Ring networks have been extensively studied and installed for communication services. In actual ring networks, nodes are usually allocated at random positions. Two nodes separated far from each other along the route of a ring network may be physically allocated near each other. Adding a supplementary link directly connecting such nodes can shorten the connection paths between the two nodes as well as between other neighboring node pairs. Aggregated length of connection paths of the network can be reduced. Although such a network can be regarded as being composed of two small ring networks sharing the supplementary link, we regarded the network as being modified from a single regular ring network by adding a supplementary link, and thus we quantitatively evaluated the effect of the network modification. We analyzed, for the first time, number of connection paths in the links of networks with or without a supplementary link. A full-mesh topology was assumed, and two types of connection were examined: a single-path and a 2-path connection. The results of this analysis were then used to evaluate system costs and reliability of the networks. The evaluation confirmed that adding a supplementary link is effective in terms of cost reduction and reliability improvement under certain conditions.