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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. 

An analysis of the structural dynamic response under uncertainty is presented. Uncertainties in load and material are modelled as intervals exploiting the interval finite element method (IFEM). To reduce overestimation and increase the computational efficiency of the solution, we do not solve the dynamic problem by an explicit step-by-step time integration scheme. Instead, our approach solves for the structural variables in the whole time domain simultaneously by an implicit scheme using discrete Fourier transform and its inverse (DFT and IDFT). Non-trivial initial conditions are handled by modifying the right-hand side of the governing equation. To further reduce overestimation, a new decomposition strategy is applied to the IFEM matrices, and both primary and derived quantities are solved simultaneously. The final solution is obtained using an iterative enclosure method, and in our numerical examples the exact solution is enclosed at minimal computational cost. 

We present an interval-based approach for parameter identification in structural static problems. Our inverse formulation models uncertainties in measurement data as interval and exploits the Interval Finite Element Method (IFEM) combined with adjoint-based optimization. The inversion consists of a two-step algorithm: first, an estimate of the parameters is obtained by a deterministic iterative solver. Then, the algorithm switches to the interval extension of the previous solver, using the deterministic estimate of the parameters as an initial guess. The formulation is illustrated in solutions of various numerical examples showing how the guaranteed interval enclosures always contain Monte Carlo predictions.