Similarity index measures the topological proximity of node pairs in a complex network. Numerous similarity indices have been defined and investigated, but the dependency of structure on the performance of similarity indices has not been sufficiently investigated. In this study, we investigated the relationship between the performance of similarity indices and structural properties of a network by employing a two-state random network. A node in a two-state network has binary types that are initially given, and a connection probability is determined from the state of the node pair. The performance of similarity indices affects the number of links and the ratio of intra-connections to inter-connections. Similarity indices have different characteristics depending on their type. Local indices perform well in small-size networks and do not depend on whether the structure is intra-dominant or inter-dominant. In contrast, global indices perform better in large-size networks, and some such indices do not perform well in an inter-dominant structure. We also found that link prediction performance and the performance of similarity are correlated in both model networks and empirical networks. This relationship implies that link prediction performance can be used as an approximation for the performance of the similarity index when metadata for node types are unavailable. This relationship may help to find the appropriate index for given networks.