Interactome networks represent sets of possible physical interactions between proteins. They lack spatio-temporal information by construction. However, the specialized functions of the differentiated cell types which are assembled into tissues or organs depend on the combinatorial arrangements of proteins and their physical interactions. Is tissue-specificity, therefore, encoded within the interactome? In order to address this question, we combined protein-protein interactions, expression data, functional annotations and interactome topology. We first identified a subnetwork formed exclusively of proteins whose interactions were observed in all tested tissues. These are mainly involved in housekeeping functions and are located at the topological center of the interactome. This 'Largest Common Interactome Network' represents a 'functional interactome core'. Interestingly, two types of tissue-specific interactions are distinguished when considering function and network topology: tissue-specific interactions involved in regulatory and developmental functions are central whereas tissue-specific interactions involved in organ physiological functions are peripheral. Overall, the functional organization of the human interactome reflects several integrative levels of functions with housekeeping and regulatory tissue-specific functions at the center and physiological tissue-specific functions at the periphery. This gradient of functions recapitulates the organization of organs, from cells to organs. Given that several gradients have already been identified across interactomes, we propose that gradients may represent a general principle of protein-protein interaction network organization.