大型随机超图核的有限尺度标度
Finite size scaling for the core of large random hypergraphs
摘要：
超图的（二）核是所有顶点只出现一次的超边的最大集合。它在诸如有效地求解GF上的大型线性系统[2]或在二进制擦除信道上使用的低密度奇偶校验码的迭代解码等任务中是重要的。类似的结构出现在各种NP难的组合优化和决策问题中，从顶点覆盖到可满足性。对于一个均匀选择的随机超图的$m=nρ$顶点和$n$超边，每个组成相同的固定数目$l\geq3$的顶点，核心的大小表现出一个一阶相变，从$o(n)$为$ρ>ρ_{\mathrm{c}}$的正分数$n$为$ρ<ρ_{\mathrm{c}}$，与过渡窗口大小$Θ(n^{-1/2})$周围$ρ_{\mathrm{c}}>0$。分析相应的“叶移除”算法，我们确定相关的有限大小缩放行为。具体地，如果$ρ$在缩放窗口（更精确地，$ρ=ρ_{\mathrm{c}}+rn^{-1/2}$）内，则具有大小为$Θ(n)$的核的概率具有严格在0和1之间的限制，以及阶数为$Θ(n^{-1/6})$的前导校正。修正承认一个尖锐的特征，在布朗运动的分布与二次移位，从它继承了与$n$缩放。这种行为预计将是普遍的组合问题的广泛收集。
英文摘要：
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.
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