It is known that random directed graphs undergo a phase transition around the point . Earlier, Łuczak and Seierstad have established that as when , the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as goes from to . By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter , we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2-cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.

中文翻译：

---

强组件的诞生

众所周知，随机有向图围绕该点发生相变。早些时候，Łuczak 和 Seierstad 已经确定，什么时候，随机有向图的强连通分量仅为循环且单个顶点的渐近概率从 1 减小到 0，如下所示来自到。通过使用分析组合学的技术，我们建立了该概率的精确极限值作为函数并提供有关其转变点周围、下方和上方的随机有向图结构的更多统计见解。我们获得了随机有向图是无环的极限概率，以及它具有一个强连通复分量且边和顶点数量之间给定差异（称为过剩）的概率。我们的结果可以扩展到具有给定过量的多个复杂组件的情况以及稀疏有向图的整个范围。我们的研究基于可以处理多种可能的有向图族的通用符号方法，以及可以系统地应用于符号方法中出现的复杂轮廓积分的鞍点方法的版本。虽然技术上最简单的模型是随机多重有向图模型，其中允许多条边，并且根据具有固定参数的泊松分布对边重数进行独立采样，我们还展示了如何系统地处理简单有向图族，其中禁止多个边，以及允许或不允许 2 个循环。当顶点数量有限时，我们的理论预测得到了数值模拟的支持，并且我们提供了本研究中出现的艾里函数积分的数值表。