Chromosome 22q11.2 deletion syndrome (22q11DS) is a genetic disease known to lead to cerebral structural alterations, which we study using the framework of the macroscopic white-matter connectome. We create weighted connectomes of 44 patients with 22q11DS and 44 healthy controls using diffusion tensor magnetic resonance imaging, and perform a weighted graph theoretical analysis. After confirming global network integration deficits in 22q11DS (previously identified using binary connectomes), we identify the spatial distribution of regions responsible for global deficits. Next, we further characterize the dysconnectivity of the deficient regions in terms of sub-network properties, and investigate their relevance with respect to clinical profiles. We define the subset of regions with decreased nodal integration (evaluated using the closeness centrality measure) as the affected core (A-core) of the 22q11DS structural connectome. A-core regions are broadly bilaterally symmetric and consist of numerous network hubs - chiefly parietal and frontal cortical, as well as subcortical regions. Using a simulated lesion approach, we demonstrate that these core regions and their connections are particularly important to efficient network communication. Moreover, these regions are generally densely connected, but less so in 22q11DS. These specific disturbances are associated to a rerouting of shortest network paths that circumvent the A-core in 22q11DS, "de-centralizing" the network. Finally, the efficiency and mean connectivity strength of an orbito-frontal/cingulate circuit, included in the affected regions, correlate negatively with the extent of negative symptoms in 22q11DS patients, revealing the clinical relevance of present findings. The identified A-core overlaps numerous regions previously identified as affected in 22q11DS as well as in schizophrenia, which approximately 30-40% of 22q11DS patients develop.