Abstract
This paper introduces a notion of a random threshold directed graph, extending the work of Reilly and Scheinerman in the undirected case and closely related to random Ferrers digraphs.We begin by presenting the main definition: $D$ is a threshold digraph provided we can find a pair of weighting functions $f,g:V(D)\to\mathbb{R}$ such that for distinct $v,w\in V(D)$ we have $v\to w$ iff $f(v)+g(w)\ge1$. We also give an equivalent formulation based on an order representation that is purely combinatorial (no arithmetic). We show that our formulations are equivalent to the definition in the work of Cloteaux, LaMar, Moseman, and Shook in which the focus is on the degree sequence, and present a new characterization theorem for threshold digraphs.We then develop the notion of a random threshold digraph formed by choosing vertex weights independently and uniformly at random from $[0,1]$. We show that this notion of a random threshold digraph is equivalent to a purely combinatorial approach, and present a formula for the probability of a digraph based on counting linear extensions of an auxiliary partially ordered set.We capitalize on this equivalence to develop exact and asymptotic properties of random threshold digraphs such as the number of vertices with in-degree (or out-degree) equal to zero, domination number, connectivity and strong connectivity, clique and independence number, and chromatic number.
Highlights
In this paper graph exclusively means a simple, undirected graph without loops or multiple edges
We begin by presenting the main definition: D is a threshold digraph provided we can find a pair of weighting functions f, g : V (D) → R such that for distinct v, w ∈ V (D) we have v → w iff f (v) + g(w) 1
We show that our formulations are equivalent to the definition in the work of Cloteaux, LaMar, Moseman, and Shook in which the focus is on the degree sequence, and present a new characterization theorem for threshold digraphs
Summary
In this paper graph exclusively means a simple, undirected graph without loops or multiple edges. We write u ∼ v to denote that uv is an edge. The term digraph means a strict directed graph without loops or multiple edges in the same direction, but antiparallel edges (u → v and v → u) may be present. We write |G| to denote the number of vertices and G the number of edges in the [di]graph G. For a digraph D, its underlying simple graph is denoted simp(D). Simp(D) is a simple graph with the same vertex set as D in which u ∼ v if and only if u → v or v → u in D.
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