Tore Opsahl

A paper called “Clustering in Weighted Networks” that I have co-authored with Pietro Panzarasa will be published in Social Networks. Unfortunately, the copyright agreement prevents me from uploading a pdf of the published paper to this blog. However, if you have access to Social Networks, you can download the paper directly. Otherwise, a preprint with the exact same text is available.

Abstract

In recent years, researchers have investigated a growing number of weighted networks where ties are differentiated according to their strength or capacity. Yet, most network measures do not take weights into consideration, and thus do not fully capture the richness of the information contained in the data. In this paper, we focus on a measure originally defined for unweighted networks: the global clustering coefficient. We propose a generalization of this coefficient that retains the information encoded in the weights of ties. We then undertake a comparative assessment by applying the standard and generalized coefficients to a number of network datasets.

Motivation

In this sample network the binary clustering coefficient is 0.33 as a third of the triplets are closed by being part of a triangle. By looking at the weights, it is possible to see that the strongest ties are in part of the closed triplets. This is not reflected in the binary clustering coefficient.

By applying the proposed generalisation of the coefficient using the arithmetic mean method for defining triplet value, the clustering coefficient increases to 0.42. This increase of this coefficient from the binary coefficient is a reflection of the fact that the strongest ties are part of the closed triplets.

Want to test it with your data?

The clustering_w function in tnet allows you to test the generalised clustering coefficient on your own dataset.

For example, to test the clustering_w function on the sample network above, you can run the following code in R:

# Load tnet
library(tnet)

# Load network
net <- cbind(
i=c(1,1,2,2,2,2,3,3,4,5,5,6),
j=c(2,3,1,3,4,5,1,2,2,2,6,5),
w=c(4,2,4,4,1,2,2,4,1,2,1,1))

# Run function
clustering_w(net, measure=c("am", "gm", "ma", "mi"))

# The output is:
#       am        gm        ma        mi
#0.4166667 0.4361302 0.3750000 0.5000000

To test in on Freeman’s third EIES network from the datasets page, you can do the following:

# Load tnet
library(tnet)

# Load network
data(Freemans.EIES)

# Run function
clustering_w(Freemans.EIES.net.3.n32, measure=c("am", "gm", "ma", "mi"))

# The output is:
#0.7378310 0.7331536 0.7410959 0.7249982