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Exponential Inequalities for the Distribution Tails of the Number of Cycles in the Erdös-Rényi Random Graphs

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Abstract

An exponential upper bound is obtained for the tail probability of the centered and normalized number of cycles in an Erdös-Rényi random graph where every edge occurs with the same probability, independently of the others.

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REFERENCES

  1. S.G. Bobkov, M.A. Danshina, and V.V. Ulyanov, “Rate of convergence to the Poisson law of the number of cycles in the generalized random graphs. I,” Operator Theory & Harmonic Analysis, OTHA 2020. Springer Proc. in Math. and Statist. 358, 109 (2021).

  2. I.S. Borisov and A.A. Bystrov, “Exponential inequalities for the distributions of canonical multiple partial sum processes,” Theory Probab. Appl. 64, 171 (2019).

    Article  MathSciNet  Google Scholar 

  3. S. Chatterjee, “The missing log in large deviations for triangle counts,” Random Struct. & Algorithms 40, 437 (2011).

    Article  MathSciNet  Google Scholar 

  4. S. Chatterjee and S.R.S. Varadhan, “The large deviation principle for the Erdös-Rényi random graphs,” European J. of Combinatorics 32, 1000 (2011).

    Article  MathSciNet  Google Scholar 

  5. B. DeMarco and J. Kahn, “Upper tails for triangles,” Random Struct. & Algorithms 40, 452 (2012).

    Article  MathSciNet  Google Scholar 

  6. P. Erdös and A. Rényi, “On the evolution of random graphs,” Publ. Math. Inst. Hungar. Acad. Sci. 5, 209 (1960).

    MathSciNet  MATH  Google Scholar 

  7. W. Höffding, “Probability inequalities for sums of bounded random variables,” J. Amer. Statist. Assoc. 58, 13–30 (1963).

    Article  MathSciNet  Google Scholar 

  8. S. Janson, K. Oleszkiewicz, and A. Ruciński, “Upper tails for subgraph counts in random graphs,” Israel J. Math. 142, 61 (2004).

    Article  MathSciNet  Google Scholar 

  9. J.H. Kim and V.H. Vu, “Divide and conquer martingales and the number of triangles in a random graph,” Random Struct & Algorithms 24, 166 (2004).

    Article  MathSciNet  Google Scholar 

  10. Q. Liu and Z. Dong, “Limit laws for the number of triangles in the generalized random graphs with random node weights,” Stat. & Probab. Letters 161, 1 (2020).

    Article  MathSciNet  Google Scholar 

  11. A.V. Logachov and A.A. Mogulskiĭ, “Exponential Chebyshev inequalities for random graphons and their applications,” Siberian Math. J. 61, 880 (2020).

    Article  MathSciNet  Google Scholar 

  12. A. Ruciński, “When are small subgraphs of a random graph normally distributed?,” Prob. Th. Rel. Fields 78, 1 (1988).

    Article  MathSciNet  Google Scholar 

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ACKNOWLEDGMENTS

The authors thank Professor I.S. Borisov for posing the problem and for valuable advice.

Funding

The work was supported by the Russian Science Foundation (grant 22-21-00414).

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Authors and Affiliations

  1. Novosibirsk State University, Novosibirsk, 630090, Russia

    A. A. Bystrov & N. V. Volodko

  2. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    N. V. Volodko

Authors
  1. A. A. Bystrov
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  2. N. V. Volodko
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Corresponding authors

Correspondence to A. A. Bystrov or N. V. Volodko.

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Bystrov, A.A., Volodko, N.V. Exponential Inequalities for the Distribution Tails of the Number of Cycles in the Erdös-Rényi Random Graphs. Sib. Adv. Math. 32, 87–93 (2022). https://doi.org/10.1134/S105513442202002X

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  • DOI: https://doi.org/10.1134/S105513442202002X

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