In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically-showing promising results. The following work investigates GNNs from a theoretical point of view and relates them to the 1-dimensional Weisfeiler-Leman graph isomorphism heuristic (1-WL). We show that GNNs have the same expressiveness as the 1-WL in terms of distinguishing non-isomorphic (sub-)graphs. Hence, both algorithms also have the same shortcomings. Based on this, we propose a generalization of GNNs, so-called k-dimensional GNNs (k-GNNs), which can take higher-order graph structures at multiple scales into account. These higher-order structures play an essential role in the characterization of social networks and molecule graphs. Our experimental evaluation confirms our theoretical findings as well as confirms that higher-order information is useful in the task of graph classification and regression.completing the equivalence. Since the power of the 1-WL has been completely characterized, see, e.g., (Arvind et al. 2015;Kiefer, Schweitzer, and Selman 2015), we can transfer these results to the case of GNNs, showing that both approaches have the same shortcomings.Going further, we leverage these theoretical relationships to propose a generalization of GNNs, called k-GNNs, which are neural architectures based on the k-dimensional WL algorithm (k-WL), which are strictly more powerful than GNNs. The key insight in these higher-dimensional variants is that they perform message passing directly between subgraph structures, rather than individual nodes. This higher-order form of message passing can capture structural information that is not visible at the node-level.Graph kernels based on the k-WL have been proposed in the past (Morris, Kersting, and Mutzel 2017). However, a key advantage of implementing higher-order message passing in GNNs-which we demonstrate here-is that we can design hierarchical variants of k-GNNs, which combine graph representations learned at different granularities in an end-to-end trainable framework. Concretely, in the presented hierarchical approach the initial messages in a k-GNN are based on the output of lower-dimensional k -GNN (with k < k), which allows the model to effectively capture graph structures of varying granularity. Many real-world graphs inherit a hierarchical structure-e.g., in a social network we must model both the ego-networks around individual nodes, as well as the coarse-grained relationships between entire communities, see, e.g., (Newman 2003)-and our experimental results demonstrate that these hierarchical k-GNNs are able to consistently outperform traditional GNNs on a variety of graph classification and regression tasks. Across twelve graph regression tasks from the QM9 benchmark, we find that our hierarchical model reduces the mean absolute error by 54.45% on average. For graph classification, we find that our hierarchical models...

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Glucose tolerance is lower in the evening and at night than in the morning. However, the relative contribution of the circadian system vs. the behavioral cycle (including the sleep/wake and fasting/ feeding cycles) is unclear. Furthermore, although shift work is a diabetes risk factor, the separate impact on glucose tolerance of the behavioral cycle, circadian phase, and circadian disruption (i.e., misalignment between the central circadian pacemaker and the behavioral cycle) has not been systematically studied. Here we show-by using two 8-d laboratory protocols-in healthy adults that the circadian system and circadian misalignment have distinct influences on glucose tolerance, both separate from the behavioral cycle. First, postprandial glucose was 17% higher (i.e., lower glucose tolerance) in the biological evening (8:00 PM) than morning (8:00 AM; i.e., a circadian phase effect), independent of the behavioral cycle effect. Second, circadian misalignment itself (12-h behavioral cycle inversion) increased postprandial glucose by 6%. Third, these variations in glucose tolerance appeared to be explained, at least in part, by different mechanisms: during the biological evening by decreased pancreatic β-cell function (27% lower earlyphase insulin) and during circadian misalignment presumably by decreased insulin sensitivity (elevated postprandial glucose despite 14% higher late-phase insulin) without change in early-phase insulin. We explored possible contributing factors, including changes in polysomnographic sleep and 24-h hormonal profiles. We demonstrate that the circadian system importantly contributes to the reduced glucose tolerance observed in the evening compared with the morning. Separately, circadian misalignment reduces glucose tolerance, providing a mechanism to help explain the increased diabetes risk in shift workers.circadian disruption | shift work | night work | glucose metabolism | diabetes I n healthy humans, there is a strong time-of-day variation in glucose tolerance, with a peak in the morning and a trough in the evening and night (1-6). Understanding the underlying mechanisms of the day/night variation in glucose tolerance is important for diurnally active individuals as well as shift workers, who are at increased risk for developing type 2 diabetes (7-9). The endogenous circadian system and circadian misalignment (i.e., misalignment between the endogenous circadian system and 24-h environmental/behavioral cycles) have been shown to affect glucose metabolism (4,(10)(11)(12)(13)(14). However, the relative and separate importance of the endogenous circadian system and circadian misalignment-after accounting for behavioral cycle effects (including the sleep/wake, fasting/feeding, and physical inactivity/activity cycles, etc.)-on 24-h variation in glucose tolerance is not well understood.Most species have evolved an endogenous circadian timing system that optimally times physiological variations and behaviors relative to the 24-h environmental cycle (15-17). The mammalian circadian system is comp...