标题： 核心如何位于地幔内部？ 显示英文标题

标题： How does the core sit inside the mantle?

摘要： 随机图$G(n,p)$的最小度为$k$的最大子图，即$k$-核心，已经被广泛研究。 在一篇具有里程碑意义的论文中，Pittel、Wormald 和 Spencer [JCTB 67 (1996) 111--151] 确定了大规模$k$-核心出现的阈值$d_k$。 此处我们推导了一个多类型Galton-Watson分支过程，该过程精确描述了对于任何$k\geq3$和固定的平均度$d=np>d_k$，$k$-核心如何嵌入随机图中。 这推广了先前关于$k$-核心内部结构的结果。 显示英文摘要

摘要： The $k$-core, defined as the largest subgraph of minimum degree $k$, of the random graph $G(n,p)$ has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [JCTB 67 (1996) 111--151] determined the threshold $d_k$ for the appearance of an extensive $k$-core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the $k$-core is embedded into the random graph for any $k\geq3$ and any fixed average degree $d=np>d_k$. This generalises prior results on, e.g., the internal structure of the $k$-core.