  Threshold graphs are recursive deterministic network models that have been
proposed for describing certain economic and social interactions. One drawback
of this graph family is that it has limited generative attachment rules. To
mitigate this problem, we introduce a new class of graphs termed Paired
Threshold (PT) graphs described through vertex weights that govern the
existence of edges via two inequalities. One inequality imposes the constraint
that the sum of weights of adjacent vertices has to exceed a specified
threshold. The second inequality ensures that adjacent vertices have a weight
difference upper bounded by another threshold. We provide a conceptually simple
characterization and decomposition of PT graphs, analyze their forbidden
induced subgraphs and present a method for performing vertex weight assignments
on PT graphs that satisfy the defining constraints. Furthermore, we describe a
polynomial-time algorithm for recognizing PT graphs. We conclude our exposition
with an analysis of the intersection number, diameter and clustering
coefficient of PT graphs.
