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Clustering Coefficients in Weighted Complex Networks

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC374315/

This article, titled “The architecture of complex weighted networks” by A. Barrat, M. Barthelemy, R. Pastor-Satorras, and A. Vespignani, discusses the structures of large and complex networks with weights defined on the edges and also defines some interesting measures for the weighted networks including the weighted clustering coefficient.  We have learned in our textbook in Chapter 3 of “Networks, Crowds and Markets” of a social network measure called the “clustering coefficient“(page 44). The clustering coefficient of a node A is defined as “the probability that two randomly selected friends of A are friends with each other.”  If a node has a high clustering coefficient, then many of its friends are also friends. If most of the nodes in the network have high clustering coefficient, then the network will probably have many edges that connect nodes to each other. Thus, we see a high prevalence of “clusters” in the network, and hence why this measure is probably called the “clustering coefficient.” In applications to real-life situations, Dr. Barrat’s paper introduces the concept of a “weighted network,” in which the dynamics of information or traffic flow specified in a network often defines weighted networks to be very  complex.

The paper discusses two examples of the weighted complex networks: the world-wide airport network (WAN) and the scientist collaboration network (SCN). The WAN network contains the world list of airport pairs connected by direct flights and the number of available seats on any given connection for the year 2002 which define the weights on the connections. The SCN is a network of scientists who have authored manuscripts submitted to the e-Print Archive relative to condensed matter physics between 1995 and 1998 where the weights are specified as the intensity of the interaction between scientists. I think the weighted networks are very useful to specify a variety of applications and the weights can be used to describe the degree of the strength between the nodes.  It is interesting to see how the paper defines the clustering coefficient in the context of the weighted networks: according to the paper, the weighted clustering coefficient of a node A defined by Dr. Barrat and his co-authors counts for each triangle formed in the neighborhood of the node A and is the average weight of the two participating edges of the node A.  So their definition considers not only the number of triangles in the neighborhood of node A but also the weight with respect to the strength of the node. Note that their definition only considers the weights of the edges adjacent to node A but not the weight of the edge which connects the neighbors of A.  I wondered why they formed their definition that way, and I think for some applications it’s more important to just consider the weights of the edges adjacent to the node but for other applications it may be necessary to also include the weight of the edge which connects the neighbors of the node. I also think it may be necessary to consider different ways to include the weights into the definition of the clustering coefficient rather than just taking the average of the weights in the neighborhood.

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