  Generating random and pseudorandom numbers with a deterministic system is a
long-standing challenge in theoretical research and engineering applications.
Several pseudorandom number generators based on the inversive congruential
method have been designed as attractive alternatives to those based on the
classical linear congruential method. This paper discloses the least period of
sequences generated by iterating an inversive pseudorandom number generator
over the ring $\mathbb{Z}_e$ by transforming it into a two-order linear
congruential recurrence relation. Depending on whether the sequence is periodic
or ultimately periodic, all states in the domain can be attributed to two types
of objects: some cycles of different lengths and one unilateral connected
digraph whose structure remains unchanged concerning parameter $e$. The graph
structure of the generator over the ring $\mathbb{Z}_e$ is precisely disclosed
with rigorous theoretical analysis and verified experimentally. The adopted
analysis methodology can be extended to study the graph structure of other
nonlinear maps.
