This study proposes a complex networks-based method to determine the connections among the stations in a streamflow monitoring network and assess the importance of the individual stations. For implementation, 13 streamflow stations in the Pyeongchang River basin in South Korea are studied, and daily flow (discharge) data are analyzed. Three different centrality measures are employed to identify the connections in the streamflow network: degree centrality, closeness centrality, and betweenness centrality. The links between the nodes can significantly change depending upon the centrality method used and the threshold considered. Therefore, an integrated centrality method is proposed using a Bayesian network. The integrated centrality results show that stations situated along the main stream in the middle of the basin have high centrality, while the tributary stations have low centrality. To assess the importance of stations, the integrated centrality is used with community-based clusters. This assessment on the importance of the individual streamflow stations through their centrality is useful to establish strategies for their effective and efficient maintenance.