Finite size scaling for the core of large random hypergraphs

The (two) core of an hyper-graph is the maximal collection of hyper-edges 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 hyper-graph of m = nρ vertices and n hyper-edges, 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 + r n −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 wide collection of combinatorial problems.