In this paper, we consider the problem of localizing disturbance inputs in first-order linear time-invariant (LTI) consensus networks using measurement-based graph-theoretic methods. We consider every node and edge of the network graph to be characterized with physical weights, and show that the resulting system dynamics can be represented in terms of an asymmetric Laplacian matrix L-m. Assuming the network graph to be divided into p coherent clusters, we next propose an input localization algorithm based on the properties of the weak nodal domains corresponding to the first p - 1 slow eigenvalues of L-m. The algorithm takes in sensor measurements of the states from selected nodes, runs a system identification routine to construct the input-output transfer matrix, and compares the signs of the residues of the component transfer functions to a nominal localization key to determine in which cluster(s) the disturbance input may have entered. We prove that for systems defined over a specific class of graphs, referred to as p-area complete graphs, the localization is unique. We also state the extension of this result for second-order synchronization networks. We illustrate the algorithms by applying them to large-scale power system networks.