Disentangling direct from indirect relationships in association networks

Naijia Xiao; Aifen Zhou; Megan L Kempher; Benjamin Y Zhou; Zhou Jason Shi; Mengting Yuan; Xue Guo; Linwei Wu; Daliang Ning; Joy Van Nostrand; Mary K Firestone; Jizhong Zhou

doi:10.1073/pnas.2109995119

Disentangling direct from indirect relationships in association networks

Proc Natl Acad Sci U S A. 2022 Jan 11;119(2):e2109995119. doi: 10.1073/pnas.2109995119.

Authors

Naijia Xiao^{1

2}, Aifen Zhou^{1

2}, Megan L Kempher^{1

2}, Benjamin Y Zhou³, Zhou Jason Shi^{1

2

4

5}, Mengting Yuan^{1

2

6}, Xue Guo^{1

2

7}, Linwei Wu^{1

2}, Daliang Ning^{1

2}, Joy Van Nostrand^{1

2

3

8}, Mary K Firestone⁹, Jizhong Zhou^{10

2

11

12}

Affiliations

¹ Institute for Environmental Genomics, University of Oklahoma, Norman, OK 73019.
² Department of Microbiology and Plant Biology, University of Oklahoma, Norman, OK 73019.
³ Glomics Inc., Norman, OK 73072.
⁴ Data Science and Biotechnology Institute, Gladstone Institutes, University of California, San Francisco, CA 94158.
⁵ Chan Zuckerberg Biohub, San Francisco, CA 94158.
⁶ Department of Environmental Science, Policy, and Management, University of California, Berkeley, CA 94704.
⁷ State Key Joint Laboratory of Environment Simulation and Pollution Control, School of Environment, Tsinghua University, Beijing 100084, China.
⁸ Office of Research and Graduate Studies, Utah State University, Logan, UT 84322.
⁹ Department of Environmental Science, Policy, and Management, University of California, Berkeley, CA 94704; mkfstone@berkeley.edu jzhou@ou.edu.
¹⁰ Institute for Environmental Genomics, University of Oklahoma, Norman, OK 73019; mkfstone@berkeley.edu jzhou@ou.edu.
¹¹ Earth and Environmental Sciences, Lawrence Berkeley National Laboratory, Berkeley, CA 94705.
¹² School of Civil Engineering and Environmental Sciences, University of Oklahoma, Norman, OK 73019.

Abstract

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.

Keywords: climate change; direct relationship; indirect relationship; network analysis; systems biology.

Publication types

Research Support, U.S. Gov't, Non-P.H.S.