随机有向图中巨型$(k_1,k_2)$ -核的产生
Birth of a giant $(k_1,k_2)$-core in the random digraph
摘要：
有向图的$(k_1,k_2)$ -核是最小入度至少为$k_1$、最小出度至少为$k_2$的最大子有向图。对于$\max\{k_1, k_2\} \geq 2$，我们建立了阈值边密度$c^*=c^*(k_1,k_2)$的存在性，使得随机有向图$D(n,m)$在具有$m$边的顶点集$[n]$上，如果$m/n> c^*$，则渐近几乎必然有一个巨大的$(k_1,k_2)$ -核，如果$m/n<c^*$，则没有$(k_1,k_2)$ -核.具体地说，表示$\text{P}(\text{Poisson}(z)\ge k)$的$p_k(z)$，我们证明了$c^*=\min\limits_{z_1,z_2}\max\left\{\tfrac{z_1}{p_{k_1}(z_1)p_{k_2-1}(z_2)}; \tfrac{z_2}{p_{k_1-1}(z_1)p_{k_2}(z_2)}\right\}$。
英文摘要：
The $(k_1,k_2)$-core of a digraph is the largest sub-digraph with minimum in-degree and minimum out-degree at least $k_1$ and $k_2$ respectively. For $\max\{k_1, k_2\} \geq 2$, we establish existence of the threshold edge-density $c^*=c^*(k_1,k_2)$, such that the random digraph $D(n,m)$, on the vertex set $[n]$ with $m$ edges, asymptotically almost surely has a giant $(k_1,k_2)$-core if $m/n> c^*$, and has no $(k_1,k_2)$-core if $m/n<c^*$. Specifically, denoting $\text{P}(\text{Poisson}(z)\ge k)$ by $p_k(z)$, we prove that $c^*=\min\limits_{z_1,z_2}\max\left\{\tfrac{z_1}{p_{k_1}(z_1)p_{k_2-1}(z_2)}; \tfrac{z_2}{p_{k_1-1}(z_1)p_{k_2}(z_2)}\right\}$.
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