Connectome network analysis across multiple species should help identify principles of brain function. Here, we examined three fundamental properties-global efficiency, global betweenness centrality, and global clustering-in the mesoscale tract-tracing data of the mouse connectome; and conducted vulnerability analysis to identify the critical regions and connections based on the loss in network function when each brain region (213) and connection (16,594) was removed. Robustness tests examining noise effects were also conducted. There were five key findings. First, we identified eight critical regions and 38 critical connections, with more central, limbic regions dominant; and with robustness analysis showing (a) the importance of connection strength; and (b) the findings being robust to noise. Second, although critical regions and connections were significantly based on their local network properties, global influences sometimes deviated from local ones (e.g., critical globally but with lower local scores), thereby revealing global-level interactions. Third, the critical components organized into two main pathways (one from piriform cortex to globus pallidus; the other, entorhinal cortex to the amygdala), and two main clusters (centred on caudoputamen and entorhinal cortex). Fourth, for brain function, all main categories from perception to action were represented: e.g., olfaction (piriform cortex), learning and memory (entorhinal cortex), affect (amygdala and caudoputamen), and cognitive and motor processing (caudoputamen, globus pallidus). Finally, the claustrum was intriguingly identified as critical, perhaps for information integration and motor translation. Vulnerability analysis provides a unique approach to characterizing the fundamental structure of nervous systems.