Human brain connectome studies aim to both explore healthy brains, and extract and analyze relevant features associated with pathologies of interest. Usually this consists of modeling the brain connectome as a graph and using graph metrics as features. A fine brain description requires graph metrics computation at the node level. Given the relatively reduced number of patients in standard cohorts, such data analysis problems fall in the high-dimension, low-sample-size framework. In this context, our goal is to provide a machine learning technique that exhibits flexibility, gives the investigator an understanding of the features and covariates, allows visualization and exploration, and yields insight into the data and the biological phenomena at stake. The retained approach is dimension reduction in a manifold learning methodology; the originality is that the investigator chooses one (or several) reduced variables. The proposed method is illustrated in two studies. The first one addresses comatose patients; the second one compares young and elderly populations. The method sheds light on the differences between brain connectivity graphs using graph metrics and potential clinical interpretations of these differences. Author Summary Human brain connectome studies aim at both exploring healthy brains, and analyzing relevant features associated to pathologies of interest. This consists in modeling the brain connectome as a graph and in using graph metrics as features. Such data analysis problems fall in the high-dimension low sample size framework. Our goal is to provide a machine learning technique that exhibits flexibility, gives the investigator grip on the features and covariates, allows visualization, and yields insight into the biological phenomena at stake. The retained approach is dimension reduction in a manifold learning methodology, the originality lying in that several reduced variables be chosen by the investigator. The method sheds light on the differences between brain connectivity graphs using graph metrics and potential clinical interpretations of theses differences.