Single-Layer Networks

Single-layer networks represent the classical foundation of network theory, providing a fundamental framework for modeling relationships between entities through a single type of connection. While simpler than their multilayer networks counterparts, they remain essential for many practical applications and serve as building blocks for more complex network structures.

Core Components

Basic Elements

• Nodes (Vertices): Discrete entities or points in the network
• Edges (Links): Connections between nodes that can be:
  • Directed (one-way relationships)
  • Undirected (bidirectional relationships)
  • Weighted or unweighted

Mathematical Properties

• Represented through adjacency matrices
• Characterized by graph theory properties such as:
  • Degree distribution
  • Path lengths
  • network density

Types of Single-Layer Networks

Based on Structure

1. Random Networks

   • Erdős-Rényi model
   • Random geometric graphs

2. Regular Networks

   • Lattices
   • Circular graphs
   • Complete graphs

3. Complex Networks

   • scale-free networks
   • small-world networks
   • hierarchical networks

Analysis Methods

Network Metrics

• Centrality Measures
  • Degree centrality
  • betweenness centrality
  • eigenvector centrality
  • Closeness centrality

Structural Properties

• community detection
• network motifs
• network topology analysis
• Path analysis

Applications

Traditional Use Cases

1. Social Networks

   • Friendship networks
   • citation networks
   • Professional relationships

2. Infrastructure

   • Power grids
   • transportation networks
   • Communication systems

3. Biological Systems

   • food webs
   • neural networks
   • genetic networks

Limitations

Inherent Constraints

• Cannot represent multiple relationship types
• Limited in modeling complex real-world systems
• network complexity representation
• Loss of multi-dimensional information

Advantages

Practical Benefits

1. Simplicity

   • Easier to analyze
   • Computationally efficient
   • Clear visualization
   • Straightforward interpretation

2. Established Methods

   • Well-developed theoretical framework
   • Robust analytical tools
   • Extensive literature base
   • Proven applications

Relationship to Modern Approaches

Evolution of Network Science

• Foundation for multilayer networks
• Basis for temporal networks
• Integration with dynamic networks
• Platform for network evolution studies

Future Relevance

Despite their limitations, single-layer networks continue to be valuable for:

• Preliminary system analysis
• Basic relationship modeling
• Educational purposes
• network optimization problems
• Prototype development

The enduring significance of single-layer networks lies in their ability to provide clear, manageable representations of relationships while serving as a stepping stone to more sophisticated network analyses.