Graph theory has been extensively used to investigate brain network topology and its changes in disease cohorts. However, many graph theoretic analysis-based brain network studies focused on the shortest paths or, more generally, cost-efficiency. In this work, we use two new concepts, connectedness and 2-connectedness, to measure different global properties compared to the previously widely adopted ones. We apply them to unravel interesting characteristics in the brain, such as redundancy design and further conduct a time-varying brain functional network analysis for characterizing the progression of Alzheimer's disease (AD). Specifically, we define different connectedness and 2-connectedness states and evaluate their dynamics in AD and its preclinical stage, mild cognitive impairment (MCI), compared to the normal controls (NC). Results indicate that, compared to MCI and NC, brain networks of AD tend to be more frequently connected at a sparse level. For MCI, we found that their brains are more likely to be 2-connected in the minimal connected state as well indicating increasing redundancy in brain connectivity. Such a redundant design could ensure maintained connectedness of the MCI's brain network in the case that pathological damages break down any link or silenced any node, making it possible to preserve cognitive abilities. Our study suggests that the redundancy in the brain functional chronnectome could be altered in the preclinical stage of AD. The findings can be successfully replicated in a retest study and with an independent MCI dataset. Characterizing redundancy design in the brain chronnectome using connectedness and 2-connectedness analysis provides a unique viewpoint for understanding disease affected brain networks.© 2021. The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.