Learning Graph Quality Metrics Report

Overview

• Total Concepts: 380
• Foundational Concepts (no dependencies): 2
• Concepts with Dependencies: 378
• Average Dependencies per Concept: 1.42

Graph Structure Validation

• Valid DAG Structure: ❌ No
• Self-Dependencies: None detected ✅
• Cycles Detected: 0

Foundational Concepts

These concepts have no prerequisites:

• 1: Function
• 18: Coordinate System

Dependency Chain Analysis

• Maximum Dependency Chain Length: 23

Longest Learning Path:

1. Coordinate System (ID: 18)
2. Number Line (ID: 19)
3. Real Numbers (ID: 20)
4. Limit (ID: 21)
5. Limit Notation (ID: 22)
6. One-Sided Limit (ID: 24)
7. Left-Hand Limit (ID: 25)
8. Two-Sided Limit (ID: 27)
9. Limit Existence (ID: 28)
10. Continuity (ID: 51)
11. Continuity at a Point (ID: 52)
12. Continuity on Interval (ID: 54)
13. Continuous Function (ID: 56)
14. Extreme Value Theorem (ID: 246)
15. Global Maximum (ID: 249)
16. Candidates Test (ID: 269)
17. Optimization (ID: 286)
18. Optimization Problem (ID: 287)
19. Objective Function (ID: 288)
20. Primary Equation (ID: 290)
21. Setting Up Optimization (ID: 291)
22. Maximizing Area (ID: 292)
23. Fencing Problem (ID: 296)

Orphaned Nodes Analysis

• Total Orphaned Nodes: 180

Concepts that are not prerequisites for any other concept:

• 9: Function Transformations
• 16: Radian Measure
• 17: Trigonometric Identities
• 30: Sum Rule for Limits
• 31: Product Rule for Limits
• 32: Quotient Rule for Limits
• 33: Constant Multiple Rule
• 34: Power Rule for Limits
• 40: Factoring for Limits
• 41: Rationalization
• 42: Complex Fractions
• 45: Sin x Over x Limit
• 46: Limit of Composition
• 47: Limits from Graphs
• 49: Numerical Estimation
• 53: Three Conditions
• 55: One-Sided Continuity
• 59: Jump Discontinuity
• 61: Essential Discontinuity
• 63: Removing Discontinuities

...and 160 more

Connected Components

• Number of Connected Components: 1

✅ All concepts are connected in a single graph.

Indegree Analysis

Top 10 concepts that are prerequisites for the most other concepts:

Rank	Concept ID	Concept Label	Indegree
1	308	Indefinite Integral	17
2	1	Function	14
3	95	Derivative Function	14
4	21	Limit	11
5	272	Curve Sketching	10
6	56	Continuous Function	9
7	115	Derivative Rules	9
8	145	Chain Rule Formula	9
9	29	Limit Laws	8
10	6	Graphing Functions	7

Outdegree Distribution

Dependencies	Number of Concepts
0	2
1	231
2	134
3	13

Recommendations

• ⚠️ Many orphaned nodes (180): Consider if these should be prerequisites for advanced concepts
• ℹ️ Long dependency chains (23): Ensure students can follow extended learning paths
• ℹ️ Consider adding cross-dependencies: More connections could create richer learning pathways

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Report generated by learning-graph-reports/analyze_graph.py