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Local Clustering Coefficient (LCC)Definition
Clustering Coefficient
The clustering coefficient C(p) is defined as follows. Suppose that a vertex v has k_{v} neighbours; then at most k_{v}(k_{v}1)/2 edges can exist between them (this occurs when every neighbour of v is connected to every other neighbour of v). Let C_{v} denote the fraction of these allowable edges that actually exist. Define C as the average of C_{v} over all v. For friendship networks, these statistic have intuitive meanings: C_{v} reflects the extent to which friends of v are also friends of each other; and thus C measures the cliquishness of a typical friendship circle. [WATTS, D. 1998] Local Clustering Coefficient The local clustering coefficient of a vertex (node) in a graph quantifies how close its neighbors are to being a clique (complete graph). The local clustering coefficient C_{i} for a vertex V_{i}
is then given by the proportion of links between the vertices within its
neighbourhood divided by the number of links that could possibly exist between
them. An undirected graph has the property that e_{ij} and e_{ji} are considered identical. Therefore, if a vertex Vi has K_{i} neighbours, K_{i}(K_{i}1)/2 edges could exist among the vertices within the neighbourhood. Thus, the local clustering coefficient for undirected graphs can be defined as Transitivity Transitivity measures the probability that the adjacent vertices of a vertex are connected. This is sometimes also called the clustering coefficient. The local transitivity of a vertex is the ratio of the triangles connected to the vertex and the triples centered on the vertex. For directed graph the direction of the edges is ignored. Weighted Transitivity There are several generalizations of transitivity to weighted graphs, and the definition by A. Barrat, is a local vertexlevel quantity, its formula is In Bipartite Networks LIEBIG, J. & RAO, A. 2014. A Clustering Coefficient to Identify Important Nodes in Bipartite Networks. arXiv preprint arXiv:1406.5814. Software
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