Universality for the directed configuration model: metric space convergence of the strongly connected components at criticality

Abstract

We consider the strongly connected components (SCCs) of a uniform directed graph on $n$ vertices with i.i.d. in- and out-degree pairs distributed as $(D^-,D^+)$, with $\mathbb E[D^+]=\mathbb E[D^-]=μ$. We condition on equal total in- and out-degree. A phase transition for the emergence of a giant SCC is known to occur at the critical value $\mathbb E[D^-D^+] = μ$. We study the model at this critical value and, additionally, require that $\mathbb E[(D^-)^i(D^+)^j]<\infty$ for all $i+j\leq 3$, and for $(i,j)=(1,3)$ and $(i,j)=(3,1)$. We show that, under these conditions, the SCCs ranked by decreasing number of edges with distances rescaled by $n^{-1/3}$ converge in distribution to a sequence of finite strongly connected directed multigraphs with edge lengths, and that these are either $3$-regular or loops. The limit objects lie in a $3$-parameter family, which contains the scaling limit of the SCCs in the directed Erdős-Rényi model at criticality as found by Goldschmidt and Stephenson (2019). This is the first universality result for the scaling limit of a critical directed graph model and the first quantitative result on the directed configuration model at criticality. As a trivial consequence, the largest SCCs at criticality contain $Θ(n^{1/3})$ vertices and edges in probability, and the diameter of the directed graph at criticality is $Ω(n^{1/3})$ in probability. We use a metric on the space of weighted multigraphs in which two multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge lengths. We use the product topology on the sequence of multigraphs. Our method of proof involves a depth-first exploration of the directed graph, resulting in a spanning forest with additional identifications, of which we study the limit under rescaling.