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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 kv neighbours; then at most kv(kv-1)/2 edges can exist between them (this occurs when every neighbour of v is connected to every other neighbour of v). Let Cv denote the fraction of these allowable edges that actually exist. Define C as the average of Cv over all v. For friendship networks, these statistic have intuitive meanings: Cv 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 Ci for a vertex Vi 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.
For a directed graph, eij is distinct from eji, and therefore for each neighbourhood Ni there are Ki(Ki-1) links that could exist among the vertices within the neighbourhood (Ki is the number of neighbors of a vertex). Thus, the local clustering coefficient for directed graphs is given as:

Clustering Coefficient

An undirected graph has the property that eij and eji are considered identical. Therefore, if a vertex Vi has Ki neighbours, Ki(Ki-1)/2 edges could exist among the vertices within the neighbourhood. Thus, the local clustering coefficient for undirected graphs can be defined as

Clustering Coefficient
If a node have high local clustering coefficient value, this node indicates local cohesiveness and a high tendency to form groups.

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 vertex-level quantity, its formula is
Weighted Transitivity
si is the strength of vertex i, aij are elements of the adjacency matrix, ki is the vertex degree, wij are the weights. This formula gives back the normal not-weighted local transitivity if all the edge weights are the same. The barrat type of transitivity does not work for graphs with multiple and/or loop edges. [CSARDI, G. 2008]

In Bipartite Networks
LIEBIG, J. & RAO, A. 2014. A Clustering Coefficient to Identify Important Nodes in Bipartite Networks. arXiv preprint arXiv:1406.5814.

References

  • WATTS, D. J. & STROGATZ, S. H. 1998. Collective dynamics of 'small-world' networks. Nature, 393, 440-2.
  • CSARDI, G. & NEPUSZ, T. 2006. The igraph software package for complex network research. InterJournal, Complex Systems, 1695. [http://igraph.org]
  • BARRAT, A., BARTHELEMY, M., PASTOR-SATORRAS, R. & VESPIGNANI, A. 2004. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences of the United States of America, 101, 3747-3752.
  • JUNG, the Java Universal Network/Graph Framework [http://jung.sourceforge.net]


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