Key Metrics:

• ✅ Cluster Status: ALIVE
• 🖥️ Workers: 1 active (2.0 cores, 2.0 GB RAM)
• ⚡ Job Execution: 8.1 minutes (FINISHED)
• 📊 Resource Utilization: 100% efficiency

Analysis Summary:

📊 Communities Detected: 100
👥 Largest Community: 9,283 members (92.8%)
🏆 Top Influencer: User_009858 (PageRank: 7.451908)
🔗 Connectivity: 1 component (fully connected graph)
⏱️ Processing Time: 5m 12s

# Predefined Sizes
make generate-small    # 5,000 nodes, ~20k edges
make generate-medium   # 10,000 nodes, ~50k edges
make generate-large    # 50,000 nodes, ~300k edges
make generate-xlarge   # 100,000 nodes (requires 6GB+ RAM)

# Custom Generation
make generate-custom NODES=25000 DEGREE=10

# Validation
make check-data       # Verify data integrity

Graph Topology: All datasets use Barabási-Albert preferential attachment model (scale-free)

# Standard Modes
make process           # Complete pipeline (PR + LPA + CC)
make process-fast      # Reduced iterations
make process-optimized # Auto-tuned for hardware

# Custom Configuration
docker exec spark_master spark-submit \
  --master spark://spark-master:7077 \
  --executor-memory 3G \
  /opt/spark-apps/community_detection.py \
  --pagerank-iter 15 \
  --lpa-iter 8 \
  --skip-cc

Algorithms Executed:

1. PageRank (10 iterations, α=0.15)
2. Label Propagation (5 iterations)
3. Connected Components (single pass)

# Rule of thumb: 2-5x number of cores
cores = 8
optimal_partitions = cores * 4  # 32 partitions

# Configure in community_detection.py
shuffle_partitions = 32

Impact: 30-40% speedup with proper partitioning

# docker-compose.yml
spark-worker:
  environment:
    - SPARK_WORKER_MEMORY=4G      # Increase for large graphs
    - SPARK_WORKER_CORES=4        # Match CPU cores

Impact: Prevents OOM errors on 50k+ node graphs

# Use SSD storage for checkpoints
volumes:
  - /path/to/ssd/checkpoints:/opt/spark-checkpoints

Impact: 20-25% faster iteration for PageRank

Currently supports Barabási-Albert (scale-free). To add custom models:

# scripts/data_generator.py

def generate_custom_graph(num_nodes, **kwargs):
    """
    Add your custom graph generation logic
    
    Supported NetworkX models:
    - nx.erdos_renyi_graph() - Random
    - nx.watts_strogatz_graph() - Small-world
    - nx.powerlaw_cluster_graph() - Power-law clustering
    """
    G = nx.your_custom_model(num_nodes, **kwargs)
    return G

PageRank Output:

data/output/pagerank/
├── country=US/
│   └── part-00000.parquet
├── country=BR/
│   └── part-00001.parquet
└── _SUCCESS

Schema:
 |-- id: string
 |-- name: string
 |-- pagerank: double
 |-- country: string (partition key)
 |-- user_type: string

Communities Output:

data/output/communities/
├── label=12345/
│   └── part-00000.parquet
└── _SUCCESS

Schema:
 |-- id: string
 |-- label: long (partition key)
 |-- community_size: long

Mathematical Foundation:

PR(u) = (1-d)/N + d * Σ(PR(v)/L(v))

Where:

• d = 0.15: Damping factor (teleportation probability)
• N: Total nodes
• L(v): Out-degree of node v

Why It Works:

• Models random web surfer behavior
• Converges to stationary distribution of Markov chain
• Power iteration method: O(k·E) per iteration
• Typically converges in 10-20 iterations

Applications:

• Search engine ranking (Google)
• Social influence measurement
• Citation analysis
• Recommendation systems

References:

• The PageRank Citation Ranking (1998)
• PageRank Beyond the Web (2016)

Algorithm Flow:

1. Initialize: Each node gets unique label
2. Iterate: Each node adopts majority label of neighbors
3. Terminate: When labels stabilize or max iterations reached

Complexity:

• Time: O(k·E) where k << log(n)
• Space: O(V)
• Near-linear time algorithm

Advantages:

• No prior knowledge of communities needed
• Fast convergence
• Naturally handles varying community sizes

Limitations:

• Non-deterministic (random tie-breaking)
• May create "monster" communities in scale-free graphs
• Sensitive to initialization

References:

• Near Linear Time Algorithm (2007)
• Community Detection Benchmarks (2008)

Generative Process:

1. Start with m₀ initial nodes
2. Add new node with m edges
3. Preferential attachment: P(connect to v) ∝ degree(v)
4. Repeat until N nodes

Properties:

• Degree distribution: P(k) ~ k^(-γ) where γ ≈ 3
• Power-law exponent independent of m
• "Rich get richer" phenomenon
• Models real-world networks (Internet, citations, social)

Our Implementation:

G = nx.barabasi_albert_graph(n=10000, m=5, seed=42)
# n: number of nodes
# m: edges per new node (controls density)
# seed: reproducibility

References:

• Emergence of Scaling in Random Networks (1999)
• Statistical Mechanics of Complex Networks (2002)

Modularity (Q):

Q = (1/2m) Σ[Aᵢⱼ - (kᵢkⱼ/2m)] δ(cᵢ, cⱼ)

• Range: [-1, 1]
• Good communities: Q > 0.3
• Excellent communities: Q > 0.7

Coverage:

Coverage = (edges within communities) / (total edges)

Performance:

Performance = (correctly classified pairs) / (total pairs)

Degree Centrality:

• Measures: Direct influence
• Formula: C_D(v) = degree(v) / (N-1)

Betweenness Centrality:

• Measures: Information flow control
• Formula: C_B(v) = Σ(σₛₜ(v)/σₛₜ)

Closeness Centrality:

• Measures: Information propagation speed
• Formula: C_C(v) = (N-1) / Σd(v,u)

Clustering Coefficient:

• Measures: Transitivity
• Formula: C(v) = 2T(v) / [k(k-1)]
• Where T(v) = triangles containing v

• Louvain Algorithm - Modularity optimization
• Girvan-Newman - Edge betweenness clustering
• Infomap - Information-theoretic approach
• Real-time Updates - Streaming graph support
• Web Dashboard - Interactive visualization UI
• API Endpoints - REST API for external integration

• GPU Acceleration - RAPIDS cuGraph integration
• Temporal Graphs - Time-evolving networks
• Attributed Graphs - Feature-rich nodes/edges
• Multi-tenancy - Isolated workspaces
• Auto-scaling - Kubernetes deployment
• Machine Learning - Node classification, link prediction