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Networks are vital tools for understanding and modeling interactions in complex systems in science and engineering, and direct and indirect interactions are pervasive in all types of networks. However, quantitatively disentangling direct and indirect relationships in networks remains a formidable task. Here, we present a framework, called iDIRECT (Inference of Direct and Indirect Relationships with Effective Copula-based Transitivity), for quantitatively inferring direct dependencies in association networks. Using copula-based transitivity, iDIRECT eliminates/ameliorates several challenging mathematical problems, including ill-conditioning, self-looping, and interaction strength overflow. With simulation data as benchmark examples, iDIRECT showed high prediction accuracies. Application of iDIRECT to reconstruct gene regulatory networks in Escherichia coli also revealed considerably higher prediction power than the best-performing approaches in the DREAM5 (Dialogue on Reverse Engineering Assessment and Methods project, #5) Network Inference Challenge. In addition, applying iDIRECT to highly diverse grassland soil microbial communities in response to climate warming showed that the iDIRECT-processed networks were significantly different from the original networks, with considerably fewer nodes, links, and connectivity, but higher relative modularity. Further analysis revealed that the iDIRECT-processed network was more complex under warming than the control and more robust to both random and target species removal ( P < 0.001). As a general approach, iDIRECT has great advantages for network inference, and it should be widely applicable to infer direct relationships in association networks across diverse disciplines in science and engineering. 

ABSTRACT Tundra ecosystems are typically carbon (C) rich but nitrogen (N) limited. Since biological N 2 fixation is the major source of biologically available N, the soil N 2 -fixing (i.e., diazotrophic) community serves as an essential N supplier to the tundra ecosystem. Recent climate warming has induced deeper permafrost thaw and adversely affected C sequestration, which is modulated by N availability. Therefore, it is crucial to examine the responses of diazotrophic communities to warming across the depths of tundra soils. Herein, we carried out one of the deepest sequencing efforts of nitrogenase gene ( nifH ) to investigate how 5 years of experimental winter warming affects Alaskan soil diazotrophic community composition and abundance spanning both the organic and mineral layers. Although soil depth had a stronger influence on diazotrophic community composition than warming, warming significantly ( P <  0.05) enhanced diazotrophic abundance by 86.3% and aboveground plant biomass by 25.2%. Diazotrophic composition in the middle and lower organic layers, detected by nifH sequencing and a microarray-based tool (GeoChip), was markedly altered, with an increase of α-diversity. Changes in diazotrophic abundance and composition significantly correlated with soil moisture, soil thaw duration, and plant biomass, as shown by structural equation modeling analyses. Therefore, more abundant diazotrophic communities induced by warming may potentially serve as an important mechanism for supplementing biologically available N in this tundra ecosystem. IMPORTANCE With the likelihood that changes in global climate will adversely affect the soil C reservoir in the northern circumpolar permafrost zone, an understanding of the potential role of diazotrophic communities in enhancing biological N 2 fixation, which constrains both plant production and microbial decomposition in tundra soils, is important in elucidating the responses of soil microbial communities to global climate change. A recent study showed that the composition of the diazotrophic community in a tundra soil exhibited no change under a short-term (1.5-year) winter warming experiment. However, it remains crucial to examine whether the lack of diazotrophic community responses to warming is persistent over a longer time period as a possibly important mechanism in stabilizing tundra soil C. Through a detailed characterization of the effects of winter warming on diazotrophic communities, we showed that a long-term (5-year) winter warming substantially enhanced diazotrophic abundance and altered community composition, though soil depth had a stronger influence on diazotrophic community composition than warming. These changes were best explained by changes in soil moisture, soil thaw duration, and plant biomass. These results provide crucial insights into the potential factors that may impact future C and N availability in tundra regions.