The produced fluid of oil well is transported to the gathering station through a multi-channel gathering and transportation pipeline network. During this process, a change of the production conditions of oil well is able to reshape the flow, flow direction, and pressure distribution across the gathering network, which consequently influences the output of oil well as well as the service life of pipelines. Therefore, it is of great significance to develop a calculation method for determining the aforementioned parameter across the gathering network. In practice, multi-port wells are publicly connected though a gathering pipe network, which has different configurations including dendritic, ring, and a combination. Compared with dendritic pipeline network, the flow, flow direction, and pressure distribution in the ring system are difficult to be calculated since such a system owns a complicated topological structure together with an uncertain flow rate in the common pipe sections. In this study, a novel calculation method based on graph theory is proposed. More specifically, the network topology is firstly stored by using an adjacency list, then the breadth-first search traversal is performed to iteratively calculate the flow, flow direction, and pressure distribution. Based on the proposed method, the associated error of calculation is small comparing with the actual production condition, thus, this study can serve as a valuable guideline for field applications.