Tore Opsahl

The other day I was asked about an application of a method proposed in a recent PRL paper called the Weighted Rich-club Effect to a social science article. In this paper, we proposed a method of analysing whether a subset of prominent nodes directed their strongest ties towards each other to a greater extent than randomly expected. This method was global (gave one results for the entire network); however, in social science papers, a regression is commonly used where each node is an observation and a number of measures are created for each node. So, the problem was whether I could redefine the global weighted rich-club measure into a local one (for each node).

So to reiterate the paper quickly:
1) we define a set of prominent nodes
2) we measure the total sum of weights attached to ties among the prominent nodes (see panel a of the figure below)
3) we sum the strongest ties in the network, where is the number of ties among the prominent nodes (see panel b of the figure below)
4) we take the ratio between the sum from point 2 to the sum from point 3, . As a result, if the prominent nodes share the strongest ties in the network, . If we the prominent nodes do not share all of the strongest nodes, we get a value lower than 1.

Schematic representation of a weighted network, with size of nodes proportional to their prominence, and width of links to their weight indicated by the corresponding numbers. Several definitions of prominence can be considered. (a) The prominent nodes and the ties among them are highlighted, giving ties and . (b) The strongest ties of the network are highlighted (6), yielding the following value for the denominator of : . We thus obtain . Adapted from Opsahl et al. (2008).

However, a problem might exist. Some definitions of prominence could be associated with the strength of ties, such as the average weight of the ties originating from a node (if a set of nodes has on average stronger ties, of course it would have stronger ties among them – everything else being equal). So therefore, we divide obtained in the real network by obtained on simulated corresponding random networks, (see the paper for this part and Colizza et al., 2006).

Now, in an effort to create a local measure, we would have to retain the research question: do prominent nodes direct their strongest ties to one another?, but rewrite it in terms of a single node. So, I thought:
1) designate prominent nodes
2) for each node, sum the weights towards prominent nodes (if a node has three ties with weights of 1, 2 and 6, and the two ties with weight of 1 and 6 are directed towards prominent nodes, then the sum is 7).
3) In case the definition of prominence is associated with the weight of ties, we need to discount for the randomly expected value. To this end, we could divide the value obtained in point 2, to the randomly expected value (average weight of ties (which is 3 in this example) multiplied with the number of ties towards prominent nodes): 6.

Let me know what you guys think?

Want to test it with your data?

First, you need to load your network in R in tnet format, and ensure that it complies with the required standard (see the tnet documentation for more information on the network structure required). The documentation also contains information on exporting network data from other programmes. Then you need to define a promience vector consisting on 1′s and 0′s where 1 signify prominence and 0 non-prominence. This vector must be of the same length as the number of nodes in the network.

# Load tnet
library(tnet)

# Load the above sample 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))

# Define prominence parameter (node 1 (A), 2 (B) and 3 (C) are designated as prominent)
prominence <- c(1,1,1,0,0,0)

# Run function
weighted_richclub_w_local(net, prominence)

The output table is like this:

     node    ratio
[1,]    1 1.000000
[2,]    2 1.454545
[3,]    3 1.000000
[4,]    4 1.000000
[5,]    5 1.333333
[6,]    6 1.000000

Nodes 1, 3, 4 and 6 get a value of 1 as they do not have a choice in their behaviour because they are not connected to both prominent and non-prominent nodes. However, node 2 and 5 do have a choice in how to distributed their efforts. For example, node 5 have two ties, one with a weight of 2 to a prominent node and one with a weight of 1 to a non-prominent node. Therefore, it can be said that node 5 preferentially direct attention to prominent nodes.

References

Colizza, V., Flammini, A., Serrano, M. A., Vespignani, A., 2006. Detecting rich-club ordering in complex networks. Nature Physics 2, 110-115. arXiv:physics/0602134

Opsahl, T., Colizza, V., Panzarasa, P., Ramasco, J. J., 2008. Prominence and control: The weighted rich-club effect. Physical Review Letters 101 (168702). arXiv:0804.0417.