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Concept
Clustering coefficient
Summary
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes (Holland and Leinhardt, 1971; Watts and Strogatz, 1998).
Two versions of this measure exist: the global and the local. The global version was designed to give an overall indication of the clustering in the network, whereas the local gives an indication of the embeddedness of single nodes.
The local clustering coefficient of a vertex (node) in a graph quantifies how close its neighbours are to being a clique (complete graph). Duncan J. Watts and Steven Strogatz introduced the measure in 1998 to determine whether a graph is a small-world network.
A graph formally consists of a set of vertices and a set of edges between them. An edge connects vertex with vertex .
The neighbourhood for a vertex is defined as its immediately connected neighbours as follows:
We define as the number of vertices, , in the neighbourhood, , of a vertex.
The local clustering coefficient for a vertex is then given by a proportion of the number of links between the vertices within its neighbourhood divided by the number of links that could possibly exist between them. For a directed graph, is distinct from , and therefore for each neighbourhood there are links that could exist among the vertices within the neighbourhood ( is the number of neighbours of a vertex). Thus, the local clustering coefficient for directed graphs is given as
An undirected graph has the property that and are considered identical. Therefore, if a vertex has neighbours, edges could exist among the vertices within the neighbourhood. Thus, the local clustering coefficient for undirected graphs can be defined as
Let be the number of triangles on for undirected graph .
Official source
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