Learning Graph Quality Metrics
Analysis Summary
Total Concepts: 200 Analysis Date: 2025-10-30
1. DAG Validation ✓
Status: ✓ PASSED
The graph is a valid Directed Acyclic Graph (DAG) with no cycles detected.
This ensures: - Concepts have clear prerequisite relationships - No circular dependencies - Linear learning progression is possible - Topological ordering can be computed
2. Foundational Concepts
Count: 10 concepts with zero dependencies
These concepts require no prerequisites and serve as entry points:
- Complex Numbers
- Sine Wave Representation
- ARM Cortex M Series
- Software Floating Point
- Fixed Point Numbers
- Performance Metrics
- Open Source Libraries
- Proprietary Libraries
- GitHub Repositories
- Compiler Optimization Levels
Analysis: Good distribution of foundational concepts across mathematical, hardware, and software engineering domains.
3. Indegree Distribution
| Dependencies | Concept Count | Percentage |
|---|---|---|
| 0 | 10 | 5.0% |
| 1 | 153 | 76.5% |
| 2 | 35 | 17.5% |
| 3 | 2 | 1.0% |
Analysis: - Most concepts (76.5%) have exactly one prerequisite - simple linear progression - 17.5% have two prerequisites - reasonable concept convergence - Only 2 concepts have three prerequisites - controlled complexity - Maximum indegree of 3 keeps learning manageable
4. Longest Dependency Chains
Top 10 concepts by chain depth:
| Concept | Chain Length |
|---|---|
| Static Linking | 7 |
| Dynamic Linking | 7 |
| Library Dependencies | 7 |
| Mixed Radix FFT | 6 |
| Bit Reversal Permutation | 6 |
| Power Of Two Constraint | 6 |
| Initialization Functions | 6 |
| Configuration Parameters | 6 |
| Header Files | 6 |
| Linking Libraries | 6 |
Analysis: - Maximum chain depth: 7 levels - Advanced topics naturally appear deeper in the dependency tree - Library integration concepts require the most prerequisites - Reasonable depth - not too shallow (would indicate poor structure) nor too deep (would be overwhelming)
5. Connectivity Analysis
Status: ⚠ 6 disconnected components detected
| Component | Size | Description |
|---|---|---|
| 1 | 188 | Main connected graph (94%) |
| 2-6 | 12 | Small isolated clusters (6%) |
Interpretation: - Main component contains 94% of concepts - excellent core connectivity - 6% of concepts are in isolated clusters - This is actually pedagogically valid - some topics (e.g., version control, licensing) are independent of core FFT/DSP material - These can be studied in parallel with the main learning path
Isolated Concepts: - Likely include: licensing, documentation, version control tools - These support topics don't require technical prerequisites
6. Graph Statistics
| Metric | Value |
|---|---|
| Total Edges (Dependencies) | 229 |
| Average Dependencies per Concept | 1.15 |
| Graph Density | 0.0058 |
| Foundational Concepts | 10 |
| Maximum Indegree | 3 |
| Maximum Chain Length | 7 |
Interpretation: - Low average dependencies (1.15) - concepts are well-isolated and focused - Low graph density (0.58%) - sparse graph indicates clear learning paths - This is ideal for educational content - not overly interdependent
7. Pedagogical Assessment
Strengths ✓
- Valid DAG structure - no circular dependencies
- Good foundational base - 10 entry points across domains
- Manageable complexity - low average dependencies
- Clear progression - longest chains are 7 levels deep
- Focused concepts - low graph density suggests well-defined topics
- Multiple entry points - learners can start with math, hardware, or software
Observations ⚠
- Minor isolation - 6% of concepts in small disconnected components
- This is acceptable for support topics like licensing and version control
- Can be addressed by adding cross-references in documentation
Recommendations
- Consider adding lightweight connections between isolated clusters and main graph
- Use the foundational concepts as course module starting points
- Organize curriculum to follow the dependency chains
- Advanced topics (chain length 6-7) should be in later weeks
8. Course Structure Recommendations
Based on the graph structure, suggested 10-week organization:
Weeks 1-2: Foundational concepts (10 concepts with zero dependencies) Weeks 3-4: Level 1-2 concepts (mathematical foundations, basic FFT) Weeks 5-6: Level 3-4 concepts (ARM architecture, signal processing) Weeks 7-8: Level 5 concepts (optimization, benchmarking methods) Weeks 9-10: Level 6-7 concepts (advanced library integration, capstone project)
Overall Quality Rating: ✓ EXCELLENT
The learning graph demonstrates: - Sound pedagogical structure - Clear prerequisite relationships - Appropriate complexity progression - Good balance of breadth and depth - Ready for taxonomy organization and visualization