Network Analysis

Network analysis examines relationships and connections between entities, revealing community structure, influence patterns, and network properties.

Network Concepts

• Nodes: Individual entities
• Edges: Connections/relationships
• Degree: Number of connections
• Centrality: Node importance measures
• Community: Densely connected groups
• Clustering Coefficient: Local density

Key Metrics

• Degree Centrality: Number of connections
• Betweenness Centrality: Control over paths
• Closeness Centrality: Average distance to others
• Eigenvector Centrality: Connections to important nodes
• Modularity: Community structure strength

Implementation with Python

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import networkx as nx
from collections import defaultdict, Counter
import seaborn as sns

# Create sample network (social network)
G = nx.Graph()

# Add nodes with attributes
nodes = [
    ('Alice', {'role': 'Manager', 'dept': 'Sales'}),
    ('Bob', {'role': 'Engineer', 'dept': 'Tech'}),
    ('Carol', {'role': 'Designer', 'dept': 'Design'}),
    ('David', {'role': 'Engineer', 'dept': 'Tech'}),
    ('Eve', {'role': 'Analyst', 'dept': 'Sales'}),
    ('Frank', {'role': 'Manager', 'dept': 'HR'}),
    ('Grace', {'role': 'Designer', 'dept': 'Design'}),
    ('Henry', {'role': 'Engineer', 'dept': 'Tech'}),
    ('Iris', {'role': 'Analyst', 'dept': 'Sales'}),
    ('Jack', {'role': 'Manager', 'dept': 'Finance'}),
]

for node, attrs in nodes:
    G.add_node(node, **attrs)

# Add edges (relationships)
edges = [
    ('Alice', 'Bob'), ('Alice', 'Carol'), ('Alice', 'Eve'),
    ('Bob', 'David'), ('Bob', 'Henry'), ('Carol', 'Grace'),
    ('David', 'Henry'), ('Eve', 'Iris'), ('Frank', 'Jack'),
    ('Grace', 'Carol'), ('Alice', 'Frank'), ('Bob', 'Carol'),
    ('Eve', 'Alice'), ('Iris', 'Eve'), ('Jack', 'Frank'),
    ('Henry', 'David'), ('Carol', 'David'),
]

G.add_edges_from(edges)

print("Network Summary:")
print(f"Nodes: {G.number_of_nodes()}")
print(f"Edges: {G.number_of_edges()}")
print(f"Density: {nx.density(G):.2%}")

# 1. Degree Centrality
degree_centrality = nx.degree_centrality(G)
print("\n1. Degree Centrality (Top 5):")
for node, score in sorted(degree_centrality.items(), key=lambda x: x[1], reverse=True)[:5]:
    print(f"  {node}: {score:.3f}")

# 2. Betweenness Centrality (control over network)
betweenness_centrality = nx.betweenness_centrality(G)
print("\n2. Betweenness Centrality (Top 5):")
for node, score in sorted(betweenness_centrality.items(), key=lambda x: x[1], reverse=True)[:5]:
    print(f"  {node}: {score:.3f}")

# 3. Closeness Centrality (average distance to others)
closeness_centrality = nx.closeness_centrality(G)
print("\n3. Closeness Centrality (Top 5):")
for node, score in sorted(closeness_centrality.items(), key=lambda x: x[1], reverse=True)[:5]:
    print(f"  {node}: {score:.3f}")

# 4. Eigenvector Centrality
try:
    eigenvector_centrality = nx.eigenvector_centrality(G, max_iter=100)
    print("\n4. Eigenvector Centrality (Top 5):")
    for node, score in sorted(eigenvector_centrality.items(), key=lambda x: x[1], reverse=True)[:5]:
        print(f"  {node}: {score:.3f}")
except:
    print("\n4. Eigenvector Centrality: Not converged")

# 5. Community Detection (using modularity)
from networkx.algorithms import community

communities = list(community.greedy_modularity_communities(G))
print(f"\n5. Community Detection:")
print(f"Number of communities: {len(communities)}")
for i, comm in enumerate(communities):
    print(f"  Community {i+1}: {list(comm)}")

# 6. Network Statistics
degrees = [G.degree(n) for n in G.nodes()]
print(f"\n6. Network Statistics:")
print(f"Average Degree: {np.mean(degrees):.2f}")
print(f"Max Degree: {max(degrees)}")
print(f"Min Degree: {min(degrees)}")
print(f"Clustering Coefficient: {nx.average_clustering(G):.3f}")
print(f"Number of Triangles: {sum(nx.triangles(G).values()) // 3}")

# Visualization
fig, axes = plt.subplots(2, 2, figsize=(15, 12))

# Network layout
pos = nx.spring_layout(G, k=0.5, iterations=50, seed=42)

# 1. Network Graph (colored by degree)
ax = axes[0, 0]
node_colors = [degree_centrality[node] for node in G.nodes()]
nx.draw_networkx_nodes(G, pos, node_color=node_colors, node_size=1000, cmap='YlOrRd', ax=ax)
nx.draw_networkx_edges(G, pos, alpha=0.5, ax=ax)
nx.draw_networkx_labels(G, pos, font_size=8, ax=ax)
ax.set_title('Network Graph (Colored by Degree Centrality)')
ax.axis('off')

# 2. Network Graph (colored by communities)
ax = axes[0, 1]
color_map = []
colors = plt.cm.Set3(np.linspace(0, 1, len(communities)))
node_to_color = {}
for i, comm in enumerate(communities):
    for node in comm:
        node_to_color[node] = colors[i]
color_map = [node_to_color[node] for node in G.nodes()]

nx.draw_networkx_nodes(G, pos, node_color=color_map, node_size=1000, ax=ax)
nx.draw_networkx_edges(G, pos, alpha=0.5, ax=ax)
nx.draw_networkx_labels(G, pos, font_size=8, ax=ax)
ax.set_title('Network Graph (Colored by Community)')
ax.axis('off')

# 3. Centrality Comparison
ax = axes[1, 0]
centrality_df = pd.DataFrame({
    'Degree': degree_centrality,
    'Betweenness': betweenness_centrality,
    'Closeness': closeness_centrality,
}).head(8)

centrality_df.plot(kind='barh', ax=ax, width=0.8)
ax.set_xlabel('Centrality Score')
ax.set_title('Top 8 Nodes - Centrality Comparison')
ax.legend(loc='lower right')
ax.grid(True, alpha=0.3, axis='x')

# 4. Degree Distribution
ax = axes[1, 1]
degree_sequence = sorted([d for n, d in G.degree()], reverse=True)
degree_count = Counter(degree_sequence)
degrees_unique = sorted(degree_count.keys())
counts = [degree_count[d] for d in degrees_unique]

ax.bar(degrees_unique, counts, color='steelblue', edgecolor='black', alpha=0.7)
ax.set_xlabel('Degree')
ax.set_ylabel('Count')
ax.set_title('Degree Distribution')
ax.grid(True, alpha=0.3, axis='y')

plt.tight_layout()
plt.show()

# 7. Path Analysis
print(f"\n7. Path Analysis:")
try:
    shortest_path = nx.shortest_path_length(G, 'Alice', 'Jack')
    print(f"Shortest path from Alice to Jack: {shortest_path}")
except nx.NetworkXNoPath:
    print("No path exists between nodes")

# 8. Connectivity Analysis
print(f"\n8. Connectivity Analysis:")
print(f"Is connected: {nx.is_connected(G)}")
num_components = nx.number_connected_components(G)
print(f"Number of connected components: {num_components}")

# 9. Similarity Measures
def jaccard_similarity(node1, node2):
    neighbors1 = set(G.neighbors(node1)) | {node1}
    neighbors2 = set(G.neighbors(node2)) | {node2}
    intersection = len(neighbors1 & neighbors2)
    union = len(neighbors1 | neighbors2)
    return intersection / union if union > 0 else 0

print(f"\n9. Node Similarity (Jaccard):")
print(f"Alice & Bob: {jaccard_similarity('Alice', 'Bob'):.3f}")
print(f"Alice & Jack: {jaccard_similarity('Alice', 'Jack'):.3f}")

# 10. Influence Score (Combination of metrics)
influence_score = {}
for node in G.nodes():
    score = (degree_centrality[node] * 0.4 +
             betweenness_centrality[node] * 0.3 +
             closeness_centrality[node] * 0.3)
    influence_score[node] = score

print(f"\n10. Influence Score (Top 5):")
for node, score in sorted(influence_score.items(), key=lambda x: x[1], reverse=True)[:5]:
    print(f"  {node}: {score:.3f}")

# Summary
print("\n" + "="*50)
print("NETWORK ANALYSIS SUMMARY")
print("="*50)
print(f"Most influential: {max(influence_score, key=influence_score.get)}")
print(f"Most connected: {max(degree_centrality, key=degree_centrality.get)}")
print(f"Network bottleneck: {max(betweenness_centrality, key=betweenness_centrality.get)}")
print(f"Closest to all: {max(closeness_centrality, key=closeness_centrality.get)}")
print("="*50)

Centrality Measures

• Degree: Direct connections only
• Betweenness: Bridges between groups
• Closeness: Access to network
• Eigenvector: Connected to important nodes
• PageRank: Random walk probability

Community Detection

• Modularity Optimization: Find dense groups
• Louvain Algorithm: Hierarchical communities
• K-clique: Overlapping communities
• Spectral: Eigenvalue-based

Applications

• Social network analysis
• Organizational structures
• Citation networks
• Recommendation networks
• Supply chain analysis

Deliverables

• Network visualization
• Centrality analysis
• Community detection results
• Connectivity metrics
• Influence rankings
• Key node identification
• Network statistics summary