The increasing amount of graph-structured data catalyzes analytics over graph databases using semantic queries. Motivated by the ubiquity of commercial cloud platforms, data owners are willing to store their graph databases remotely. However, data privacy has emerged as a widespread concern since the cloud platforms are not fully trusted. One viable solution is to encrypt sensitive data before outsourcing, which inevitably hinders data retrieval. To enable queries over encrypted data, searchable symmetric encryption (SSE) has been introduced. Yet, the most well-studied class of SSE schemes focuses on retrieving textual files given keywords, which cannot be applied to graph databases directly. This paper extends our preliminary work (FC′17) and proposes GraphShield, a structured encryption scheme for graphs. Beyond shortest distance queries, GraphShield can support other classic graph-based queries (e.g., maximum flow) and more complicated analytics (e.g., PageRank). Technically, we incorporate a suite of (efficient) cryptographic primitives and tailor some extra secure protocols for facilitating graph analytics. Our scheme also allows updates on the encrypted graph with forward privacy guaranteed. We formalize the security model and prove the adaptive security with reasonable leakage. Finally, we implement our scheme on various real-world datasets, and the experiment results demonstrate its practicality and scalability.