## 1 Introduction  
Open-source software (OSS) refers to software whose source code is publicly available, allowing anyone to freely access, use, modify, and distribute it [1]. This openness serves as a critical foundation for fostering continuous improvement and innovation through global collaboration. OSS creates an open ecosystem where developers from diverse backgrounds share knowledge and work together [2]. GitHub, as a leading platform for OSS development, has become essential for enabling such collaboration by providing systematic project and version management tools [3]. Through GitHub, developers can replicate projects, work independently on their own branches, and contribute back to the original project [4]. Collaboration on GitHub involves processes such as branch creation, commits, Pull Requests (PRs), and merges [5]. These processes go beyond mere technical procedures; they are inherently interaction-driven activities. For example, branch creation allows developers to work independently while avoiding code conflicts, and commits ensure transparency by recording changes made by team members. PRs facilitate communication and feedback between reviewers and contributors, ultimately enhancing the quality of the code. Finally, the merge process represents the culmination of collaborative efforts, integrating code changes into the project after careful review. These interaction-rich processes are vital not only for the technical success of a project but also for fostering effective communication and collaboration among team members [6]. While such a collaborative environment can drive active participation during the early stages of a project, contributor engagement often diminishes over time [7,8]. This decline poses a significant threat to the long-term development and sustainability of OSS projects. Reduced contributor engagement can hinder maintenance and progress, making it imperative for project maintainers to develop strategies that encourage sustained participation [9, 10, 11]. This challenge is further amplified in larger and more complex projects, where the loss of key contributors can disrupt the overall collaboration network. Thus, understanding the mechanisms that promote long-term contributor retention is not just beneficial--it is essential for maintaining project continuity and innovation. Previous research has primarily focused on quantitative metrics such as activity frequency and contribution counts to understand contributor engagement. While these analyses capture basic patterns of activity, they often fall short of explaining how dynamic behavioral patterns and interactions influence long-term participation. Particularly in GitHub's collaborative workflows--such as branch creation, PR submission, and merging--the complex interplay of technical contributions and social interactions requires deeper exploration. For instance, the timing and outcomes of PR reviews can significantly impact a contributor's willingness to continue engaging with a project. Addressing these gaps necessitates analyzing not only contributors' actions but also their roles and positions within the project network. This study aims to address these limitations by leveraging advanced methodologies to analyze contributor behavior and its impact on sustained engagement. Specifically, we employ model checking--a formal verification technique--to quantitatively evaluate contributors' dynamic behavioral patterns and their effects on participation duration. Model checking is particularly advantageous for analyzing detailed aspects of collaboration, such as PR processing times. However, behavioral analysis alone is insufficient to capture contributors' roles or influence within the project. To address this, we integrate k-core decomposition, a method for assessing network centrality, to understand how contributors' roles within the collaboration network affect their engagement.