Period Segmentation in Transition Network Analysis: A Topological Data Analysis Approach

Hitoshi Inoue; Koichi Yasutake

cs.CYSep 25, 2026

Period Segmentation in Transition Network Analysis: A Topological Data Analysis Approach

Authors: Hitoshi Inoue, Koichi Yasutake

Organizations: Nakamura Gakuen University, Fukuoka, Japan · Hiroshima University, Hiroshima, Japan

Abstract

Temporal dynamics in learning behavior can be revealed through period segmentation in Transition Network Analysis (TNA). Cristea et al. demonstrated that segmenting courses into halves and quarters reveals how learning strategies evolve and relate to academic performance. Building on this approach, we investigate whether Topological Data Analysis (TDA), specifically connected components ( $β_0$ ) change points from Zigzag Persistent Homology, can provide data-driven period boundaries that identify intervention windows. Analyzing 22 courses from the Open University Learning Analytics Dataset, we find that $β_0$ -based segmentation captures greater between-period variation than time-based segmentation (median variance ratio (VR) = 4.10 $\times$ ; 21/22 courses show VR $>$ 1.1). Permutation tests confirmed significance ( $p < 0.05$ ) in 18% of individual courses, with 73% showing positive improvement over random breakpoints. Only one course showed better performance with time-based segmentation. However, analysis of academic outcomes reveals a key insight: final-period behavior shows lower correlation with outcomes in $β_0$ -based segmentation than in time-based segmentation, reflecting a marked behavioral collapse where engaged learners rapidly disengage. We interpret this as evidence that final-period behavior reflects rather than causes outcomes: students who will pass maintain engagement, while those who will fail disengage. This reframes the value of $β_0$ : rather than improving prediction, change points identify intervention windows. These are periods where behavioral structure shifts and targeted support may be most effective.

Figures & tables

Metric	Value
VR $>$ 1.1 ( $\beta_{0}$ better)	21/22 (95%)
VR $\approx$ 1 (0.9–1.1)	0/22 (0%)
VR $<$ 0.9 (Time better)	1/22 (5%)
Median VR	4.10 $\times$
Mean VR	13.95 $\times$
Min VR	0.67 $\times$

Table 1: Variance Ratio Summary Across 22 Courses (4-Period Comparison)

Metric	Value
Individual course $p<0.05$	4/22 (18%)
Individual course $p<0.10$	5/22 (23%)
Courses with positive improvement	16/22 (73%)
Mean improvement	127.5%
Median improvement	30.3%

Table 2: Permutation Test Validation (1,000 iterations per course). Improvement = (observed total variance

-

random total variance) / random total variance

\times

100, where total variance measures differentiation across all 9 transition types.

Figure 1: Period segmentation comparison: $\beta_{0}$ -based phases (background colors) vs. time-based quarters (dashed lines). Solid black lines indicate cost-minimizing $\beta_{0}$ change points. AAA_2014J shows highest VR (142.88 $\times$ , $p<0.001$ ); BBB_2014J shows VR = 16.22 $\times$ ( $p=0.005$ ); FFF_2013J shows VR = 6.74 $\times$ ( $p=0.016$ ); BBB_2013B shows VR $<$ 1 (0.67 $\times$ ) where time-based performs better. Breakpoints: AAA_2014J [6,10,37], BBB_2014J [14,16,35], FFF_2013J [27,30,35], BBB_2013B [3,9,17].

Figure 2: Transition probability comparison across four courses spanning the VR range. Top row: High $\rightarrow$ High persistence; Bottom row: High $\rightarrow$ Low disengagement. Blue bars: Time-based (Q1–Q4); Red bars: $\beta_{0}$ -based (P1–P4). High-VR courses (AAA_2014J, BBB_2014J) show dramatic P4 collapse with significant permutation tests ( $p<0.01$ ). FFF_2013J shows clear differentiation ( $p=0.016$ ). BBB_2013B (VR $<$ 1) shows greater variation in time-based Q4.

Figure 3: TNA-style state transition networks for BBB_2014J with cost-minimizing breakpoints [14, 16, 35]. Top row: Time-based segmentation (Q1–Q4); Bottom row: $\beta_{0}$ -based segmentation (P1–P4). Node colors indicate engagement states (H=High/green, M=Medium/orange, L=Low/red); edge widths are proportional to transition probabilities. The P4 network reveals a “collapse” pattern where the High $\rightarrow$ Low transition dominates.

Course	Method	Weeks	H $\rightarrow$ H	H $\rightarrow$ L	$r$
AAA_2014J	Time (Q4)	28–37	0.55	0.22	+0.52***
(VR = 142.88 $\times$ )	$\beta_{0}$ (P4)	37–37	0.12	0.72	+0.45***
BBB_2014J	Time (Q4)	28–37	0.58	0.25	+0.62***
(VR = 16.22 $\times$ )	$\beta_{0}$ (P4)	35–37	0.28	0.58	+0.55***
FFF_2013J	Time (Q4)	28–37	0.52	0.28	+0.55***
(VR = 6.74 $\times$ )	$\beta_{0}$ (P4)	35–37	0.35	0.48	+0.52***

Table 3: Final-Period Comparison for Representative Courses. H

\rightarrow

H = High-to-High persistence; H

\rightarrow

L = High-to-Low disengagement;

r

= point-biserial correlation with success. ***

p<.001

. Bold values highlight P4 collapse patterns (H

\rightarrow

<

0.35 or H

\rightarrow

>

0.5).

Change-Point Detection

Change-point detection identifies when the statistical behavior or underlying regime of a sequence changes.

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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

Nakamura Gakuen University, Japan · Hiroshima University, Japan

Educational Data Mining Online Learning

Open paper AskPDF arXiv

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 structural indicator with direct implications for AI-augmented learning analytics design.

Abstract

Change-Point Detection

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