Hrushovski's amalgamation construction can be used to join a

collection of finite graphs to produce a ``generic'' of this

collection. The choice of the collection and the way they are joined

are determined by a real-valued parameter α. Classical results

have shown that for α irrational in (0,1), the model theory

of the resulting structure is very well-behaved.

This dissertation examines analogous constructions for rational "r."

Depending on the way in which the parameter's control of the

construction is defined, the model theory of the resulting generic

will be either very well-behaved or very wild. We characterize when

each of these situations occurs.