Random Threshold Digraphs

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.

Oriented Threshold Graphs

“…Their generalization to directed graphs is focused on degree sequences and unique realizations. This work is extended by Reilly, Scheinerman, and Zhang, [6]. These extensions generalize definitions of simple threshold graphs into directed graphs and demonstrate their equivalence with the definitions of Cloteaux, et al These definitions deal predominately with directed graphs in which 2-cycles (multiedges in the underlying graph) are permitted in order to obtain unique realizability.…”

Threshold numbers of some complete multipartite graphs and their complements

“…They defined these graphs as follows. A graph G is said to be a threshold graph if we can assign a real number r v to each vertex v and there is a real number θ such that for any vertex subset U of G, ∑ v∈U r v ≤ θ if and only if U is independent in G. As one of the fundamental classes of graphs, properties of threshold graphs have been extensively studied (see [5,6,7,8,10,12,14] and [18]), and since then many applications of these graphs have been found in various areas, such as scheduling theory, resource allocation and parallel processes (see [1,4,11,13,15] and [16]).…”