The strong component structure of the barely subcritical directed configuration model

TL;DR

This paper analyzes the structure of the largest strongly connected components in the barely subcritical directed configuration model, revealing they are mostly cycles or isolated vertices, with an asymptotic distribution for cycle sizes.

Contribution

It provides a detailed description of the component structure in the barely subcritical regime and extends known results to the directed configuration model.

Findings

All large components are cycles or isolated vertices.

Asymptotic distribution of the size of the kth largest cycle.

Analogue of a known binomial digraph result for the directed configuration model.

Abstract

We study the behaviour of the largest components of the directed configuration model in the barely subcritical regime. We show that with high probability all strongly connected components in this regime are either cycles or isolated vertices and give an asymptotic distribution of the size of the $k$th largest cycle. This gives a configuration model analogue of a result of {\L}uczak and Seierstad for the binomial random digraph.