Global CPN Measures for Curriculum Analysis

• Global CPN measures are macro-level descriptors that model academic curricula as directed acyclic graphs using key invariants such as breadth, depth, and flux.
• They employ topological stratification to assign courses into sequential layers, ensuring robustness through invariance under transitive reduction.
• Empirical analyses using these measures enable cross-institutional curriculum comparisons, diagnose bottlenecks, and guide effective curriculum design.

Global CPN measures are rigorously defined, macro-level quantitative descriptors developed for course-prerequisite networks (CPNs), encapsulating the structure and knowledge flow of academic curricula represented as directed acyclic graphs (DAGs). Unlike micro-scale measures (e.g., node centralities), these invariants—breadth, depth, and flux—are crafted for whole-curriculum comparison, diagnosis, and design. They are derived from the topological stratification of the CPN, generalizing classical topological ordering to provide a more flexible and realistic account of curricular levels.

1. Concept and Motivation

Course-prerequisite networks model curricula as DAGs, with courses as nodes and directed edges encoding prerequisite relationships. Traditional analyses have emphasized micro-properties (individual courses) or meso-structures (communities, layers), but lacked tools to compare or characterize curricula at the macro (global) scale. The breadth, depth, and flux measures address this deficiency by quantifying:

• The width of parallel curricular options (breadth)
• The vertical structure or sequential complexity (depth)
• The net flow of prerequisite relationships between levels (flux)

A key design principle is invariance under transitive reduction: these measures depend only on essential prerequisite links, not on administrative redundancies.

2. Topological Stratification: The Analytical Foundation

The theoretical foundation is topological stratification, which assigns each node (course) to a unique level (stratum) based on prerequisite relations. The process is recursive:

• First stratum $\mathcal{S}_1$: all nodes with in-degree zero (no prerequisites).
• After removing $\mathcal{S}_1$ and their outgoing edges, the next stratum $\mathcal{S}_2$ is formed from nodes now lacking prerequisites, and so forth, until all nodes are assigned (total strata $T$).
• This yields a partition:

$$\mathcal{V} = \mathcal{S}_1 \sqcup \mathcal{S}_2 \sqcup \cdots \sqcup \mathcal{S}_T$$

This structure reflects the minimal "years" or "terms" necessary to progress through the curriculum, while grouping courses without mutual prerequisites in the same stratum.

3. Breadth: Quantifying Parallelism in Curricula

Breadth ($B$) measures how many courses are, on average, available at each curricular level: $$B(\mathcal{G}) = \frac{1}{T} \sum_{t=1}^T n_t = \frac{n}{T}$$ where $n$ is the total number of courses, $T$ is the number of strata, and $n_t = |\mathcal{S}_t|$ is the number of courses in stratum $t$.

Interpretively, higher breadth indicates greater student choice or interdisciplinary exposure at each progression stage. For fixed $n$, breadth is maximized when $T$ is minimized, corresponding to highly parallelized curricula.

4. Depth: Measuring Sequential Complexity

Depth ($D$) captures the typical "vertical" journey a student undertakes through the curriculum: $$D(\mathcal{G}) = \frac{1}{|\Omega|} \sum_{t=1}^T t\, |\Omega_t|$$ Here, $\Omega$ is the set of terminal nodes (courses with no postrequisites), and $\Omega_t = \Omega \cap \mathcal{S}_t$ those in stratum $t$. Thus, $D$ is the expected stratum of a randomly chosen terminal course, signifying the average sequential length required to reach curricular endpoints.

Breadth and depth are inversely correlated: curricula with many parallel options (large $B$) tend to have shallower depth (small $D$), and vice versa.

5. Flux: Characterizing Knowledge Flow

Flux ($\Phi$) provides a per-stratum measure of the net flow of prerequisite links: $\Phi_t = \frac{L_t^{t+1} - L_{t-1}^t}{n_t}$

$$\Phi(\mathcal{G}) = \frac{1}{T} \sum_{t=1}^T \Phi_t$$

where $L_{t-1}^t$ is the number of links from $\mathcal{S}_{t-1}$ to $\mathcal{S}_t$, and $L_t^{t+1}$ those from $\mathcal{S}_t$ to $\mathcal{S}_{t+1}$. Positive local flux indicates strata serving mainly as sources (providing foundational knowledge), while negative values indicate net sinks (absorbing knowledge as advanced courses). Flux, by design, only counts essential ("between adjacent strata") links and thus remains unaffected by transitive reduction.

6. Empirical Observations and Applications

The measures have been computed for real CPNs from Cyprus University of Technology, Caltech, and Johns Hopkins University, showing:

• Breadth increases with university size; depth decreases, reflecting greater parallelism in larger curricula.
• Flux values are negative across cases, indicating that most advanced courses are net knowledge sinks.
• Random DAG models (Erdős–Rényi) and stratified random models (Karrer–Newman) tend to underestimate breadth and overestimate depth compared to real CPNs, highlighting structural constraints present in authentic curricula.

Macro-scale metrics enable:

• Cross-institutional curriculum comparison
• Diagnostic detection of bottlenecks or excessive sequential constraints
• Policy evaluation for curriculum design, balancing choice and depth according to educational objectives

Applications include advising, detecting gatekeeper courses, predicting graduation distributions, and analyzing interdepartmental knowledge flows.

7. Theoretical Advances and Future Directions

Global CPN measures formalize the macro-architecture of curricula and are robust to variations in modeling conventions due to their invariance property. The inadequacy of standard random models for real curriculum networks suggests a need for new models capturing departmental clustering and modularity.

A plausible implication is that these measures could inform algorithmic curriculum optimization, such as maximizing breadth for timely graduation or enforcing desired depth profiles for various programs. Ongoing research may focus on extending these concepts to dynamically evolving curricula or integrating them with data-driven curricular outcome analyses.

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Measure	Formula	Interpretation
Breadth ($B$)	$B(\mathcal{G}) = \frac{n}{T}$	Courses per stratum (curricular width)
Depth ($D$)	$$D(\mathcal{G}) = \frac{1}{|\Omega|} \sum_{t=1}^T t\,|\Omega_t|$$	Strata to terminal courses (vertical complexity)
Flux ($\Phi$)	$$\Phi(\mathcal{G}) = \frac{1}{T} \sum_{t=1}^T \frac{L_t^{t+1} - L_{t-1}^t}{n_t}$$	Net knowledge flow per stratum

Global CPN measures—breadth, depth, and flux—not only facilitate systematic curriculum analysis and comparison but also highlight the complex design trade-offs and structural features pivotal in higher education planning.