Finite size scaling for the core of large random hypergraphs

Abstract

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=nρ 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>ρ_c to a positive fraction of n for ρ<ρ_c, with a transition window size Θ(n^−1/2) around ρ_c>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=ρ_c+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.