Upper tails of subgraph counts in directed random graphs

TL;DR

This paper investigates the probability of large deviations in the count of fixed oriented subgraphs within directed Erdős-Rényi graphs, extending the upper tail problem to directed settings and providing bounds and exact solutions for specific subgraphs.

Contribution

It adapts existing methods to directed graphs, establishing bounds and exact solutions for the upper tail probabilities of various oriented subgraphs.

Findings

Bounds for the variational problem differ by at most a factor of 2.

Exact solutions obtained for triangles, stars, directed cycles, and balanced digraphs.

Provides a framework for analyzing upper tail probabilities in directed random graphs.

Abstract

The upper tail problem in a sparse Erd\H{o}s-R\'enyi graph asks for the probability that the number of copies of some fixed subgraph exceeds its expected value by a constant factor. We study the analogous problem for oriented subgraphs in directed random graphs. By adapting the proof of Cook, Dembo, and Pham, we reduce this upper tail problem to the asymptotic of a certain variational problem over edge weighted directed graphs. We give upper and lower bounds for the solution to the corresponding variational problem, which differ by a constant factor of at most $2$. We provide a host of subgraphs where the upper and lower bounds coincide, giving the solution to the upper tail problem. Examples of such digraphs include triangles, stars, directed $k$-cycles, and balanced digraphs.