The Fork and Pull-Request model is widely used in collaborative development of open source software (OSS), fostering innovation through independent repository copies, but it can also lead to inefficiencies and fragmentation. A key underexplored aspect is the integration effectiveness—the degree to which distributed original commits across forks are effectively integrated back into the main repository. It plays a critical role in OSS project productivity but remains poorly understood. In response, we introduce convergence entropy, a novel metric that quantifies the integration effectiveness by measuring the similarity between distributions of original and merged commits across forks, adjusted for integration ratio. This metric highlights not only the volume of contributions but also their diversity and coordination, offering a unique lens to understand forking practices. Moreover, we explore the relationship between convergence entropy and three dimensions of OSS project productivity, showing significant correlations. We also observe that other factors can alter this dynamic.

In the realm of open source software (OSS) development, the resolution of issues is not just a technical task but a pivotal activity that drives ongoing enhancement and secures a project's sustainability. Contributing to issues is the majority form for external contributors to take part in OSS projects. Although the significance of external contributors is recognized, their contributions in issue process still lacks full quantification and clarity, and the correlation between their contributions and the project productivity remains unclear. In response, we propose issue entropy, a novel metric that quantifies the complexity of event sequence in issue process applied to study the external contributions. The metric applies principles of information theory to scrutinize granular details within a project's issue-related activities, providing a unique lens to understand and assess external contributions. To explore the correlation between external contributions and project productivity, we employed issue entropy as a novel way to examine external contributions, and analyzed its correlation with new bugs, commits, and bug fix time, serving as proxies for OSS project productivity. Our findings reveal a strong positive correlation with new bugs, variable relationship with commit volume based on company sponsorship, and significant negative correlation with bug fix time. We also observe the significant interactions between external contributions and other factors, such as project age and the number of files. Moreover, issue entropy offers a new perspective on the health of the OSS project ecosystem, potentially supporting further research and practical applications.

This work presents four novel entropy-based indices for measuring the community evolution of developer social networks (DSNs) in open source software (OSS) projects. The proposed indices offer a quantitative measure of community split, shrink, merge, and expand events. The indices have proven properties like monotonicity, and they have defined maximum and minimum values that signify meaningful scenarios. These indices can be combined to describe complex community evolution events such as emergence and extinction. Expanding upon these indices, this research proposes a novel machine learning approach, leveraging shapelet mining, to unearth representative patterns of community evolution. The results from real-world OSS projects show that these indices effectively capture various community evolution behaviors with a 94.1% accuracy compared to existing work. They also predict OSS team productivity with a 0.718 accuracy. With the shapelet mining and learning framework, the indices can identify patterns of community evolution and predict the survival of OSS projects with 93% accuracy 3 months before the projects’ last observed commits. The findings highlight the potential of these entropy-based indices for understanding OSS project status and predicting future trends, which are valuable for supporting future research on DSNs and OSS communities.