  An important knowledge dimension of science and technology is the extent to
which their development is cumulative, that is, the extent to which later
findings build on earlier ones. Cumulative knowledge structures can be studied
using a network approach, in which nodes represent findings and links represent
knowledge flows. Of particular interest to those studies is the notion of
network paths and path length. Starting from the Price model of network growth,
we derive an exact solution for the path length distribution of all unique
paths from a given initial node to each node in the network. We study the
relative importance of the average in-degree and cumulative advantage effect
and implement a generalization where the in-degree depends on the number of
nodes. The cumulative advantage effect is found to fundamentally slow down path
length growth. As the collection of all unique paths may contain many
redundancies, we additionally consider the subset of the longest paths to each
node in the network. As this case is more complicated, we only approximate the
longest path length distribution in a simple context. Where the number of all
unique paths of a given length grows unbounded, the number of longest paths of
a given length converges to a finite limit, which depends exponentially on the
given path length. Fundamental network properties and dynamics therefore
characteristically shape cumulative structures in those networks, and should
therefore be taken into account when studying those structures.
