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This paper proposes a maximum-flow-based complex network approach for the analysis of the vulnerability of power systems. A new centrality index is proposed, taking into consideration the maximum flow from the source (generator) nodes to the sink (load) nodes, for assessing the network. The Max-Flow Min-Cut Theorem, also known as Ford-Fulkerson Theorem, is used for evaluating the capacity of links. The proposed methodology is then used to identify vulnerable lines of the IEEE 118 bus system and its effectiveness is demonstrated through simulation studies.