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On the Method of Typical Bounded Differences

Part of: Combinatorial probability

Published online by Cambridge University Press:  27 August 2015

LUTZ WARNKE
LUTZ WARNKE*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: L.Warnke@dpmms.cam.ac.uk)
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Abstract

Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that functions of random variables are typically near their means. Of particular importance is the case where f(X) is a function of independent random variables X = (X1, . . ., Xn). Here the well-known bounded differences inequality (also called McDiarmid's inequality or the Hoeffding–Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |f(X) − f(X′)| ⩽ ck whenever X, X′ differ only in Xk. While this is easy to check, the main disadvantage is that it considers worst-case changes ck, which often makes the resulting bounds too weak to be useful.

In this paper we prove a variant of the bounded differences inequality which can be used to establish concentration of functions f(X) where (i) the typical changes are small, although (ii) the worst case changes might be very large. One key aspect of this inequality is that it relies on a simple condition that (a) is easy to check and (b) coincides with heuristic considerations as to why concentration should hold. Indeed, given an event Γ that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where Γ occurs. The point is that the resulting typical changes ck are often much smaller than the worst case ones.

To illustrate its application we consider the reverse H-free process, where H is 2-balanced. We prove that the final number of edges in this process is concentrated, and also determine its likely value up to constant factors. This answers a question of Bollobás and Erdős.

MSC classification

Primary: 60C05: Combinatorial probability
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 2 , March 2016 , pp. 269 - 299
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Achlioptas, D. (2000) Setting 2 variables at a time yields a new lower bound for random 3-SAT. In Proc. 32nd Annual ACM Symposium on Theory of Computing: STOC'00, ACM, pp. 28–37.Google Scholar
[2] Alon, N., Kim, J. H. and Spencer, J. (1997) Nearly perfect matchings in regular simple hypergraphs. Israel J. Math. 100 171–187.CrossRefGoogle Scholar
[3] Azuma, K. (1967) Weighted sums of certain dependent random variables. Tôhoku Math. J. (2) 19 357–367.Google Scholar
[4] Bapst, V., Coja-Oghlan, A., Hetterich, S., Rassmann, F. and Vilenchik, D. The condensation phase transition in random graph coloring. Comm. Math. Phys., to appear (Proceedings version appeared in RANDOM 2014). arXiv:1404.5513.Google Scholar
[5] Bohman, T. (2009) The triangle-free process. Adv. Math. 221 1653–1677.Google Scholar
[6] Bohman, T., Frieze, A. and Lubetzky, E. (2015) Random triangle removal. Adv. Math. 280 379–438.CrossRefGoogle Scholar
[7] Bohman, T. and Keevash, P. (2010) The early evolution of the H-free process. Invent. Math. 181 291–336.CrossRefGoogle Scholar
[8] Bollobás, B. (1988) The chromatic number of random graphs. Combinatorica 8 49–55.Google Scholar
[9] Bollobás, B. (2012) Personal communication.Google Scholar
[10] Bollobás, B. and Riordan, O. (2015) An old approach to the giant component problem. J. Combin. Theory Ser. B 113 236–260.Google Scholar
[11] Bollobás, B. and Brightwell, G. (1992) The height of a random partial order: Concentration of measure. Ann. Appl. Probab. 2 1009–1018.Google Scholar
[12] Bollobás, B. and Riordan, O. (2009) Random graphs and branching processes. In Handbook of Large-Scale Random Networks, Vol. 18 of Bolyai Society Mathematical Studies, pp. 15–115.Google Scholar
[13] Borgs, C., Chayes, J. T., Mertens, S. and Pittel, B. (2004) Phase diagram for the constrained integer partitioning problem. Random Struct. Alg. 24 315–380.Google Scholar
[14] Bushaw, N., Collares Neto, M., Morris, R. and Smith, P. (2015) The sharp threshold for maximum-size sum-free subsets in even-order abelian groups. Combin. Probab. Comput. 24 609–640.Google Scholar
[15] Chalker, T. K., Godbole, A. P., Hitczenko, P., Radcliff, J. and Ruehr, O. G. (1999) On the size of a random sphere of influence graph. Adv. Appl. Probab. 31 596–609.CrossRefGoogle Scholar
[16] Chung, F. and Lu, L. (2006) Concentration inequalities and martingale inequalities: A survey. Internet Math. 3 79–127.Google Scholar
[17] Dembo, A. and Zeitouni, O. (1998) Large Deviations Techniques and Applications, Springer.CrossRefGoogle Scholar
[18] Dubhashi, D. P. and Panconesi, A. (2009) Concentration of Measure for the Analysis of Randomized Algorithms, Cambridge University Press.Google Scholar
[19] Erdős, P., Suen, S. and Winkler, P. (1995) On the size of a random maximal graph. Random Struct. Alg. 6 309–318.Google Scholar
[20] Freedman, D. A. (1975) On tail probabilities for martingales. Ann. Probab. 3 100–118.CrossRefGoogle Scholar
[21] de Graaf, M. and Manthey, B. (2014) Probabilistic analysis of power assignments. In Mathematical Foundations of Computer Science 2014, Vol. 8635 of Lecture Notes in Computer Science, Springer, pp. 201–212.Google Scholar
[22] Grable, D. A. (1998) A large deviation inequality for functions of independent, multi-way choices. Combin. Probab. Comput. 7 57–63.Google Scholar
[23] Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13–20.Google Scholar
[24] Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.Google Scholar
[25] Janson, S. (1990) Poisson approximation for large deviations. Random Struct. Alg. 1 221–229.CrossRefGoogle Scholar
[26] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience.Google Scholar
[27] Janson, S. and Ruciński, A. (2004) The deletion method for upper tail estimates. Combinatorica 24 615–640.Google Scholar
[28] Janson, S. and Warnke, L. The lower tail: Poisson approximation revisited. Random Struct. Alg., to appear. arXiv:1406.1248.Google Scholar
[29] Kim, J. H. (1995) On Brooks' theorem for sparse graphs. Combin. Probab. Comput. 4 97–132.CrossRefGoogle Scholar
[30] Kim, J. H. and Vu, V. H. (2000) Concentration of multivariate polynomials and its applications. Combinatorica 20 417–434.Google Scholar
[31] Kim, J. H. and Vu, V. H. (2004) Divide and conquer martingales and the number of triangles in a random graph. Random Struct. Alg. 24 166–174.Google Scholar
[32] Krivelevich, M., Lubetzky, E. and Sudakov, B. (2013) Longest cycles in sparse random digraphs. Random Struct. Alg. 43 1–15.Google Scholar
[33] Makai, T. (2015) The reverse H-free process for strictly 2-balanced graphs. J. Graph Theory 79 125–144.Google Scholar
[34] McDiarmid, C. (1989) On the method of bounded differences. In Surveys in Combinatorics, Vol. 141 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 48–188.Google Scholar
[35] McDiarmid, C. (1998) Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics, Vol. 16 of Algorithms and Combinatorics, Springer, pp. 195–248.Google Scholar
[36] McKay, B. and Skerman, F. (2013) Degree sequences of random digraphs and bipartite graphs. arXiv:1302.2446.Google Scholar
[37] Milman, V. D. and Schechtman, G. (1986) Asymptotic Theory of Finite-Dimensional Normed Spaces, Vol. 1200 of Lecture Notes in Mathematics, Springer.Google Scholar
[38] Osthus, D. and Taraz, A. (2001) Random maximal H-free graphs. Random Struct. Alg. 18 61–82.Google Scholar
[39] Riordan, O. and Warnke, L. (2015) The Janson inequalities for general up-sets. Random Struct. Alg. 46 391–395.CrossRefGoogle Scholar
[40] Riordan, O. and Warnke, L. (2015) The evolution of subcritical Achlioptas processes. Random Struct. Alg. 47 174–203.Google Scholar
[41] Rödl, V. and Thoma, L. (1996) Asymptotic packing and the random greedy algorithm. Random Struct. Alg. 8 161–177.Google Scholar
[42] Schudy, W. and Sviridenko, M. (2011) Bernstein-like concentration and moment inequalities for polynomials of independent random variables: Multilinear case. arXiv:1109.5193.Google Scholar
[43] Schudy, W. and Sviridenko, M. (2012) Concentration and moment inequalities for polynomials of independent random variables. In Proc. 23rd Annual ACM–SIAM Symposium on Discrete Algorithms: SODA'12, SIAM, pp. 437–446.CrossRefGoogle Scholar
[44] Shamir, E. and Spencer, J. (1987) Sharp concentration of the chromatic number on random graphs Gn,p . Combinatorica 7 121–129.Google Scholar
[45] Spencer, J. (1990) Counting extensions. J. Combin. Theory Ser. A 55 247–255.Google Scholar
[46] Spencer, J. (1995) Asymptotic packing via a branching process. Random Struct. Alg. 7 167–172.CrossRefGoogle Scholar
[47] Steiger, W. L. (1969) A best possible Kolmogoroff-type inequality for martingales and a characteristic property. Ann. Math. Statist. 40 764–769.CrossRefGoogle Scholar
[48] Talagrand, M. (1996) A new look at independence. Ann. Probab. 24 1–34.Google Scholar
[49] Vu, V. H. (2000) On the concentration of multivariate polynomials with small expectation. Random Struct. Alg. 16 344–363.Google Scholar
[50] Vu, V. H. (2002) Concentration of non-Lipschitz functions and applications. Random Struct. Alg. 20 262–316.CrossRefGoogle Scholar
[51] Warnke, L. (2014) The Cℓ -free process. Random Struct. Alg. 44 490–526.CrossRefGoogle Scholar
[52] Warnke, L. (2014) When does the K 4-free process stop? Random Struct. Alg. 44 355–397.CrossRefGoogle Scholar
[53] Wormald, N. C. (1999) Models of random regular graphs. In Surveys in Combinatorics, Vol. 267 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 239–298.Google Scholar