Learning Graph Quality Metrics Report

Overview

Total Concepts: 600
Foundational Concepts (no prerequisites, other concepts depend on them): 12
Terminal Nodes (nothing depends on them, but have prerequisites): 214
Orphaned Nodes (completely disconnected, no edges): 0
Concepts with Dependencies: 588
Average Dependencies per Concept: 1.83

Graph Structure Validation

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

Foundational Concepts

These concepts have no prerequisites:

1: Semiconductor Definition
11: Bandgap Energy
18: Crystal Lattice
39: Covalent Bonding in Semiconductors
40: Ionic Bonding
41: Metallic Bonding
42: Van der Waals Forces
58: Wave-Particle Duality
79: Spin Quantum Number
91: Free Electron Model
107: Fermi Energy
110: Maxwell-Boltzmann Distribution

Dependency Chain Analysis

Maximum Dependency Chain Length: 32

Longest Learning Path:

Wave-Particle Duality (ID: 58)
de Broglie Wavelength (ID: 59)
Schrodinger Equation (ID: 61)
Time-Independent Schrodinger Equation (ID: 63)
Periodic Potential (ID: 89)
Kronig-Penney Model (ID: 85)
Bloch Theorem (ID: 86)
Band Gap Origin (ID: 90)
Nearly-Free Electron Model (ID: 92)
E-k Dispersion Relation (ID: 95)
Effective Mass Approximation (ID: 96)
Density of States 3D (ID: 112)
Density-of-States Effective Mass (ID: 99)
Effective Density of States (ID: 116)
Conduction Band Effective DOS (ID: 117)
Intrinsic Carrier Concentration (ID: 119)
Law of Mass Action (ID: 122)
Minority Carriers (ID: 145)
Drift Current (ID: 154)
Drift Velocity (ID: 155)
Carrier Mobility (ID: 156)
Diffusion Coefficient (ID: 177)
Minority Carrier Diffusion Length (ID: 196)
Ideal Diode Equation (ID: 240)
Shockley Equation (ID: 241)
Solar Cell I-V Curve (ID: 448)
Short-Circuit Current (ID: 449)
Fill Factor Solar Cell (ID: 451)
Power Conversion Efficiency (ID: 452)
Shockley-Queisser Limit (ID: 453)
Multi-Junction Solar Cell (ID: 455)
Tandem Solar Cell (ID: 456)

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: 214 (35.7% of all concepts)
Healthy Range: 5-40% of total concepts

Concepts at the end of learning paths:

2: Conductor vs Insulator vs Semiconductor
6: Germanium Material Properties
13: Indirect Bandgap
17: Forbidden Energy Gap
20: Primitive Cell
24: Body-Centered Cubic Structure
30: Crystal Directions
32: First Brillouin Zone
33: Wigner-Seitz Cell
38: Electron Diffraction
43: Hybridization SP3
48: Frenkel Defects
49: Schottky Defects
51: Edge Dislocations
52: Screw Dislocations
53: Stacking Faults
54: Grain Boundaries
57: Dangling Bonds
62: Time-Dependent Schrodinger Equation
66: Quantum Normalization

...and 194 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	18	Crystal Lattice	23
2	222	P-N Junction	17
3	11	Bandgap Energy	16
4	231	Depletion Width	15
5	361	MOSFET Structure	15
6	156	Carrier Mobility	14
7	323	Bipolar Junction Transistor	14
8	44	Point Defects	12
9	108	Fermi Level	12
10	144	Majority Carriers	10

Outdegree Distribution

Dependencies	Number of Concepts
0	12
1	188
2	317
3	79
4	4

Recommendations

✅ Terminal node percentage (35.7%): Within healthy range (5-40%)
✅ DAG structure verified: Graph supports valid learning progressions
ℹ️ Long dependency chains (32): Ensure students can follow extended learning paths

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