Does each cognitive task elicit a new cognitive network each time in the brain? Recent data suggest that pre-existing repertoires of a much smaller number of canonical network components are selectively and dynamically used to compute new cognitive tasks. To this end, we propose a novel method (graph-ICA) that seeks to extract these canonical network components from a limited number of resting state spontaneous networks. Graph-ICA decomposes a weighted mixture of source edge-sharing subnetworks with different weighted edges by applying an independent component analysis on cross-sectional brain networks represented as graphs. We evaluated the plausibility in our simulation study and identified 49 intrinsic subnetworks by applying it in the resting state fMRI data. Using the derived subnetwork repertories, we decomposed brain networks during specific tasks including motor activity, working memory exercises, and verb generation, and identified subnetworks associated with performance on these tasks. We also analyzed sex differences in utilization of subnetworks, which was useful in characterizing group networks. These results suggest that this method can effectively be utilized to identify task-specific as well as sex-specific functional subnetworks. Moreover, graph-ICA can provide more direct information on the edge weights among brain regions working together as a network, which cannot be directly obtained through voxel-level spatial ICA.