Local Structure Theorems for Erdős–Rényi Graphs and Their Algorithmic Applications

  • Jan Dreier
  • Philipp Kuinke
  • Ba Le Xuan
  • Peter Rossmanith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10706)


We analyze local properties of sparse Erdős–Rényi graphs, where d(n)/n is the edge probability. In particular we study the behavior of very short paths. For \(d(n)=n^{o(1)}\) we show that \(G(n,d(n)/n)\) has asymptotically almost surely (a.a.s.) bounded local treewidth and therefore is a.a.s. nowhere dense. We also discover a new and simpler proof that \(G(n,d/n)\) has a.a.s. bounded expansion for constant d. The local structure of sparse Erdős–Rényi graphs is very special: The r-neighborhood of a vertex is a tree with some additional edges, where the probability that there are m additional edges decreases with m. This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter. Finally, experiments suggest that preferential attachment graphs might have similar properties after deleting a small number of vertices.


Graph theory Random graphs Sparse graphs Graph algorithms 


  1. 1.
    Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, Cambridge (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Erdős, P., Rényi, A.: On random graphs. Publ. Math. 6, 290–297 (1959)MathSciNetMATHGoogle Scholar
  3. 3.
    Fagin, R.: Probabilities on finite models. J. Symb. Log. 41(1), 50–58 (1976)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Grohe, M.: Logic, graphs, and algorithms (2007)Google Scholar
  5. 5.
    Coja-Oghlan, A., Taraz, A.: Colouring random graphs in expected polynomial time. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 487–498. Springer, Heidelberg (2003). CrossRefGoogle Scholar
  6. 6.
    Dawar, A., Grohe, M., Kreutzer, S.: Locally excluding a minor. In: 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007), pp. 270–279, July 2007Google Scholar
  7. 7.
    Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. J. ACM 49(6), 716–752 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gao, Y.: Treewidth of Erdős–Rényi random graphs, random intersection graphs, and scale-free random graphs. Discrete Appl. Math. 160(4–5), 566–578 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Nešetřil, J., de Mendez, P.O.: Grad and classes with bounded expansion I. Decompositions. Eur. J. Comb. 29(3), 760–776 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dvořák, Z., KráÎ, D., Thomas, R.: Testing first-order properties for subclasses of sparse graphs. J. ACM 60(5), 36:1–36:24 (2013)MathSciNetGoogle Scholar
  11. 11.
    Grohe, M., Kreutzer, S., Siebertz, S.: Deciding first-order properties of nowhere dense graphs. In: Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC 2014, pp. 89–98. ACM, New York (2014)Google Scholar
  12. 12.
    Nešetřil, J., de Mendez, P.O., Wood, D.R.: Characterisations and examples of graph classes with bounded expansion. Eur. J. Comb. 33(3), 350–373 (2012). Topological and Geometric Graph TheoryMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fountoulakis, N., Friedrich, T., Hermelin, D.: On the average-case complexity of parameterized clique. Theoret. Comput. Sci. 576, 18–29 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Frick, M., Grohe, M.: Deciding first-order properties of locally tree-decomposable structures. J. ACM 48(6), 1184–1206 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dawar, A., Grohe, M., Kreutzer, S.: Locally excluding a minor. In: Proceedings of the 22nd IEEE Symposium on Logic in Computer Science (LICS 2007), Wroclaw, Poland, 10–12 July 2007, pp. 270–279. IEEE Computer Society (2007)Google Scholar
  16. 16.
    Grohe, M.: Generalized model-checking problems for first-order logic. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 12–26. Springer, Heidelberg (2001). CrossRefGoogle Scholar
  17. 17.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999). American Association for the Advancement of ScienceMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Cohen, R., Havlin, S.: Scale-free networks are ultrasmall. Phys. Rev. Lett. 90, 058701 (2003)CrossRefGoogle Scholar
  19. 19.
    Kamrul, M.H., Hassan, M.Z., Pavel, N.I.: Dynamic scaling, data-collapse and self-similarity in Barabási–Albert networks. J. Phys. A: Math. Theoret. 44(17), 175101 (2011)CrossRefGoogle Scholar
  20. 20.
    Klemm, K., Eguíluz, V.M.: Growing scale-free networks with small-world behavior. Phys. Rev. E 65, 057102 (2002)CrossRefGoogle Scholar
  21. 21.
    Demaine, E.D., Reidl, F., Rossmanith, P., Villaamil, F.S., Sikdar, S., Sullivan, B.D.: Structural sparsity of complex networks: random graph models and linear algorithms. CoRR abs/1406.2587 (2014)Google Scholar
  22. 22.
    Diestel, R.: Graph Theory. Springer, Berlin (2010)CrossRefMATHGoogle Scholar
  23. 23.
    Nešetřil, J., de Mendez, P.O.: Sparsity: Graphs, Structures, and Algorithms. Springer, Berlin (2014)MATHGoogle Scholar
  24. 24.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Chen, J., Kanj, I.A., Meng, J., Xia, G., Zhang, F.: On the effective enumerability of NP problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 215–226. Springer, Heidelberg (2006). CrossRefGoogle Scholar
  26. 26.
    Nešetřil, J., de Mendez, P.O.: Grad and classes with bounded expansion II. Algorithmic aspects. Eur. J. Comb. 29(3), 777–791 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (2012)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Jan Dreier
    • 1
  • Philipp Kuinke
    • 1
  • Ba Le Xuan
    • 2
  • Peter Rossmanith
    • 1
  1. 1.Theoretical Computer Science, Department of Computer ScienceRWTH Aachen UniversityAachenGermany
  2. 2.The Sirindhorn International Thai-German Graduate School of EngineeringKing Mongkut’s University of Technology North BangkokBangkokThailand

Personalised recommendations