Examination of the cortico-cortical network of mammals has unraveled key topological features and their role in the function of the healthy and diseased brain. Recent findings from social and biological networks pinpoint the significant role of weak connections in network coherence and mediation of information from segregated parts of the network. In the current study, inspired by such findings and proposed architectures pertaining to social networks, we examine the structure of weak connections in the macaque cortico-cortical network by employing a tract-tracing dataset. We demonstrate that the cortico-cortical connections as a whole, as well as connections between segregated communities of brain areas, comply with the architecture suggested by the so-called strength-of-weak-ties hypothesis. However, we find that the wiring of these connections is not optimal with respect to the aforementioned architecture. This configuration is not attributable to a trade-off with factors known to constrain brain wiring, i.e., wiring cost and efficiency. Lastly, weak connections, but not strong ones, appear important for network cohesion. Our findings relate a topological property to the strength of cortico-cortical connections, highlight the prominent role of weak connections in the cortico-cortical structural network and pinpoint their potential functional significance. These findings suggest that certain neuroimaging studies, despite methodological challenges, should explicitly take them into account and not treat them as negligible.