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Sustainable development is a practical path to optimize industrial structures and enhance investment efficiency. Investigating risk contagion within ESG industries is a crucial step towards reducing systemic risks and fostering the green evolution of the economy. This research constructs ESG industry indices, taking into account the possibility of extreme tail risks, and employs VaR and CoVaR as measures of tail risk. The TENET network approach is integrated to to capture the structural evolution and direction of information flow among ESG industries, employing information entropy to quantify the topological characteristics of the network model, exploring the risk transmission paths and evolution patterns of ESG industries in an extreme tail risk event. Finally, Mantel tests are conducted to examine the existence of significant risk spillover effects between ESG and traditional industries. The research finds strong correlations among ESG industry indices during stock market crash, Sino–US trade frictions, and the COVID-19 pandemic, with industries such as the COAL, CMP, COM, RT, and RE playing key roles in risk transmission within the network, transmitting risks to other industries. Affected by systemic risk, the information entropy of the TENET network significantly decreases, reducing market information uncertainty and leading market participants to adopt more uniform investment strategies, thus diminishing the diversity of market behaviors. ESG industries show resilience in the face of extreme risks, demonstrating a lack of significant risk contagion with traditional industries. [ABSTRACT FROM AUTHOR]

Shannon entropy is used to measure information uncertainty, while the information dimension is used to measure information complexity. Given two probability distributions, the difference can be measured by relative entropy. However, the existing relative entropy does not consider the effect of information dimension. To improve the existing entropy, a new relative entropy is presented in this paper. The information fractal dimension is considered in the proposed relative entropy. The new relative entropy is more generalized than the initial relative entropy. When dimension is not considered, it will degenerate to the initial relative entropy. Another interesting point is that the new relative entropy may have negative values when calculating. The physical meaning is still under exploration. Finally, some application examples are provided to exemplify the utilization of the proposed relative entropy. [ABSTRACT FROM AUTHOR]