The rise of the Internet of Things and edge computing has shifted computing resources closer to end-users, benefiting numerous delay-sensitive, computation-intensive applications. To speed up computation, distributed computing is a promising technique that allows parallel execution of tasks across multiple compute nodes. However, current research predominantly revolves around the master-worker paradigm, limiting resource sharing within one-hop neighborhoods. This limitation can render distributed computing ineffective in scenarios with limited nearby resources or constrained/dynamic connectivity. In this paper, we address this limitation by introducing a new distributed computing framework that extends resource sharing beyond one-hop neighborhoods through exploring layered network structures. Our framework involves transforming the network graph into a sink tree and formulating a joint optimization problem based on the layered tree structure for task allocation and scheduling. To solve this problem, we propose two exact methods that find optimal solutions and three heuristic strategies to improve efficiency and scalability. The performances of these methods are analyzed and evaluated through theoretical analyses and comprehensive simulation studies. The results demonstrate their promising performances over the traditional distributed computing and computation offloading strategies.