扩张子上随机有向图的局部性
Locality of Random Digraphs on Expanders
摘要：
研究了平均度有界的扩张子序列{依概率局部收敛}上的随机有向图。我们证明了一个巨大的强连通分量的存在的阈值，以及与巨大的扇入节点或节点与巨大的扇出节点的渐近分数是局部的，在这个意义上，他们是相同的两个序列具有相同的局部极限。有向图具有蝴蝶结结构，除了消失的一部分节点之外，所有的节点要么位于唯一的强连通巨人及其扇入和扇出中，要么位于具有小扇入和小扇出的集合中。所有的本地量表示的限制有根图上的渗流，没有任何结构上的限制，允许，特别是非树状图。{In在建立这些结果的过程中，我们把以前关于巨球大小的局部性的结果推广到平均度有界且可能具有非树型极限的扩张子上。我们还证明了，无论序列的局部收敛，巨的唯一性和它的相对大小的收敛无定向渗流意味着蝴蝶结结构的定向渗流。我们的方法的应用表明，键渗流和随机有向图的优先连接图的临界阈值是$p_c=0$，与无限级相变在$p_c$。
英文摘要：
We study random digraphs on sequences of expanders with bounded average degree {which converge locally in probability}. We prove that the threshold for the existence of a giant strongly connected component, as well as the asymptotic fraction of nodes with giant fan-in or nodes with giant fan-out are local, in the sense that they are the same for two sequences with the same local limit. The digraph has a bow-tie structure, with all but a vanishing fraction of nodes lying either in the unique strongly connected giant and its fan-in and fan-out, or in sets with small fan-in and small fan-out. All local quantities are expressed in terms of percolation on the limiting rooted graph, without any structural assumptions on the limit, allowing, in particular, for non tree-like graphs. {In the course of establishing these results, we generalize previous results on the locality of the size of the giant to expanders of bounded average degree with possibly non-tree like limits. We also show that regardless of the local convergence of a sequence, the uniqueness of the giant and convergence of its relative size for unoriented percolation imply the bow-tie structure for directed percolation.} An application of our methods shows that the critical threshold for bond percolation and random digraphs on preferential attachment graphs is $p_c=0$, with an infinite order phase transition at $p_c$.
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