The knowledge of the underlying topology is essential for understanding and manipulating power grids, water distribution networks, biological networks. At times, the topology may be reported (or recorded) erroneously, mostly owing to human mistakes in reporting. The networks can be represented as a graph in which entities are represented as nodes and interactions among nodes as edges. This work focuses on the study of a specific type of error in topology that occurs when the incidence of an edge in the network is incorrectly reported. We propose a methodology to detect, isolate, and rectify this type of error using a single noisy measurement of flows along all the edges of a conserved network. We first show that this type of error generates specific error signatures, which enables error diagnosis, when the data is noise-free. An approach based on a series of statistical tests is developed to handle noisy data for online error detection and rectification. Simulation studies are performed to test the robustness of the proposed methodology.