  We use directed graphs called "syzygy quivers" to study the asymptotic growth
rates of the dimensions of the syzygies of representations of finite
dimensional algebras. For any finitely generated representation of a monomial
algebra, we show that this growth rate is poly-exponential, i.e. the product of
a polynomial and an exponential function, and give a procedure for computing
the corresponding degree and base from a syzygy quiver. We characterize the
growth rates arising in this context: The bases of the occurring exponential
functions are the real, nonnegative algebraic integers $b$ whose irreducible
polynomial over $\mathbb{Q}$ has no root with with modulus larger than $b$.
Moreover, we show that these growth rates are invariant under stable derived
equivalences.
