Katz similarity index comparisons

In this paper we compare the Katz similarity index to two other node similarity metrics, the standard distance and the resistance distance, for both path and cycle graphs. We consider how Katz similarity varies as the parameter $\alpha$ and the size of the graph vary. We also characterize when the three metrics give rise to different orderings of vertex pairs. In particular, we find that for all admissible values of $\alpha$ the Katz similarity index, the resistance distance, and the distance give the same ordering for node pairs in a cycle graph of arbitrary length, but the same is not true for the path graph.