Understanding the Clustering Coefficient in Cliques

In graph theory, the concept of a clique and its clustering coefficient is fundamental in studying graph structures, especially in the context of social networks and network science. This article will delve into the clustering coefficient of a clique, providing clear explanations and calculations.

What is a Clique?

A clique is a subset of vertices in a graph such that every two distinct vertices are adjacent. In simpler terms, a clique is a fully connected subgraph. For example, imagine a group of friends where every member knows and is known by every other member in the group.

Clustering Coefficient of a Clique

The clustering coefficient of a clique is a measure of how closely connected the vertices of the clique are. In a clique, every vertex is connected to every other vertex.

Formal Definition and Calculation

The clustering coefficient (C_v) for a vertex (v) in a graph is defined as:

[ C_v frac{2E_v}{k_v(k_v - 1)} ]

where:

(E_v) is the number of edges between the neighbors of (v) (k_v) is the number of neighbors of (v)

Clustering Coefficient of a Clique in Depth

In a clique of size (n):

Each vertex has (k_v n - 1) neighbors The number of edges between the neighbors of any vertex is (E_v frac{n - 1}{2} cdot (n - 2)), since every pair of neighbors is also connected

Calculating the Clustering Coefficient for a Clique

Substituting these values into the clustering coefficient formula gives:

[ C_v frac{2 cdot frac{(n-1)(n-2)}{2}}{(n-1)(n-2)} 1 ]

Since this holds true for every vertex in the clique, the overall clustering coefficient of the clique is also 1. This indicates that the vertices in a clique are perfectly clustered together, forming a fully connected subgraph.

Clustering Coefficient in General Graphs

The clustering coefficient of a vertex (v) in a graph can be calculated differently:

Count the number (k_v) of vertices adjacent to (v) Count the number (e_v) of edges occurring between the vertices adjacent to (v) Form the quotient (q frac{e_v}{frac{1}{2} k_v(k_v - 1)}), where (frac{1}{2}) is used for undirected graphs since each edge is counted twice.

A low clustering coefficient of a vertex indicates that the neighborhood graph of the vertex is fairly sparse. A clustering coefficient of 1 indicates the neighborhood graph is itself a clique. This concept is particularly useful in analyzing social networks and other complex network structures.

Further Reading and Examples

To further explore the application of clustering coefficient in different contexts, refer to graph theory resources or related research papers on social network analysis.

For a deeper dive into the topic, check out the clustering coefficient on Wikipedia for additional insights and examples.