The densest subgraph problem in sparse random graphs

Justin Salez (1) , Venkat Anantharam (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 EECS - Department of Electrical Engineering and Computer Science [Berkeley]

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].

Domaines: Probabilités [math.PR]

Justin Salez, Venkat Anantharam. The densest subgraph problem in sparse random graphs. The Annals of Applied Probability, 2016, 26 (1), pp.305-327. ⟨hal-01361829⟩