Graph Thinking – Non-Linear Problem Solving

Mental model for visualizing complex relationships and connections between ideas, concepts, or data points. Evolved from Graph-of-Thought (GoT) reasoning that mirrors human cognition.

When to Use This Skill

• Mapping feature dependencies in product development
• Analyzing stakeholder relationships
• Understanding system architectures
• Exploring interconnected concepts
• Designing recommendation systems or knowledge graphs
• Identifying opportunity areas through network analysis

Core Concepts

Graph Elements

Graph-of-Thought (GoT) Reasoning

Traditional (Chain-of-Thought):
A → B → C → D → Conclusion

Graph-of-Thought:
    ┌─── B ───┐
    │         │
A ──┼─── C ───┼──→ Synthesis → Conclusion
    │         │
    └─── D ───┘
         ↑
    Feedback Loop

GoT enables:

• Combining arbitrary thoughts into synergistic outcomes
• Distilling networks of thoughts for clarity
• Enhancing ideas using feedback loops
• Non-linear exploration of solution spaces

Fundamental Principles

First Principles Thinking

Break down complex problems into fundamental truths:

Surface Level:
"We need more marketing"
    ↓
Why?
    ↓
"Not enough customers"
    ↓
Why?
    ↓
Root Truth:
"Value proposition unclear to target audience"

Second-Order Thinking

Demand deeper analysis by asking “And then what?”:

Decision: Reduce prices by 20%

First-order:  More sales
Second-order: Lower margins → Less R&D budget
Third-order:  Competitors catch up → Price war
Fourth-order: Race to bottom → Industry commoditization

Non-Linear Processing

Unlike sequential thinking:

Analysis Framework

Double Diamond Model

Apply divergent and convergent thinking cycles:

    DISCOVER          DEFINE          DEVELOP          DELIVER
   (Diverge)        (Converge)       (Diverge)       (Converge)
       /\              \/              /\              \/
      /  \            /  \            /  \            /  \
     /    \          /    \          /    \          /    \
    /      \        /      \        /      \        /      \
   /        \      /        \      /        \      /        \

   Explore         Focus on         Generate         Focus on
   problem         specific         diverse          optimal
   space           challenges       solutions        implementation

Step 1: Map the Nodes

Identify all relevant elements:

Product Launch Analysis:

Nodes:
├── Stakeholders
│   ├── Customers
│   ├── Engineering
│   ├── Marketing
│   └── Leadership
├── Features
│   ├── Core functionality
│   ├── Nice-to-haves
│   └── Technical debt
├── Constraints
│   ├── Timeline
│   ├── Budget
│   └── Resources
└── Dependencies
    ├── External APIs
    ├── Infrastructure
    └── Regulatory

Step 2: Define Relationships (Edges)

Document connections between nodes:

Edge Types:
├── Dependency:    A requires B
├── Influence:     A affects B
├── Correlation:   A and B move together
├── Conflict:      A competes with B
└── Synergy:       A enhances B

Step 3: Identify Clusters and Patterns

Find highly connected groups:

High Centrality (Critical Nodes):
├── Authentication service → 12 dependencies
├── Database layer → 8 dependencies
└── API gateway → 6 dependencies

Clusters:
├── User-facing features (tightly coupled)
├── Backend services (loosely coupled)
└── Third-party integrations (isolated)

Step 4: Analyze Pathways

Trace routes through the network:

User Journey Graph:

Landing Page
    ↓
[Sign Up] ←→ [Social Login]
    ↓
Onboarding
    ↓           ↓
Quick Start   Full Setup
    ↓           ↓
    └─────┬─────┘
          ↓
    First Value
          ↓
    ↙    ↓    ↘
Churn  Retain  Upgrade

Output Template

After completing analysis, document as:

## Graph Thinking Analysis

**Subject:** [What you're analyzing]

**Analysis Date:** [Date]

### Node Map

| Category | Nodes   | Centrality     |
| -------- | ------- | -------------- |
| [Cat 1]  | [Nodes] | [High/Med/Low] |
| [Cat 2]  | [Nodes] | [High/Med/Low] |

### Relationship Matrix

| From | To  | Relationship | Strength |
| ---- | --- | ------------ | -------- |
| [A]  | [B] | [Type]       | [1-5]    |

### Key Insights

1. **Clusters identified:** [Description]
2. **Critical paths:** [Description]
3. **Bottlenecks:** [Description]
4. **Opportunities:** [Description]

### Recommendations

| Priority | Action   | Rationale |
| -------- | -------- | --------- |
| High     | [Action] | [Why]     |
| Medium   | [Action] | [Why]     |

Application Examples

Feature Dependency Mapping

Feature: Real-time Collaboration

Dependencies:
├── WebSocket infrastructure
│   ├── Connection management
│   └── Message queuing
├── Conflict resolution
│   ├── Operational transforms
│   └── CRDT implementation
├── Presence indicators
│   └── User state sync
└── Permissions
    ├── Document access
    └── Cursor visibility

Stakeholder Analysis

                    HIGH INFLUENCE
                          │
    Keep Satisfied        │        Manage Closely
    ┌─────────────────────┼─────────────────────┐
    │                     │                     │
    │   Executives        │    Product Owner    │
    │   Compliance        │    Key Customers    │
    │                     │                     │
LOW ──────────────────────┼────────────────────── HIGH
INTEREST                  │                      INTEREST
    │                     │                     │
    │   General Users     │    Power Users      │
    │   IT Support        │    Dev Team         │
    │                     │                     │
    └─────────────────────┼─────────────────────┘
    Monitor               │        Keep Informed
                          │
                   LOW INFLUENCE

System Architecture Analysis

Microservice Graph:

API Gateway [Centrality: 0.95]
    │
    ├── Auth Service [0.82]
    │   └── User DB
    │
    ├── Product Service [0.71]
    │   ├── Catalog DB
    │   └── Search Index
    │
    ├── Order Service [0.68]
    │   ├── Order DB
    │   └── Payment Gateway (external)
    │
    └── Notification Service [0.45]
        └── Email Provider (external)

Critical Path: Gateway → Auth → Product → Order
Bottleneck: Auth Service (single point of failure)

Best Practices

Do

• Visualize relationships – Draw the graph, don’t just describe it
• Iterate continuously – Graphs evolve as understanding deepens
• Measure centrality – Identify the most critical nodes
• Look for clusters – Natural groupings reveal system structure
• Trace pathways – Understand how information/value flows

Avoid

• Over-connecting – Not everything relates to everything
• Ignoring edge types – Different relationships have different meanings
• Static thinking – Graphs change over time
• Missing feedback loops – Circular dependencies are significant
• Forgetting weights – Some relationships are stronger than others

Integration with Other Methods

Tools

Visualization

• Mermaid – Code-based diagrams in markdown
• Graphviz – Programmatic graph generation
• Excalidraw – Hand-drawn style diagrams
• Miro/FigJam – Collaborative whiteboarding

Analysis

• Gephi – Network analysis and visualization
• Neo4j – Graph database for complex queries
• NetworkX – Python library for graph algorithms

Resources

• Graph of Thoughts: ArXiv Paper
• Neo4j Graph Database Use Cases
• Network Science by Albert-László Barabási