## 2 Background and Related Work
### Contributor Behavior and PR Management in Open-Source Projects  
Research on contributor retention and behavior in open-source projects has gained increasing attention, particularly as platforms like GitHub have become central to software development. Several studies have explored how technical processes, such as PR management, along with social and collaborative factors, impact long-term contributor engagement. Ref. [17] investigated the influence of social and technical factors in evaluating contributions on GitHub. Their study highlighted how contributors' social interactions within the community, combined with the technical nature of their contributions, affected their standing and retention in the project. They found that contributors who received positive feedback and frequent social interaction were more likely to continue contributing, underscoring the importance of timely communication and recognition in retaining contributors. Ref. [18] conducted an exploratory study on the pull-based software development model, focusing on how PRs function as a critical interaction mechanism between contributors and project maintainers. Their findings revealed that the efficiency of PR processing, particularly the time taken to review and merge contributions, plays a significant role in determining contributor retention. Contributors whose PRs were merged quickly tended to return and contribute more frequently, while delays in processing often led to disengagement. This directly aligns with the findings in this study, which demonstrate that prompt PR feedback and successful merges positively impact long-term participation. Ref. [19] focused on modeling the early behaviors of contributors in free/libre open-source software (FLOSS) communities to predict their long-term participation. They found that contributors who quickly became integrated into the project by having their contributions accepted or merged early were more likely to continue participating over time. This study complements our analysis by showing that early success in the form of PR merges can serve as a critical motivation for continued involvement. Ref. [20] examined the classification of developers into core and peripheral roles, showing that core developers--who are more central in the project's network--tend to have longer participation durations. Their study emphasized the importance of network centrality in understanding contributor behavior, which aligns with our use of k-core decomposition to analyze how being centrally positioned in the collaboration network influences long-term engagement. Network centrality metrics have been widely used to understand the structural importance of nodes in a network. Ref. [21] provided an introduction to centrality measures, highlighting their relevance in assessing influence and connectivity. Ref. [22] discussed various indices, including their theoretical underpinnings and computational complexities. Recent work by [23] introduced extensions to centrality metrics to account for group interactions. These studies underscore the importance of using diverse metrics to gain a comprehensive understanding of network dynamics. In this study, k-core centrality was chosen for its ability to capture the cohesive substructures critical to collaborative OSS environments. In addition to the studies mentioned above, this research introduces a unique approach by incorporating model checking techniques alongside SHAP analysis and k-core decomposition. By utilizing model checking, this study ensures the reliability of predictive models used to assess contributor behaviors, providing a more rigorous understanding of how various factors interact to influence long-term engagement. This method allows for the verification of model accuracy and robustness, setting this research apart from previous studies that may not have applied such in-depth validation techniques. Overall, existing research highlights the importance of efficient PR management, timely feedback, and network integration in retaining contributors.