Twisted supersymmetry and the topology of theory space

We present examples of four dimensional, non-supersymmetric field theories in which ultraviolet supersymmetry breaking effects, such as bose-fermi splittings and the vacuum energy, are suppressed by $(\alpha/4 \pi)^{N}$, where $\alpha$ is a weak coupling factor and $N$ can be made arbitrarily large. The particle content and interactions of these models are conveniently represented by a graph with sites and links, describing the gauge theory space structure. While the theories are supersymmetric ``locally'' in theory space, supersymmetry can be explicitly broken by topological obstructions.