The Annals of Applied Probability

The densest subgraph problem in sparse random graphs

Venkat Anantharam and Justin Salez

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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

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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.

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