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Learning Graph Quality Metrics Report

Overview

  • Total Concepts: 307
  • Foundational Concepts (no prerequisites, other concepts depend on them): 4
  • Terminal Nodes (nothing depends on them, but have prerequisites): 117
  • Orphaned Nodes (completely disconnected, no edges): 0
  • Concepts with Dependencies: 303
  • Average Dependencies per Concept: 1.80

Graph Structure Validation

  • Valid DAG Structure: ✅ Yes
  • Self-Dependencies: None detected ✅
  • Cycles Detected: 0

Foundational Concepts

These concepts have no prerequisites:

  • 1: Real Number System
  • 183: Angle Definition
  • 279: Vector Definition
  • 290: Matrix Definition

Dependency Chain Analysis

  • Maximum Dependency Chain Length: 19

Longest Learning Path:

  1. Real Number System (ID: 1)
  2. Order of Operations (ID: 3)
  3. Variables and Expressions (ID: 4)
  4. Input and Output (ID: 7)
  5. Function Definition (ID: 8)
  6. Sequence Definition (ID: 127)
  7. Geometric Sequence (ID: 131)
  8. Exponential Function Def (ID: 138)
  9. Natural Base e (ID: 144)
  10. Radian Measure (ID: 185)
  11. Unit Circle Definition (ID: 193)
  12. Unit Circle Coordinates (ID: 194)
  13. Sine Function (ID: 198)
  14. Sine Graph (ID: 209)
  15. Amplitude (ID: 211)
  16. Sinusoidal Functions (ID: 217)
  17. Sinusoidal Modeling (ID: 225)
  18. Periodic Data Analysis (ID: 226)
  19. Period from Data (ID: 231)

Terminal Nodes Analysis

Terminal nodes are concepts that nothing else depends on but have prerequisites. They represent natural endpoints of learning paths — culminating or specialized concepts.

  • Total Terminal Nodes: 117 (38.1% of all concepts)
  • Healthy Range: 5-40% of total concepts

Concepts at the end of learning paths:

  • 15: Set-Builder Notation
  • 18: Piecewise Functions
  • 26: Constant Functions
  • 29: Bounded Functions
  • 37: Rate of Change Units
  • 38: Secant Line
  • 41: Concavity from Rates
  • 42: Increasing Rate of Change
  • 43: Decreasing Rate of Change
  • 45: Slope-Intercept Form
  • 46: Point-Slope Form
  • 47: Standard Form of a Line
  • 48: Parallel Lines
  • 49: Perpendicular Lines
  • 56: Factored Form
  • 58: Axis of Symmetry
  • 59: Parabola
  • 69: Monomial
  • 70: Binomial
  • 71: Trinomial

...and 97 more

Orphaned Nodes Analysis

Orphaned nodes are completely disconnected concepts with no inbound AND no outbound edges. These indicate a quality problem — every concept should connect to the graph.

  • Total Orphaned Nodes: 0

✅ No orphaned nodes detected. All concepts are connected to the graph.

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 20 Graphing Functions 24
2 8 Function Definition 15
3 198 Sine Function 14
4 199 Cosine Function 14
5 66 Polynomial Definition 13
6 138 Exponential Function Def 9
7 5 Coordinate Plane 8
8 9 Function Notation 7
9 44 Linear Function Definition 7
10 79 Zeros of Polynomials 7

Outdegree Distribution

Dependencies Number of Concepts
0 4
1 117
2 148
3 23
4 12
5 2
6 1

Recommendations

  • ✅ Terminal node percentage (38.1%): Within healthy range (5-40%)
  • ✅ DAG structure verified: Graph supports valid learning progressions
  • ℹ️ Long dependency chains (19): Ensure students can follow extended learning paths

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