Essential proteins are those that are indispensable to cellular survival and development. Existing methods for essential protein identification generally rely on knock-out experiments and/or the relative density of their interactions (edges) with other proteins in a Protein-Protein Interaction (PPI) network. Here, we present a computational method, called EW, to first rank protein-protein interactions in terms of their Edge Weights, and then identify sub-PPI-networks consisting of only the highly-ranked edges and predict their proteins as essential proteins. We have applied this method to publicly-available PPI data on Saccharomyces cerevisiae (Yeast) and Escherichia coli (E. coli) for essential protein identification, and demonstrated that EW achieves better performance than the state-of-the-art methods in terms of the precision-recall and Jackknife measures. The highly-ranked protein-protein interactions by our prediction tend to be biologically significant in both the Yeast and E. coli PPI networks. Further analyses on systematically perturbed Yeast and E. coli PPI networks through randomly deleting edges demonstrate that the proposed method is robust and the top-ranked edges tend to be more associated with known essential proteins than the lowly-ranked edges.