Participation-Sensitive Convergence and the Fragment First, Converge Later Pattern in Asynchronous Online Learning: A Topological Analysis Across 22 OULAD Courses

Hitoshi Inoue, Koichi Yasutake

Asynchronous online learning offers temporal flexibility at a structural cost: learning communities tend to fragment rather than cohere. $β_0$, the number of disconnected behavioral clusters from Zigzag Persistent Homology, serves as a cohort-level indicator of this structure. Two questions remained unverified at scale: (1) does apparent $β_0$ convergence reflect genuine behavioral alignment or learner dropout? and (2) do assessment deadlines produce reproducible fragmentation-convergence cycles? We address both across all 22 OULAD courses (N > 22,000; 857 week-pairs). Changes in $β_0$ strongly co-vary with active learner changes (pooled r = 0.387; median per-course r_delta = 0.459, 20/22 courses), identifying $β_0$ as a participation-sensitive indicator: $β_0$ and active learner counts co-respond to deadline events rather than one causing the other. Deadlines produced fragmentation in 82.6% of assessments and the full Fragment First, Converge Later (FFCL) cycle in 60.2%. 3-phase analysis confirmed structural fragmentation as the dominant long-term trajectory (90.9% of courses), moderated by curriculum structure. These findings establish $β_0$ as a participation-sensitive…