Adjacency Matrix and Networks

Introduction to Adjacency Matrices

• Networks are visually appealing but can be difficult to analyze mathematically.
• Adjacency Matrix: A tool to convert networks into a matrix format for easier mathematical manipulation.

Constructing an Adjacency Matrix

• Use a matrix format labeled with network points, e.g., ABCDE.
• Fill the matrix with 0s and 1s:
  • 1: If two points are connected.
  • 0: If two points are not connected.

Example Construction

1. A and B: Not connected, so 0 at intersection.
2. A and C: Connected, so 1 at intersection.
3. B and D: Not connected, so 0 at intersection.
4. B and E: Connected, so 1 at intersection.
5. E and E: Connected via a loop, so 1 at intersection.

Characteristics of Adjacency Matrix

• Self-Loops: A point connected to itself gets a 1 if there is a loop, otherwise 0.
• Symmetry:
  • If a diagonal line is drawn through the matrix, the matrix is symmetrical.
  • This symmetry implies a certain structure in the network, likened to a 'butterfly' print.
  • Example: AC = CA, AE = EA.

Summary

• Adjacency matrices are a simplified representation of networks.
• Useful for performing mathematical operations and analyses on network data.
• Symmetry is a notable feature in these matrices when dealing with simple networks.