  We study the growth of networks from a set of isolated ground nodes by the
addition of one new node per time step and also of a fixed number of directed
edges leading from the new node to randomly selected nodes already in the
network. A fixed-width time window is used so that, in general, only nodes that
entered the network within the latest window may receive new incoming edges.
The resulting directed network is acyclic at all times and allows some of the
ground nodes, then called sinks, to be reached from some of the non-ground
nodes. We regard such networks as representative of abstract systems of
partially ordered constituents, for example in some of the domains related to
technological evolution. Two properties of interest are the number of sinks
that can be reached from a randomly chosen non-ground node (its reach) and, for
a fixed sink, the number of nonoverlapping directed paths through which the
sink can be reached, at a given time, from some of the latest nodes to have
entered the network. We demonstrate, by means of simulations and also of
analytic characterizations, that reaches are distributed according to a power
law and that the desired directed paths are expected to occur in very small
numbers, perhaps indicating that recovering sinks late in the process of
network growth is strongly sensitive to accidental path disruptions.
