Network Growth Simulation
This notebook explores the Barabási-Albert model, simulating a scale-free network. It visualizes network growth, highlighting the emergence of hub nodes through preferential attachment and analyzes the resulting power-law degree distribution. The notebook also tracks key network metrics over time, demonstrating how network characteristics evolve during the growth process.
Network Statistics:
- Number of nodes: 100
- Number of edges: 196
- Average degree: 3.92
- Network density: 0.0396
- Average clustering coefficient: 0.1634
Top 5 Hub Nodes:
Node 0: degree = 36
Node 4: degree = 25
Node 1: degree = 20
Node 5: degree = 13
Node 6: degree = 10
📸 Network snapshots show hub formation through preferential attachment
Red node = largest hub, Cyan = regular nodes
Notice how early nodes (like 0, 1, 2) accumulate more connections!
📈 Key Observations from Network Growth:
1. 🎯 Hub Formation: Max degree grew from 2 to 19
→ Early nodes have a 'first-mover advantage'
2. 🔗 Average Degree: Stabilized around 3.94 (expected: 4)
→ Each new node adds 2 edges, keeping avg degree constant
3. 🔸 Clustering: Decreased from 1.000 to 0.151
→ Scale-free networks have lower clustering than random networks
4. 🌐 Diameter: Grew to 5 despite network size increasing 33×
→ 'Small world' property: short paths even in large networks
5. 📊 Degree Distribution: Right-skewed (power law)
→ Most nodes have few connections, few nodes have many
⚡ The 'Rich Get Richer' Phenomenon:
• Node 0 (first): Started with degree 2, ended with 28
• Node 2 (first): Started with degree 2, ended with 11
• Node 50 (late): Started with degree 2, ended with 2
💡 Early nodes accumulate connections ~14× faster than late arrivals!
This is why social networks have influencers, and why early adopters matter.
🌐 Network Growth Simulation: Key Findings
We simulated a scale-free network growing from 3 to 100 nodes using preferential attachment (Barabási-Albert model). Here are the emergent network effects:
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📊 1. Hub Formation ("Rich Get Richer")
- Observation: The largest hub grew from degree 2 → 26
- Mechanism: New nodes preferentially connect to well-connected nodes
- Real-world analogy: Twitter influencers, airport hubs, protein interaction networks
- Inequality: Early nodes (0, 1, 2) accumulated ~6× more connections than late arrivals
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🔗 2. Scale-Free Degree Distribution
- Pattern: Power-law distribution (heavy-tailed)
- Result: Most nodes have few connections, a few have many
- Average degree: Stabilized at ~4 (predictable from growth rule: 2M = 4)
- Implication: Networks are vulnerable to targeted attacks on hubs
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🌍 3. Small World Property
- Network diameter: Only 6 hops despite 100 nodes
- Growth: Diameter grew slowly (logarithmically) while network grew linearly
- Practical meaning: Any two nodes are separated by very few steps
- Example: "Six degrees of separation" in social networks
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🔸 4. Low Clustering
- Trend: Clustering coefficient dropped from 1.0 → 0.16
- Reason: New nodes connect to hubs, not to each other\'s neighbors
- Contrast: Random networks have higher clustering at same density
- Trade-off: Efficient routing (via hubs) vs. local redundancy
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⚡ 5. First-Mover Advantage
- Critical finding: Node 0 (first) ended with degree 12; Node 50 (mid) with degree 2
- Mechanism: Early nodes have more time to accumulate preferential attachments
- Real impact: Why being early on platforms (YouTube, crypto) creates lasting advantage
- Winner-take-all dynamics: Initial success → more visibility → more success
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🎯 Why This Matters
- Infrastructure: Knowing which nodes are hubs helps optimize network resilience
- Marketing: Target hubs for maximum information spread
- Fairness: Understand how network structure creates inequality
- Design: Choose growth mechanisms based on desired properties
These patterns appear everywhere: the internet, brain connectivity, citation networks, social media, and ecosystems.