We study herewith some aspects related to predictability of the long-term global behavior of the star topology based infrastructures when all the nodes, including the central node, are assumed to function reliably, faultlessly and synchronously. In particular, we use the nonlinear complex systems concepts and methodology, coupled with those of computational complexity, to show that, simple as the star topology is, determining and predicting the longterm global behavior of the star-based infrastructures are computationally challenging tasks. More formally, determining various configuration space properties of the appropriate star network abstractions is shown to be hard in general. We particularly focus herein on the computational (in)tractability of counting the "fixed point" configurations of a class of formal discrete dynamical systems defined over the star interconnection topology.