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 $l\geq3$ of vertices, the size of the core exhibits for large $n$ a first-order phase transition, changing from $o(n)$ for $ρ>ρ_{\mathrm{c}}$ to a positive fraction of $n$ for $ρ<ρ_{\mathrm{c}}$, with a transition window size $Θ(n^{-1/2})$ around $ρ_{\mathrm{c}}>0$. 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, $ρ=ρ_{\mathrm{c}}+rn^{-1/2}$), 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.