A Classification of Isomorphism-Invariant Random Digraphs

TL;DR

This paper classifies isomorphism-invariant random digraphs based on where randomness is introduced, extending graph classifications to directed graphs and exploring their relationships with various random digraph models.

Contribution

It introduces a comprehensive classification scheme for isomorphism-invariant random digraphs, including direction randomness, and analyzes their interrelations.

Findings

Classification of random digraphs into ARD, VRD, VARD, and DRD.

Identification of relationships between different classes of random digraphs.

Analysis of random nearest neighbor digraphs within the classification framework.

Abstract

We classify isomorphism-invariant random digraphs according to where randomness resides, namely, arcs, vertices, and vertices and arcs together which in turn yield arc random digraphs (ARD), vertex random digraphs (VRD) and vertex-arc random digraphs (VARD), respectively. This digraph classification can be viewed as an extension of the classification of isomorphism-invariant random graphs. We introduce randomness in the direction of the edges of a given graph and obtain direction random digraphs (DRD) as well. We classify DRDs according to which component is random in addition to the direction and study the relations of DRDs with VARDs, VRDs and ARDs. We also consider random nearest neighbor digraphs and determine their membership with respect to these digraph families.