Evidence for interregional structural asymmetries has been previously reported for brain anatomic regions supporting well-described functional lateralization. Here, we aimed to investigate whether the two brain hemispheres demonstrate dissimilar general structural attributes implying different principles on information flow management. Common left hemisphere/right hemisphere structural network properties are estimated and compared for right-handed healthy human subjects and a nonhuman primate, by means of 3 different diffusion-weighted magnetic resonance imaging fiber tractography algorithms and a graph theory framework. In both the human and the nonhuman primate, the data support the conclusion that, in terms of the graph framework, the right hemisphere is significantly more efficient and interconnected than the left hemisphere, whereas the left hemisphere presents more central or indispensable regions for the whole-brain structural network than the right hemisphere. From our point of view, in terms of functional principles, this pattern could be related with the fact that the left hemisphere has a leading role for highly demanding specific process, such as language and motor actions, which may require dedicated specialized networks, whereas the right hemisphere has a leading role for more general process, such as integration tasks, which may require a more general level of interconnection.