In this paper, we give an algorithm to publish the number of paths and Katz centrality under the local differential privacy (LDP), providing a thorough theoretical analysis. Although various works have already introduced subgraph counting algorithms under LDP, they have primarily concentrated on subgraphs of up to five nodes. The challenge in extending this to larger subgraphs is the cumulative and exponential growth of noise as the subgraph size increases in any publication under LDP. We address this issue by proposing an algorithm to publish the number of paths that start at every node in the graph, leading to an algorithm that publishes the Katz centrality of all nodes. This algorithm employs multiple rounds of communication and the clipping technique. Both our theoretical and experimental assessments indicate that our algorithm exhibits acceptable bias and variance, considerably less than an algorithm that bypasses clipping. Furthermore, our Katz centrality estimation is able to recall up to 90% of the nodes with the highest Katz centrality.