We introduce a simple model of static networks, where nodes are located on a ring structure, and two accompanying dynamic rules of repeated averaging on periodic node states. We assume nodes can interact with neighbors, and will add long-range links randomly. The number of long-range links, E, controls structures of these networks, and we show that there exist many types of fixed points, when E is varied. When E is low, fixed points are mostly diverse states, in which node states are diversely populated; on the other hand, when E is high, fixed points tend to be dominated by converged states, in which node states converge to one value. Numerically, we observe properties of fixed points for various E's, and also estimate points of the transition from diverse states to converged states for four different cases. This kind of simple network models will help us understand how diversities that we encounter in many systems of complex networks are sustained, even when mechanisms of averaging are at work,and when they break down if more long-range connections are added.