Learning representation of the enclosing subgraph of node pairs is recognized as an efficient approach for link-oriented prediction tasks in network applications. The core challenge within this subgraph encoding approach is how to effectively distinguish and then properly aggregate the contribution of nodes in the subgraph into a single vector to indicate the relation between the target node pair. In this work, we propose a novel sphere-based subgraph encoding architecture, namely BS-SubGNN, to address the challenge. In detail, we design two key building blocks, including Bicentric Sphere Node Labeling (BSNL) and Bicentric Sphere Subgraph Pooling (BSSP) to assist message passing in BS-SubGNN. BSNL endows each node a label according to the sphere it belongs to in the subgraph to distinguish the contribution of nodes, while BSSP adopts an attention mechanism to aggregate the contribution of nodes in each sphere. Theoretically, we prove that BS-SubGNN can unify existing node distance labeling methods, and yield discriminative node features with less time complexity. We evaluate the performance of BS-SubGNN in link prediction tasks over a variety of network types, including undirected networks, attribute networks, directed networks, and signed directed networks. Our experimental results demonstrate that BS-SubGNN consistently achieves significant performance improvements over the above diverse types of networks. In particular, compared to those methods with a requisite of multi-hop neighborhood information, BS-SubGNN can obtain better performance even when only one-hop neighborhood information of the node pair is utilized.