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Graph analysts cannot directly obtain the global structure in decentralized social networks, and analyzing such a network requires collecting local views of the social graph from individual users. Since the edges between users may reveal sensitive social interactions in the local view, applying differential privacy in the data collection process is often desirable, which provides strong and rigorous privacy guarantees. In practical decentralized social graphs, different edges have different privacy requirements due to the distinct sensitivity levels. However, the existing differentially private analysis of social graphs provide the same protection for all edges. To address this issue, this work proposes a fine-grained privacy notion as well as novel algorithms for private graph analysis. We first design a fine-grained relationship differential privacy (FGR-DP) notion for social graph analysis, which enforces different protections for the edges with distinct privacy requirements. Then, we design algorithms for triangle counting and k-stars counting, respectively, which can accurately estimate subgraph counts given fine-grained protection for social edges. We also analyze upper bounds on the estimation error, including k-stars and triangle counts, and show their superior performance compared with the state-of-the-arts. Finally, we perform extensive experiments on two real social graph datasets and demonstrate that the proposed mechanisms satisfying FGR-DP have better utility than the state-of-the-art mechanisms due to the finer-grained protection.