The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 26, Number 1 (2016), 305-327.
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
Abstract
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
Source
Ann. Appl. Probab. Volume 26, Number 1 (2016), 305-327.
Dates
Received: January 2014
Revised: July 2014
First available in Project Euclid: 5 January 2016
Permanent link to this document
https://projecteuclid.org/euclid.aoap/1452003240
Digital Object Identifier
doi:10.1214/14-AAP1091
Mathematical Reviews number (MathSciNet)
MR3449319
Zentralblatt MATH identifier
1336.60010
Subjects
Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]
Secondary: 90B15: Network models, stochastic
Keywords
Maximum subgraph density load balancing local weak convergence objective method unimodularity pairing model
Citation
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. https://projecteuclid.org/euclid.aoap/1452003240.
References
- [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.Mathematical Reviews (MathSciNet): MR1825411
- [2] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508.Mathematical Reviews (MathSciNet): MR2354165
Zentralblatt MATH: 1131.60003
Digital Object Identifier: doi:10.1214/EJP.v12-463 - [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.Mathematical Reviews (MathSciNet): MR2023650
Zentralblatt MATH: 1037.60008
Digital Object Identifier: doi:10.1007/978-3-662-09444-0_1 - [4] Aldous, D. J. (2001). The $\zeta(2)$ limit in the random assignment problem. Random Structures Algorithms 18 381–418.Mathematical Reviews (MathSciNet): MR1839499
- [5] Aldous, D. J. and Bandyopadhyay, A. (2005). A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 1047–1110.Mathematical Reviews (MathSciNet): MR2134098
Zentralblatt MATH: 1105.60012
Digital Object Identifier: doi:10.1214/105051605000000142
Project Euclid: euclid.aoap/1115137969 - [6] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 no. 23, 13 pp. (electronic).Mathematical Reviews (MathSciNet): MR1873300
Zentralblatt MATH: 1010.82021
Digital Object Identifier: doi:10.1214/EJP.v6-96 - [7] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.Mathematical Reviews (MathSciNet): MR1700749
- [8] Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311–316.Mathematical Reviews (MathSciNet): MR595929
Zentralblatt MATH: 0457.05038
Digital Object Identifier: doi:10.1016/S0195-6698(80)80030-8 - [9] Bordenave, C. (2012). Lecture notes on random graphs and probabilistic combinatorial optimization. Available at http://www.math.univ-toulouse.fr/~bordenave/coursRG.pdf.
- [10] Bordenave, C., Lelarge, M. and Salez, J. (2013). Matchings on infinite graphs. Probab. Theory Related Fields 157 183–208.Mathematical Reviews (MathSciNet): MR3101844
Zentralblatt MATH: 1278.60018
Digital Object Identifier: doi:10.1007/s00440-012-0453-0 - [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.Mathematical Reviews (MathSciNet): MR2482873
- [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.Mathematical Reviews (MathSciNet): MR2858395
Digital Object Identifier: doi:10.1137/1.9781611973082.93 - [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.Mathematical Reviews (MathSciNet): MR2187483
- [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.Mathematical Reviews (MathSciNet): MR2743258
- [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.Mathematical Reviews (MathSciNet): MR1389831
Zentralblatt MATH: 0863.60099
Digital Object Identifier: doi:10.1214/aoap/1034968065
Project Euclid: euclid.aoap/1034968065 - [18] Janson, S. (2009). The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205–225.Mathematical Reviews (MathSciNet): MR2497380
Zentralblatt MATH: 1216.05145
Digital Object Identifier: doi:10.1017/S0963548308009644 - [19] Khandwawala, M. and Sundaresan, R. (2014). Belief propagation for optimal edge cover in the random complete graph. Ann. Appl. Probab. 24 2414–2454.Mathematical Reviews (MathSciNet): MR3262507
Zentralblatt MATH: 06371854
Digital Object Identifier: doi:10.1214/13-AAP981
Project Euclid: euclid.aoap/1409058036 - [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.Mathematical Reviews (MathSciNet): MR3185378
- [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.Mathematical Reviews (MathSciNet): MR3205213
Digital Object Identifier: doi:10.1137/1.9781611973099.23 - [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.Mathematical Reviews (MathSciNet): MR1166614
- [23] Lyons, R. (2005). Asymptotic enumeration of spanning trees. Combin. Probab. Comput. 14 491–522.Mathematical Reviews (MathSciNet): MR2160416
Zentralblatt MATH: 1076.05007
Digital Object Identifier: doi:10.1017/S096354830500684X - [24] Salez, J. (2013). Weighted enumeration of spanning subgraphs in locally tree-like graphs. Random Structures Algorithms 43 377–397.Mathematical Reviews (MathSciNet): MR3094425
- [25] Salez, J. and Shah, D. (2009). Belief propagation: An asymptotically optimal algorithm for the random assignment problem. Math. Oper. Res. 34 468–480.Mathematical Reviews (MathSciNet): MR2554069
Zentralblatt MATH: 1230.68183
Digital Object Identifier: doi:10.1287/moor.1090.0380 - [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.Mathematical Reviews (MathSciNet): MR1940139
Zentralblatt MATH: 1032.60007
Digital Object Identifier: doi:10.1007/978-3-0348-8211-8_14 - [28] van der Hofstad, R. (2013). Random graphs and complex networks. Available at http://www.win.tue.nl/~rhofstad/NotesRGCN.html.
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
Bayati, Mohsen, Gamarnik, David, and Tetali, Prasad, The Annals of Probability, 2013 - Community detection in sparse random networks
Verzelen, Nicolas and Arias-Castro, Ery, The Annals of Applied Probability, 2015 - The importance sampling technique for understanding rare events in Erdős–Rényi random graphs
Bhamidi, Shankar, Hannig, Jan, Lee, Chia Ying, and Nolen, James, Electronic Journal of Probability, 2015
- Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
Bayati, Mohsen, Gamarnik, David, and Tetali, Prasad, The Annals of Probability, 2013 - Community detection in sparse random networks
Verzelen, Nicolas and Arias-Castro, Ery, The Annals of Applied Probability, 2015 - The importance sampling technique for understanding rare events in Erdős–Rényi random graphs
Bhamidi, Shankar, Hannig, Jan, Lee, Chia Ying, and Nolen, James, Electronic Journal of Probability, 2015 - Moderate Deviations in a Random Graph and for the Spectrum of Bernoulli Random Matrices
Döring, Hanna and Eichelsbacher, Peter, Electronic Journal of Probability, 2009 - Estimating and understanding exponential random graph models
Chatterjee, Sourav and Diaconis, Persi, The Annals of Statistics, 2013 - Capacity of an associative memory model on random graph architectures
Löwe, Matthias and Vermet, Franck, Bernoulli, 2015 - Scaling Limits for Critical Inhomogeneous Random Graphs with Finite Third Moments
Bhamidi, Shankar, van der Hofstad, Remco, and van Leeuwaarden, Johan, Electronic Journal of Probability, 2010 - Community detection in dense random networks
Arias-Castro, Ery and Verzelen, Nicolas, The Annals of Statistics, 2014 - Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs and percolation
Krokowski, Kai, Reichenbachs, Anselm, and Thäle, Christoph, The Annals of Probability, 2017 - Critical Random Graphs: Limiting Constructions and Distributional Properties
Addario-Berry, Louigi, Broutin, Nicolas, and Goldschmidt, Christina, Electronic Journal of Probability, 2010