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We define the neighbourhood N for a vertex as its immediately connected neighbours as follows:
The degree of a vertex is the number of vertices, , in its neighbourhood .
The clustering coefficient for a vertex is the proportion 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. Thus, the clustering coefficient 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 clustering coefficient for undirected graphs can be defined as:
Let be the number of triangles on for undirected graph . That is, is the number of subgraphs of with 3 edges and 3 vertices, one of which is . Let be the number of triples on . That is, is the number of subgraphs (not necessarily induced) with 2 edges and 3 vertices, one of which is and such that is incident to both edges. Then we can also define the clustering coefficient as:
It is simple to show that the two preceding definitions are the same, since .
These measures are 1 if every neighbour connected to is also connected to every other vertex within the neighbourhood, and 0 if no vertex that is connected to connects to any other vertex that is connected to .
The clustering coefficient for the whole system is given by Watts and Strogatz as the average of the clustering coefficient for each vertex:
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