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The multivariate and discrete data are commonly used to monitor product defects and epidemic diseases. It is difficult to model their complex structure and design a suitable control chart to monitor them in the area of statistical process control. To monitor the tri-variate Poisson process, this paper establishes a one-parameter copula function to describe the spatial correlation and designs a control chart based on the log-likelihood ratio test. The Markov chain is employed to approximate the average run length and to measure the performance of the control chart. Simulation results show that the proposed chart is efficient for detecting upward shifts and can achieve better monitoring performances when the target mean shift in control chart design is equal to a true mean shift. Compared with D chart, the proposed chart achieves a better performance when the correlation level is high.

Although physical exercise has been recommended as a useful means of enhancing the mental health of adolescents, the exact mechanisms through which physical exercise plays a role are unclear. Both physical exercise and mental health are complex concepts with multiple facets, and traditional methods may constrain the manifestations of their mapping relationships. This research aimed to find the bridging connections between physical exercise and mental health. Mental health and physical exercise behaviors were assessed using the Symptom Checklist 90 (SCL‐90) and the Adolescent Physical Activity Questionnaire (PAQ‐A) in 9072 Chinese adolescents, respectively. Network analysis was utilized to construct the mental health‐physical exercise network and to analyze the relationships between individual physical exercise behaviors and mental health symptoms. Core and bridging nodes were identified based on expected influence (EI) and bridge expected influence (BEI). Gender differences were also examined. The results revealed specific and distinct pathways between physical exercise and mental health (e.g., winter sports–obsessive‐compulsive symptoms, winter sports–phobia). For both males and females, anxiety, depression, interpersonal sensitivity, ball sports, and evening activity were the most central symptoms/behaviors, reflecting their relative significance in their respective associations. The nodes with the highest BEI were obsessive‐compulsive symptoms and physical education, showing negative associations with nodes in the other community. Furthermore, in the male group, somatization and winter sports stood out as the most positive bridge nodes. Conversely, in the female group, interpersonal sensitivity and sports games were the most positive bridge nodes. These findings illuminate the pathways linking physical exercise and mental health, supporting the implementation of physical exercise in a more elaborate way. [ABSTRACT FROM AUTHOR]

In many real-world networks, interactions between nodes are weighted to reflect their strength, such as predator–prey interactions in the ecological network and passenger numbers in airline networks. These weighted networks are prone to cascading effects caused by minor perturbations, which can lead to catastrophic outcomes. This vulnerability highlights the importance of studying weighted network resilience to prevent system collapses. However, due to many variables and weight parameters coupled together, predicting the behavior of such a system governed by a multi-dimensional rate equation is challenging. To address this, we propose a dimension reduction technique that simplifies a multi-dimensional system into a one-dimensional state space. We applied this methodology to explore the impact of weights on the resilience of four dynamics whose weights are assigned by three weight assignment methods. The four dynamical systems are the biochemical dynamical system (B), the epidemic dynamical system (E), the regulatory dynamical system (R), and the birth–death dynamical system (B D). The results show that regardless of the weight distribution, for B , the weights are negatively correlated with the activities of the network, while for E , R , and B D , there is a positive correlation between the weights and the activities of the network. Interestingly, for B , R , and B D , the change in the weights of the system has little impact on the resilience of the system. However, for the E system, the greater the weights the more resilient the system. This study not only simplifies the complexity inherent in weighted networks but also enhances our understanding of their resilience and response to perturbations. [ABSTRACT FROM AUTHOR]