Large graph datasets have become invaluable assets for studying problems in business applications and scientific research. These datasets, collected and owned by data owners, may also contain privacy-sensitive information. When using public clouds for elastic processing, data owners have to protect both data ownership and privacy from curious cloud providers. We propose a cloud-centric framework that allows data owners to efficiently collect graph data from the distributed data contributors, and privately store and analyze graph data in the cloud. Data owners can conduct expensive operations in untrusted public clouds with privacy and scalability preserved. The major contributions of this work include two privacy-preserving approximate eigen decomposition algorithms (the secure Lanczos and Nystrom methods) for spectral analysis of large graph matrices, and a personalized privacy-preserving data submission method based on differential privacy that allows for the trade-off between data sparsity and privacy. For a N-node graph, the proposed approach allows a data owner to finish the core operations with only O(N) client-side costs in computation, storage, and communication. The expensive O(N²) operations are performed in the cloud with the proposed privacy-preserving algorithms. We prove that our approach can satisfactorily preserve data privacy against the untrusted cloud providers. We have conducted an extensive experimental study to investigate these algorithms in terms of the intrinsic relationships among costs, privacy, scalability, and result quality.