The Annals of Applied Probability

The densest subgraph problem in sparse random graphs

Venkat Anantharam and Justin Salez

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We determine the asymptotic behavior of the maximum subgraph density of large random graphs with a prescribed degree sequence. The result applies in particular to the Erdős–Rényi model, where it settles a conjecture of Hajek [IEEE Trans. Inform. Theory 36 (1990) 1398–1414]. Our proof consists in extending the notion of balanced loads from finite graphs to their local weak limits, using unimodularity. This is a new illustration of the objective method described by Aldous and Steele [In Probability on Discrete Structures (2004) 1–72 Springer].

Article information

Ann. Appl. Probab. Volume 26, Number 1 (2016), 305-327.

Received: January 2014
Revised: July 2014
First available in Project Euclid: 5 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]
Secondary: 90B15: Network models, stochastic

Maximum subgraph density load balancing local weak convergence objective method unimodularity pairing model


Anantharam, Venkat; Salez, Justin. The densest subgraph problem in sparse random graphs. Ann. Appl. Probab. 26 (2016), no. 1, 305--327. doi:10.1214/14-AAP1091.

Export citation


  • [1] Agarwal, R. P., Meehan, M. and O’Regan, D. (2001). Fixed Point Theory and Applications. Cambridge Tracts in Mathematics 141. Cambridge Univ. Press, Cambridge.
  • [2] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508.
  • [3] Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin.
  • [4] Aldous, D. J. (2001). The $\zeta(2)$ limit in the random assignment problem. Random Structures Algorithms 18 381–418.
  • [5] Aldous, D. J. and Bandyopadhyay, A. (2005). A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 1047–1110.
  • [6] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 no. 23, 13 pp. (electronic).
  • [7] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [8] Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311–316.
  • [9] Bordenave, C. (2012). Lecture notes on random graphs and probabilistic combinatorial optimization. Available at
  • [10] Bordenave, C., Lelarge, M. and Salez, J. (2013). Matchings on infinite graphs. Probab. Theory Related Fields 157 183–208.
  • [11] Cain, J. A., Sanders, P. and Wormald, N. (2007). The random graph threshold for $k$-orientability and a fast algorithm for optimal multiple-choice allocation. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms 469–476. ACM, New York.
  • [12] Fernholz, D. and Ramachandran, V. (2007). The $k$-orientability thresholds for $G_{n,p}$. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms 459–468. ACM, New York.
  • [13] Fountoulakis, N., Khosla, M. and Panagiotou, K. (2011). The multiple-orientability thresholds for random hypergraphs. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms 1222–1236. SIAM, Philadelphia, PA.
  • [14] Gamarnik, D., Nowicki, T. and Swirszcz, G. (2006). Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method. Random Structures Algorithms 28 76–106.
  • [15] Gao, P. and Wormald, N. C. (2010). Load balancing and orientability thresholds for random hypergraphs [extended abstract]. In STOC’10—Proceedings of the 2010 ACM International Symposium on Theory of Computing 97–103. ACM, New York.
  • [16] Hajek, B. (1990). Performance of global load balancing by local adjustment. IEEE Trans. Inform. Theory 36 1398–1414.
  • [17] Hajek, B. (1996). Balanced loads in infinite networks. Ann. Appl. Probab. 6 48–75.
  • [18] Janson, S. (2009). The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205–225.
  • [19] Khandwawala, M. and Sundaresan, R. (2014). Belief propagation for optimal edge cover in the random complete graph. Ann. Appl. Probab. 24 2414–2454.
  • [20] Leconte, M., Lelarge, M. and Massoulié, L. (2012). Convergence of multivariate belief propagation, with applications to cuckoo hashing and load balancing. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms 35–46. SIAM, Philadelphia, PA.
  • [21] Lelarge, M. (2012). A new approach to the orientation of random hypergraphs. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms 251–264. ACM, New York.
  • [22] Łuczak, T. (1992). Sparse random graphs with a given degree sequence. In Random Graphs, Vol. 2 (Poznań, 1989). Wiley-Intersci. Publ. 165–182. Wiley, New York.
  • [23] Lyons, R. (2005). Asymptotic enumeration of spanning trees. Combin. Probab. Comput. 14 491–522.
  • [24] Salez, J. (2013). Weighted enumeration of spanning subgraphs in locally tree-like graphs. Random Structures Algorithms 43 377–397.
  • [25] Salez, J. and Shah, D. (2009). Belief propagation: An asymptotically optimal algorithm for the random assignment problem. Math. Oper. Res. 34 468–480.
  • [26] Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Inc., Boston, MA.
  • [27] Steele, J. M. (2002). Minimal spanning trees for graphs with random edge lengths. In Mathematics and Computer Science, II (Versailles, 2002). Trends Math. 223–245. Birkhäuser, Basel.
  • [28] van der Hofstad, R. (2013). Random graphs and complex networks. Available at