Modern graphs are large, often containing billions of nodes and edges that demand huge amount of processing for analysis purposes. The algorithms processing these graphs often run for long time and consume substantial amount of energy. However, not all edges in the graphs are equally important. Some edges play critical role in maintaining the community and other interesting structures in the graph, while the rest are less important for analysis. Identifying edges as important and unimportant allows one to apply elastic fidelity computing when processing edges of low importance, hence saving significant amount of energy while processing large graphs. In this paper we propose a novel technique for identifying important edges in a graph using a fast method that exploits locality sensitive hashing. We then propose a framework for energy-efficient computing that applies elastic fidelity computing when processing edges of low importance and applies full fidelity computing when processing important edges. This allows the framework to deliver good results while saving energy when processing a large number of low-importance edges. Our proposed technique reduces the power consumption by 3-30% while still producing results that are within acceptable range of the full-accuracy results.