Question

Difference between global and local clustering coeffien

Answer 1

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;[1] Watts and Strogatz, 1998[2]).

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 global clustering coefficient is based on triplets of nodes. A triplet is three nodes that are connected by either two (open triplet) or three (closed triplet) undirected ties. A triangle graph therefore includes three closed triplets, one centered on each of the nodes (n.b. this means the three triplets in a triangle come from overlapping selections of nodes). The global clustering coefficient is the number of closed triplets (or 3 x triangles) over the total number of triplets (both open and closed). The first attempt to measure it was made by Luce and Perry (1949).[3] This measure gives an indication of the clustering in the whole network (global), and can be applied to both undirected and directed networks (often called transitivity, see Wasserman and Faust, 1994, page 243[4]).

The global clustering coefficient is defined as:

{\displaystyle C={\frac {\mbox{number of closed triplets}}{\mbox{number of all triplets (open and closed)}}}} {\displaystyle C={\frac {\mbox{number of closed triplets}}{\mbox{number of all triplets (open and closed)}}}}.
The number of closed triplets has also been referred to as 3 × triangles in the literature, so:

{\displaystyle C={\frac {3\times {\mbox{number of triangles}}}{\mbox{number of all triplets}}}} {\displaystyle C={\frac {3\times {\mbox{number of triangles}}}{\mbox{number of all triplets}}}}.
A generalisation to weighted networks was proposed by Opsahl and Panzarasa (2009),[5] and a redefinition to two-mode networks (both binary and weighted) by Opsahl (2009).[6]