Random Threshold Graphs

Abstract

We introduce a pair of natural, equivalent models for random threshold graphs and use these models to deduce a variety of properties of random threshold graphs. Specifically, a random threshold graph $G$ is generated by choosing $n$ IID values $x_1,\ldots,x_n$ uniformly in $[0,1]$; distinct vertices $i,j$ of $G$ are adjacent exactly when $x_i + x_j \ge 1$. We examine various properties of random threshold graphs such as chromatic number, algebraic connectivity, and the existence of Hamiltonian cycles and perfect matchings.

Random Threshold Digraphs

“…The goal of this paper is to extend the work on random threshold graphs [12] (see also [11]) to directed graphs. (See also [11] and [13] which consider a variant of the random threshold model for bipartite graphs; such graphs are known as random difference graphs.…”

“…We use the transformation T to create a natural model of random threshold graphs. One simply selects a point in x ∈ [0, 1] n uniformly at random and apply T to yield a random threshold graph; see [11] and [12]. Specifically, we define the random graph G n to be a random variable taking values in G n such that…”

“…Of course, if G is not a threshold graph, Pr[G n = G] = 0. In [11,12] an equivalent, purely combinatorial model for random threshold graphsbased on creation sequences-provides a route to proving various exact and asymptotic results about random threshold graphs.…”

Paired threshold graphs

“…The problem of analyzing complex behaviors of large social, economic and biological networks based on generative recursive and probabilistic models has been the subject of intense research in graph theory, machine learning and statistics. In these settings, one often assumes the existence of attachment and preference rules for network formation, or imposes constraints on subgraph structures as well as vertex and edge features that govern the creation of network communities [1,2,3,4,5]. Models of this type have been used to predict network dynamics and topology fluctuations, infer network community properties and preferences, determine the bottlenecks and rates of spread of information and commodities and elucidate functional and structural properties of individual network modules [6,7,8].…”

Properties of Random Difference Graphs

“…Difference graphs were formally defined by Hammer, Peled, and Sun [5] in 1990, although they had been independently explored prior to that point [3], a result of the multiple equivalent characterizations for this class of graphs [6]. They are closely related to threshold graphs, which can be viewed as difference graphs without the bipartitioning; this similarity enables us to adapt the techniques used in the study of random threshold graphs, notably those of Reilly and Scheinerman [7]. Here, we focus on difference graphs with a specified bipartition, also known as bipartite threshold graphs (Diaconis, Holmes, Janson [4]).…”