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In his 2018 paper, Herlihy introduced an atomic protocol for multi-party asset swaps across different blockchains. Practical implementation of this protocol is hampered by its intricacy and computational complexity, as it relies on elaborate smart contracts for asset transfers, and specifying the protocol’s steps on a given digraph requires solving an NP-hard problem of computing longest paths. Herlihy left open the question whether there is a simple and efficient protocol for cross-chain asset swaps in arbitrary digraphs. Addressing this, we study HTLC-based protocols, in which all asset transfers are implemented with standard hashed time-lock smart contracts (HTLCs). Our main contribution is a full characterization of swap digraphs that have such protocols, in terms of so-called reuniclus graphs. We give an atomic HTLC-based protocol for reuniclus graphs. Our protocol is simple and efficient. We then prove that non-reuniclus graphs do not have atomic HTLC-based swap protocols. 

Brain age (BA), distinct from chronological age (CA), can be estimated from MRIs to evaluate neuroanatomic aging in cognitively normal (CN) individuals. BA, however, is a cross-sectional measure that summarizes cumulative neuroanatomic aging since birth. Thus, it conveys poorly recent or contemporaneous aging trends, which can be better quantified by the (temporal) pace P of brain aging. Many approaches to map P, however, rely on quantifying DNA methylation in whole-blood cells, which the blood–brain barrier separates from neural brain cells. We introduce a three-dimensional convolutional neural network (3D-CNN) to estimate P noninvasively from longitudinal MRI. Our longitudinal model (LM) is trained on MRIs from 2,055 CN adults, validated in 1,304 CN adults, and further applied to an independent cohort of 104 CN adults and 140 patients with Alzheimer’s disease (AD). In its test set, the LM computes P with a mean absolute error (MAE) of 0.16 y (7% mean error). This significantly outperforms the most accurate cross-sectional model, whose MAE of 1.85 y has 83% error. By synergizing the LM with an interpretable CNN saliency approach, we map anatomic variations in regional brain aging rates that differ according to sex, decade of life, and neurocognitive status. LM estimates of P are significantly associated with changes in cognitive functioning across domains. This underscores the LM’s ability to estimate P in a way that captures the relationship between neuroanatomic and neurocognitive aging. This research complements existing strategies for AD risk assessment that estimate individuals’ rates of adverse cognitive change with age.