Clustering Coefficient Calculator

Historical Background

The concept of clustering coefficient emerged in graph theory and network science to describe how nodes cluster together in graphs representing social networks, transportation systems, and other structures. It provides a numerical value reflecting the degree to which nodes tend to form tightly connected groups.

Calculation Formula

The formula to calculate the clustering coefficient is simple:

\[ C = \frac{CT}{AT} \]

where:

• C is the clustering coefficient,
• CT is the number of closed triplets,
• AT is the number of all triplets (closed and open).

Example Calculation

If a graph has 12 closed triplets and 30 total triplets, the clustering coefficient is:

\[ C = \frac{12}{30} = 0.4 \]

Importance and Usage Scenarios

Clustering coefficients are essential in social network analysis, biological network studies, and in various other applications where the structure of relationships between nodes is significant. It helps in understanding the local cohesiveness of networks and the potential for the formation of tightly-knit communities.

Common FAQs

1. What is a triplet in graph theory?

   • A triplet is a set of three nodes that are interconnected. A closed triplet means all three nodes are directly connected to each other, forming a triangle. An open triplet is a set of three nodes with only two direct connections.

2. What does a high clustering coefficient indicate?

   • A high clustering coefficient indicates that the nodes in a graph tend to form tightly connected clusters or communities.

3. Can the clustering coefficient be used to study social networks?

   • Yes, it can be used to understand social interactions and the likelihood of forming tightly-knit groups or communities.