The Annals of Applied Probability

Finite size scaling for the core of large random hypergraphs

Amir Dembo and Andrea Montanari

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The (two) core of a hypergraph is the maximal collection of hyperedges within which no vertex appears only once. It is of importance in tasks such as efficiently solving a large linear system over GF[2], or iterative decoding of low-density parity-check codes used over the binary erasure channel. Similar structures emerge in a variety of NP-hard combinatorial optimization and decision problems, from vertex cover to satisfiability.

For a uniformly chosen random hypergraph of m= vertices and n hyperedges, each consisting of the same fixed number l≥3 of vertices, the size of the core exhibits for large n a first-order phase transition, changing from o(n) for ρ>ρc to a positive fraction of n for ρ<ρc, with a transition window size Θ(n−1/2) around ρc>0. Analyzing the corresponding “leaf removal” algorithm, we determine the associated finite-size scaling behavior. In particular, if ρ is inside the scaling window (more precisely, ρ=ρc+rn−1/2), the probability of having a core of size Θ(n) has a limit strictly between 0 and 1, and a leading correction of order Θ(n−1/6). The correction admits a sharp characterization in terms of the distribution of a Brownian motion with quadratic shift, from which it inherits the scaling with n. This behavior is expected to be universal for a wide collection of combinatorial problems.

Article information

Ann. Appl. Probab., Volume 18, Number 5 (2008), 1993-2040.

First available in Project Euclid: 30 October 2008

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60F17: Functional limit theorems; invariance principles
Secondary: 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35] 94A29: Source coding [See also 68P30]

Core random hypergraph random graph low-density parity-check codes XOR-SAT finite-size scaling


Dembo, Amir; Montanari, Andrea. Finite size scaling for the core of large random hypergraphs. Ann. Appl. Probab. 18 (2008), no. 5, 1993--2040. doi:10.1214/07-AAP514.

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