossona de Mendez, Modeling limits in hereditary classes: reduction and application to trees, Electron. J. Combin. 23 (2016), no. 2, Paper 2.52, 33. MR 3522136
Alexander Scott, Szemerédi’s regularity lemma for matrices and sparse graphs, Combin. Probab. Comput. 20 (2011), no. 3, 455–466. MR 2784637, DOI 10.1017/S0963548310000490
Endre Szemerédi, Regular partitions of graphs, Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976) Colloq. Internat. CNRS, vol. 260, CNRS, Paris, 1978, pp. 399–401 (English, with French summary). MR 540024
P. J. Wolfe and S. C. Olhede, Nonparametric graphon estimation, preprint, 2013. arXiv:1309.5936