Recent functional magnetic resonance imaging (fMRI) studies on autism spectrum condition (ASC) have identified dysfunctions in specific brain networks involved in social and non-social cognition that persist into adulthood. Although increasing numbers of fMRI studies have revealed atypical functional connectivity in the adult ASC brain, such functional alterations at the network level have not yet been fully characterized within the recently developed graph-theoretical framework. Here, we applied a graph-theoretical analysis to resting-state fMRI data acquired from 46 adults with ASC and 46 age- and gender-matched controls, to investigate the topological properties and organization of autistic brain network. Analyses of global metrics revealed that, relative to the controls, participants with ASC exhibited significant decreases in clustering coefficient and characteristic path length, indicating a shift towards randomized organization. Furthermore, analyses of local metrics revealed a significantly altered organization of the hub nodes in ASC, as shown by analyses of hub disruption indices using multiple local metrics and by a loss of "hubness" in several nodes (e.g., the bilateral superior temporal sulcus, right dorsolateral prefrontal cortex, and precuneus) that are critical for social and non-social cognitive functions. In particular, local metrics of the anterior cingulate cortex consistently showed significant negative correlations with the Autism-Spectrum Quotient score. Our results demonstrate altered patterns of global and local topological properties that may underlie impaired social and non-social cognition in ASC.