  The multiplicative attribute graph (MAG) model was introduced by Kim and
Leskovec as a mathematically tractable model of certain classes of real-world
networks. It is an instance of hidden graph models, and implements the
plausible idea that network structure is collectively shaped by attributes
individually associated with nodes. These authors have studied several aspects
of this model, including its connectivity, the existence of a giant
component,its diameter and the degree distribution. This was done in the
asymptotic regime when the number of nodes and the number of node attributes
both grow unboundedly large, the latter scaling with the former under a natural
admissibility condition. In the same setting, we explore the existence (or
equivalently, absence) of isolated nodes, a property not discussed in the
original paper. The main result of the paper is a {\em zero-one} law for the
absence of isolated nodes; this zero-one law coincides with that obtained by
Kim and Leskovec for graph connectivity (although under slightly weaker
assumptions). We prove these results by applying the method of first and second
moments in a non-standard way to multiple sets of counting random variables
associated with the number of isolated nodes.
