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. A team led by researchers at the University of Oklahoma has developed 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.
Overview of iDIRECT
(A) An association network contains both direct (blue) and indirect (red) associations. Indirect associations include spurious links (solid lines) and overestimated direct links (dotted lines). (B) iDIRECT uses a copula-based addition ⊕ to combine association between two nodes through different paths, ensuring the interaction strengths to be within the range [0,1]. (C) iDIRECT introduces a transitivity matrix Ti,kj (association between k and j excluding paths passing i) and uses SikTi,kj to calculate indirect association strength between i and j, eliminating spurious self-looping paths like i–k–i–j. (D) iDIRECT uses nonlinear solvers to obtain direct association strengths of each link, without inverting the ill-conditioned association matrix. (E) Overall workflow for iDIRECT.