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Recent neuroimaging studies have shown that functional connectomes are unique to individuals, i.e., two distinct fMRIs taken over different sessions of the same subject are more similar in terms of their connectomes than those from two different subjects. In this study, we present new results that identify specific parts of resting state and task-specific connectomes that are responsible for the unique signatures. We show that a very small part of the connectome can be used to derive features for discriminating between individuals. A network of these features is shown to achieve excellent training and test accuracy in matching imaging datasets. We show that these features are statistically significant, robust to perturbations, invariant across populations, and are localized to a small number of structural regions of the brain. Furthermore, we show that for task-specific connectomes, the regions identified by our method are consistent with their known functional characterization. We present a new matrix sampling technique to derive computationally efficient and accurate methods for identifying the discriminating sub-connectome and support all of our claims using state-of-the-art statistical tests and computational techniques.