How does brain network architecture balance cooperation and competition between distributed circuits? Here we use computational whole-brain modeling to examine the dynamical and computational relevance of cooperative and competitive interactions in the mammalian connectome. Across human, macaque and mouse, we show that to faithfully reproduce brain activity, model architecture consistently combines modular cooperative interactions with diffuse, long-range competitive interactions. Across species, competitive interactions preferentially link regions characterized by opposite profiles of cytoarchitecture, gene expression and receptor expression. The model with competitive interactions provides superior subject specificity, consistently outperforming the cooperative-only model and exhibiting excellent fit to the spatiotemporal properties of the living brain. These properties were not explicitly optimized, instead emerging spontaneously. Competitive interactions in the generative connectivity produce more synergistic and hierarchical dynamics, leading to enhanced performance for neuromorphic computing. Altogether, this work provides a generative link among network architecture, dynamical properties and computational performance in the mammalian brain.