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Hypergraph Basics

A hypergraph is a generalization of a graph where edges (called hyperedges) can connect any number of vertices, not just two. This makes hypergraphs particularly powerful for modeling complex relationships in real-world scenarios.

What is a Hypergraph?

Traditional Graph vs Hypergraph

In a traditional graph: - Vertices (nodes) represent entities - Edges connect exactly two vertices - Relationships are pairwise only

# Example: Social network friendship
edges = [
    (Alice, Bob),      # Alice and Bob are friends
    (Bob, Charlie),    # Bob and Charlie are friends
    (Alice, Charlie)   # Alice and Charlie are friends
]

In a hypergraph: - Vertices represent entities - Hyperedges can connect any number of vertices - Can model group relationships naturally

# Example: Group activities
hyperedges = [
    (Alice, Bob),              # Alice and Bob have coffee
    (Alice, Bob, Charlie),     # All three work on a project
    (Bob, Charlie, David, Eve) # Team meeting
]

Key Concepts

Vertices

Vertices represent the fundamental entities in your data:

from hyperdb import HypergraphDB

hg = HypergraphDB()

# Add vertices with attributes
hg.add_v(1, {"name": "Alice", "age": 30, "role": "Engineer"})
hg.add_v(2, {"name": "Bob", "age": 25, "role": "Designer"})
hg.add_v(3, {"name": "Charlie", "age": 35, "role": "Manager"})

Hyperedges

Hyperedges represent relationships between multiple vertices:

# Binary relationship (like traditional edge)
hg.add_e((1, 2), {"type": "collaboration", "project": "WebApp"})

# Multi-way relationship (unique to hypergraphs)
hg.add_e((1, 2, 3), {"type": "team", "project": "Database"})

# Even larger groups
hg.add_e((1, 2, 3, 4, 5), {"type": "department", "name": "Engineering"})

Real-World Applications

1. Academic Collaboration Networks

Model research papers and their authors:

# Authors as vertices
hg.add_v("alice", {"name": "Dr. Alice Smith", "field": "ML"})
hg.add_v("bob", {"name": "Dr. Bob Jones", "field": "NLP"})
hg.add_v("charlie", {"name": "Dr. Charlie Brown", "field": "Vision"})

# Papers as hyperedges connecting all co-authors
hg.add_e(("alice", "bob"), {
    "title": "Deep Learning for NLP",
    "year": 2023,
    "venue": "ICML"
})

hg.add_e(("alice", "bob", "charlie"), {
    "title": "Multimodal AI Systems", 
    "year": 2024,
    "venue": "NeurIPS"
})

2. E-commerce Transactions

Model shopping baskets with multiple items:

# Products as vertices
hg.add_v("laptop", {"category": "Electronics", "price": 999})
hg.add_v("mouse", {"category": "Electronics", "price": 25})
hg.add_v("keyboard", {"category": "Electronics", "price": 75})

# Shopping baskets as hyperedges
hg.add_e(("laptop", "mouse", "keyboard"), {
    "transaction_id": "T001",
    "customer": "John Doe",
    "total": 1099,
    "date": "2024-01-15"
})

3. Social Group Activities

Model group events and activities:

# People as vertices
people = ["alice", "bob", "charlie", "david", "eve"]
for person in people:
    hg.add_v(person, {"type": "person"})

# Group activities as hyperedges
hg.add_e(("alice", "bob", "charlie"), {
    "activity": "Study Group",
    "subject": "Machine Learning",
    "location": "Library"
})

hg.add_e(("bob", "david", "eve"), {
    "activity": "Basketball Game",
    "location": "Gym",
    "time": "Saturday 3pm"
})

Advantages of Hypergraphs

1. Natural Group Modeling

  • No need to artificially decompose group relationships into pairwise connections
  • Preserves the semantic meaning of multi-way interactions

2. Information Preservation

  • Traditional graphs lose information when representing group relationships
  • Hypergraphs maintain the original group structure

3. Flexible Queries

  • Find all groups containing a specific member
  • Discover overlapping communities naturally
  • Analyze group sizes and compositions

Mathematical Properties

Degree Concepts

In hypergraphs, we have several types of degrees:

# Vertex degree: number of hyperedges containing the vertex
vertex_degree = hg.d_v(vertex_id)

# Hyperedge size: number of vertices in the hyperedge  
edge_size = hg.d_e(edge_id)

# Get all neighbors of a vertex (vertices connected via any hyperedge)
neighbors = hg.N_v(vertex_id)

Incidence Relationships

# Get all hyperedges containing a vertex
incident_edges = hg.N_e(vertex_id)

# Get all vertices in a hyperedge
vertices_in_edge = hg.N_v_of_e(edge_id)

Comparison with Other Data Structures

Feature Graph Hypergraph Database Table
Relationship Type Pairwise Multi-way Rows & Columns
Group Modeling Artificial Natural Complex JOINs
Query Flexibility Medium High Very High
Mathematical Theory Rich Growing Different Domain
Visualization Easy Challenging Tabular

Getting Started with Hypergraph-DB

Now that you understand the basics of hypergraphs, let's explore how to use Hypergraph-DB effectively:

  1. Quick Start Guide - Basic operations and examples
  2. API Reference - Complete function documentation
  3. Advanced Examples - Complex use cases

Best Practices

  • Use meaningful attributes for both vertices and hyperedges
  • Consider the semantic meaning when deciding between multiple binary edges vs. one hyperedge
  • Leverage the natural group structure in your data
  • Use visualization to understand complex relationships