Threshold graphs were introduced by Chv´atal and Hammer in [4, 5]; see also [6, 13]. There areseveral, logically equivalent ways to define this family of graphs, but the one we choose workswell for developing a model of random graphs. A simple graph G is a threshold graph if we canassign weights to the vertices such that a pair of distinct vertices is adjacent exactly when thesum of their assigned weights is or exceeds a specified threshold. Without loss of generality, thethreshold can be taken to be 1 and the weights can be restricted to lie in the interval [0,1]; seeDefinition 2.1. References [2, 9, 16] provide an extensive introduction to this class of graphs.If we choose the weights for the vertices at random, we induce a probability measure onthe set of threshold graphs and thereby create a notion of a random threshold graph. Giventhat we may assume the weights lie in [0,1] it is natural to take the weights independentlyand...