Acta Mathematica Sinica, English Series

, Volume 34, Issue 10, pp 1626–1634 | Cite as

Large Deviations in Generalized Random Graphs with Node Weights

  • Qun Liu
  • Zhi Shan DongEmail author


Generalized random graphs are considered where the presence or absence of an edge depends on the weights of its nodes. Our main interest is to investigate large deviations for the number of edges per node in such a generalized random graph, where the node weights are deterministic under some regularity conditions, as well as chosen i.i.d. from a finite set with positive components. When the node weights are random variables, obstacles arise because the independence among edges no longer exists, our main tools are some results of large deviations for mixtures. After calculating, our results show that the corresponding rate functions for the deterministic case and the random case are very different.


Large deviations mixture generalized random graphs 

MR(2010) Subject Classification

05C80 60F10 


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We thank the referees for their time and comments.


  1. [1]
    Biggins, J. D.: Large deviations for mixtures. Electron. Commun. Probab., 9, 60–71 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Biggins, J. D., Penman, D. B.: Large deviations in randomly coloured random graphs. Electron. Commun. Probab., 14, 290–301 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Bollobás, B., Janson, S., Riordan, O.: The phase transition in inhomogeneous random graphs. Random Struct. Alg., 31, 3–122 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution. J. Stat. Phys., 124, 1377–1397 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Chung, F., Lu, L.: The volume of the giant component of a random graph with given expected degrees. SIAM J. Discrete Math., 20(2), 395–411 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Chen, X.: Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs, AMS. 157, (2010)CrossRefGoogle Scholar
  7. [7]
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, Jones and Bartlett Publishers, Boston, MA, 1993zbMATHGoogle Scholar
  8. [8]
    Erdos, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci., 5, 17–61 (1960)MathSciNetzbMATHGoogle Scholar
  9. [9]
    Hu, Z. S., Bi, W., Feng, Q. Q.: Limit laws in the generalized random graphs with random vertex weights. Statistics and Probability letters, 89, 65–76 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Norros, I., Reittu, H.: On a conditionally Poisson graph process. Adv. Appl. Probab., 38(1), 59–75 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Penman, D. B.: Ramdom Graphs with Correlation Structure, Ph. D thesis, University of Sheffield, 1998Google Scholar
  12. [12]
    Van der Hofstad, R.: Random Graphs and Complex Networks, Cambridge Series in Statistical and Probabilistic Mathematics, Volume 1, 2017CrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics School and Institute of Jilin UniversityChangchunP. R. China

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