2024/03/04 by Remco van der Hofstad, Manish Pandey, van der Hofstad, Remco +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #05C80 #Advanced Graph Theory Research #Complex Network Analysis Techniques #FOS: Mathematics #G.2.2 #G.3 #Graph theory and applications #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2403.02137
openalex publication_date 2024/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently, the first author showed that the giant in random undirected graphs is `almost' local. This means that, under a necessary and sufficient condition, the limiting proportion of vertices in the giant converges in probability to the survival probability of the local limit. We extend this result to the setting of random digraphs, where connectivity patterns are significantly more subtle. For this, we identify the precise version of local convergence for digraphs that is needed. We also determine bounds on the number of strongly connected components, and calculate its asymptotics explicitly for locally tree-like digraphs, as well as for other locally converging digraph sequences under the `almost-local' condition for the strong giant. The fact that the number of strongly connected components is \em not local once more exemplifies the delicate nature of strong connectivity in random digraphs.