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Against the background of seeking to achieve carbon neutrality, relationships among renewable-energy companies around the world have become multiple and complex. In this work, the Pearson, Kendall, tail, and partial correlation coefficients were applied to 51 global companies—including solar and wind firms, independent power plants, and utilities—to explore the linear, nonlinear, extreme-risk, and direct relations between them. Sample data from 7 August 2015 to 6 August 2021 were considered, and three sub-periods were extracted from these sample data by analysis of the evolution of multiple correlations combined with event analysis. A four-layer correlation network model was then constructed. The main results are as follows. (1) The multiple relations among the selected firms underwent dramatic changes during two external shocks (the China–US trade war and the COVID-19 pandemic). (2) The extreme-risk network layer verified that the trade war mainly affected the relationships among companies in the solar industries of China and the US. (3) During the COVID-19 pandemic period, the linear and direct relationships among wind firms from Canada, Spain, and Germany were significantly increased. In this sub-period, edge-weight distributions of the four different layers were heterogeneous and varied from power-law features to Gaussian distributions. (4) During all the sub-periods, most companies had similar numbers of neighbors, while the numbers of neighbors of a few companies varied greatly in the four different layers. These findings provide a useful reference for stakeholders and may help them understand the connectedness and evolution of global renewable-energy markets. [ABSTRACT FROM AUTHOR]