Alzheimer's disease (AD) is a progressive and disabling neurodegenerative disorder that accounts for approximately 50% to 80% of all dementia cases. AD is histopathologically defined by the presence of amyloid-β plaques and tau-related neurofibrillary tangles 
. These plaques and tangles have been associated with local synaptic dysfunction, suggesting that AD is a dysconnectivity disease 
. In addition, specific patterns of cortical atrophy have been associated with AD, including memory related structures such as the hippocampus and other medial temporal lobe regions, and also the precuneus, cingulate and prefrontal areas (e.g., 
). However, clinical phenotypes often present with more complex cognitive deficits besides memory complaints and this might not be fully explained by atrophy patterns alone 
. Precise associations between clinical phenotypes and different pathological processes are difficult to study, partly because local functional and structural disruptions can have unpredictable, widespread effects in complexly interconnected brain networks 
. Graph theory provides tools to investigate the connectivity structure of such complex networks (i.e., graphs) that can be obtained with functional and also structural neuroimaging techniques 
Coordinated patterns of cortical morphology in structural magnetic resonance imaging (sMRI) scans have been described as graphs (e.g., 
). The nodes in these structural graphs represent cortical areas that are considered to be connected when they covary in thickness or volume across subjects 
, or when they show structural similarity within single-subjects 
. Such graphs can be concisely quantified with graph theoretical properties. In agreement with other types of brain graphs (e.g., functional graphs that are derived from functional synchronisation patterns or anatomical graphs that are derived from diffusion tensor imaging, i.e., DTI) structural graphs have a non-trivial organisation of connectivity. Importantly, several properties of structural graphs are altered in AD 
. Furthermore, these alterations are heterogeneously distributed across the brain, indicating that specific cortical areas contribute more to disease modifications of global network measurements. Structural graph disturbances have been interpreted to reflect decreased information processing efficiency, possibly mirroring functional disruptions in AD. However, the methodology of these previous grey matter graph studies restricted the investigation of graph properties to group-level analyses, and therefore the relationship between grey matter graph alterations and disease severity in individual patients still remains to be established.
Furthermore, while most functional studies have reported that graph topologies move towards more random connectivity configurations in AD 
, it is still unclear whether this also occurs in grey matter graphs. It has been proposed that the loss of highly interconnected areas renders graph topologies more random 
. Moreover, highly interconnected areas might be specifically targeted by the disease, because these areas have been associated with increased amyloid-β deposition in AD 
and with increased vulnerability for activity-dependent degeneration 
. Examining these topological alterations in single-subject grey matter graphs might provide more insight into the association of functional disruptions and coordinated changes in cortical morphology.
The present study addresses these questions by investigating graph properties of single-subject
grey matter graphs for the first time in AD using a recently developed method 
. Presently, we expected that if structural graphs are related to functional disruptions then AD structural graphs would be characterised by a more random topology than those from control subjects. We also expected that the contribution of local disruptions would be heterogeneously distributed across the cortex, with preferential involvement of highly interconnected areas. Furthermore, we hypothesised that within the AD group, a more random graph topology would be related to more severe cognitive decline.