Scientists from the University of Oklahoma and University of California at Berkeley have developed a framework iDIRECT (Inference of Direct and Indirect Relationships with Effective Copula-based Transitivity) for disentangling direct and indirect relationships in association networks.

Despite the importance of networks in understanding the behavior and features of complex systems, recreating networks from large-scale experimental data is intricate.

In systems biology and microbial ecology, i.e., the study of microbes in the environment and their interactions with each other, the difficulty of unwinding direct and indirect relationships, or the capacity of one element in a system to influence another, either directly or indirectly, can exacerbate the complexities of reconstructing these networks.

“By tackling several mathematical challenges, this study provides a conceptual framework for disentangling direct and indirect relationships in association networks,” said Jizhong Zhou, leading the research team and currently the director of the Institute for Environmental Genomics at the University of Oklahoma. “The application of our iDIRECT framework to synthetic gene expression and microbial community data demonstrates that the framework is a powerful, robust, and reliable tool for network inference. It will greatly enhance our capability to discern network interactions in various complex systems and will allow scientists to address research questions which could not be approached previously.”

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 ikij. (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.
Image Source: Disentangling direct from indirect relationships in association networks

The iDIRECT framework, in particular, decreases network reconstruction’s mathematical problems, such as ill-conditioning, self-looping, and interaction strength overload. The researchers exhibited significant prediction accuracies with the iDIRECT framework using simulation data as a benchmark. 

“Application of iDIRECT to reconstruct gene regulatory networks in Escherichia coli (one of the most common types of bacteria), also revealed considerable prediction power,” said Mary Firestone, the project collaborator at University of California at Berkeley. “In addition, applying iDIRECT to highly diverse grassland soil microbial communities in response to climate warming showed that the iDIRECT-processed networks were more complex under warming than under control conditions and more robust to both random and target species removal.”

“As a general approach, iDIRECT has advantages in that it should be widely applicable to infer direct relationships in association networks across diverse disciplines in science and engineering,” she added.

Story Source: Xiao, N., Zhou, A., Kempher, M. L., Zhou, B. Y., Shi, Z. J., Yuan, M., … & Zhou, J. (2022). Disentangling direct from indirect relationships in association networks. Proceedings of the National Academy of Sciences119(2). DOI: 10.1073/pnas.2109995119

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