Tore Opsahl

In the Weighted Rich-club paper, we proposed a measure to study whether, and the extent to which, of “prominent” nodes (e.g., the ones with high (out-)degree or node (out-)strength for (directed) networks) exchange among themselves the strongest ties in a network. Since certain ways of defining the prominent nodes can be associated with having strong ties, we divided the measure calculated on the observed network by the measure calculated on an ensemble of random networks (similar to the topological rich-club measure that we built on; Colizza, 2006). To ensure that the random networks were comparable to the observed network, we constrained the random networks so that each node maintained its prominence (e.g., degree or node strength). When prominence was defined as degree, this was straight-forward by either globally reshuffling the weights in the network (the topology remained, and hence, the degree of nodes) or reshuffling ties while maintaining nodes’ degree using Molloy and Reed’s (1995) method.

However, when prominence was defined as node strength, we could not use either of these methods for undirect networks (For directed networks, it is possible to redirect a node’s out-going ties randomly in the network, thus maintaining the node strength). Therefore, for undirected networks, we created directed networks by linking connected nodes with two directed ties. The weight of an undirected tie is duplicated to the two directed ones. A transformed network based on this process is is illustrated in the first panel of this figure. Then we reshuffled the weights attached to the outgoing ties of a node, thus, maintaining the sum of weights on outgoing ties. For example, in the second panel of the figure (the weight of a directed tie is placed close to the origin of the tie in this figure) we have randomly the weights of node E’s outgoing ties, 1 and 5. This method has been refered to as the directed weight reshuffling method (Serrano et al., 2007). However, this method breaks the weight symmetry of ties.

The appropriateness of this method for undirected networks depends on the research setting and how tie weights are defined (I would like to thank Tom Snijders for highlighting this during my viva). For example, its applicability to undirected transportation networks is justified by the typically directed nature of traffic flows (although the US airport network displays a high symmetry; Barrat et al., 2004). Conversely, in an undirected collaboration network this might not be appropriate. More generally, for one-mode projections of two-mode networks, it might be more appropriate to reshuffle the two-mode network before projecting it onto a one-mode network (see randomisation without structural zeros in Rao et al., 1996; Snijders, 1991).

To test this method, I have applied it to the scientific collaboration network used in the Weighted Rich-club paper. Although the two-mode structure of this network is not publically available, Mark Newman sent me a copy of it (Thanks!).

To calculate using this null model, I first randomised the two-mode network by reshuffling ties while maintaining authors’ and papers’ degree (similarly to the method proposed by Molloy and Reed, 1995, for one-mode networks). One step of this method is exemplified in the diagram below. I then projected the randomised two-mode network onto a weighted one-mode network using the same method in the paper (for more details, see my post on projecting two-mode networks). Finally, I calculated the weighted-rich club effect, , on the one-mode projections. The is the average over many random networks.

The result from this procedure is the displayed in the following diagram (the bars refer to the 95% confidence interval, see Section 3.2.2 of my thesis for more details):

Weighted rich-club coefficient for Newman's (2001) scientific collaboration network. is the average measured on 1,000 one-mode projections of randomised two-mode network.

The diagram suggest that, unlike what we found using the Directed weight reshuffle (see Section 3.3 of my thesis), when prominence is defined as having co-authored more than 20 articles, there is a negative and significant weighted rich-club effect. Thus, this finding is a substantiation of the speculation that in certain social networks a negative effect exists among the very prominent people due to some form of competition. This might account for the reluctance of highly productive authors to establish strong ties among themselves, as is suggested by the lack of interaction among the two most productive authors in the Figure below: Barabasi and Newman are not connected.¹

Subset of the prominent nodes () in the network science collaboration network (Newman, 2006). Only ties among the prominent nodes are shown. The size of the nodes corresponds to the strength or number of co-authored papers, and the width of each tie is proportional to its weight. This figure is adapted from Figure 8B in my thesis, which in turn was based on Figure 3B in Opsahl et al. (2008).

The null model used in this post could also be used in a number of other frameworks. For example, it could be used in the topological rich-club (Colizza, 2006; Zlatic et al., 2008). In particular, it might be more appropriate for Zlatic et al. (2008) as they defined prominence as node strength.
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¹ Although Albert-László Barabási and Mark Newman have written a book together with Duncan Watts, it did not form part of this dataset. Nevertheless, this collaboration would add only a weak tie between these authors.

Want to test it with your data?

The following code was used to produced the above diagrams:

# Load tnet
library(tnet)

# Download data
data(Newman.Condmat.95.99)

# Run function
weighted_richclub_tm(Newman.Condmat.95.99.net.2mode)

References

Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A., 2004. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences 101 (11), 3747-3752. arXiv:cond-mat/0311416

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

Molloy, M., Reed, B., 1995. A critical point for random graphs with a given degree sequence. Random Structures and Algorithms 6, 161-180.

Newman, M. E. J., 2001. Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Physical Review E 64, 016132.

Newman, M. E. J., 2006. Finding community structure in networks using the eigenvectors of matrices. Physical Review E 76 (036104). arXiv:physics/0605087

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.

Rao, A. R., Jana, R., Bandyopadhyay, S., 1996. A markov chain monte carlo method for generating random (0, 1)-matrices with given marginals. Sankhya A 58, 225- 242.

Serrano, M. A., Boguna, M., Vespignani, A., 2007. Patterns of dominant flows in the world trade web. Journal of Economic Interaction and Coordination 2, 111-124. arXiv:0704.1225.

Snijders, T. A. B., 1991. Enumeration and simulation methods for 0-1 matrices with given marginals. Psychometrika 56 (3), 397-417.

Zlatic, V., Bianconi, G., Diaz-Guilera, A., Garlaschelli, D., Rao, F., Caldarelli, G., 2008. On the rich-club effect in dense and weighted networks. arXiv:0807.0793.