## Thesis: Appendix B – Appendix to Prominence and Control: The Weighted Rich-club

This chapter is based on an article co-authored with Vittoria Colizza, Pietro Panzarasa, and Jose J. Ramasco (see Opsahl et al., 2008).

B.1 Directed Weight reshuffle when prominence is defined in terms of degree

In Figure 7, for the sake of clarity, we only show the results obtained with the two randomisation procedures. We chose the first two models, the Weight reshuffle and the Weight & Tie reshuffle, as they are the ones that lead to the most randomised networks. However, when prominence is defined in terms of the node degree k, all three randomisation procedures introduced in Chapter 3 are considered to be appropriate since they all preserve the degree distribution $P(k)$ of the observed network. More specifically, in addition to the Weight and Weight & tie reshuffling procedures, the Directed weight reshuffling is also appropriate. This allows us to compare the results obtained with the three randomisation procedures on the empirical networks. In Figure 21 we report results obtained with all three reshuffling procedures. It shows that the three randomisation procedures give similar results in all networks under study. This strengthens the results reported in the chapter when prominence is defined in terms of node degree, and also provides support to the application of the Directed Weight reshuffle in the investigation of the weighted rich-club ordering when prominence is defined in terms of node strength and node average weight.

Figure 21: Weighted rich-club ordering among the most connected nodes in: the US Airport Network (top); the Scientific Collaboration Network (middle); and the Online Social Network (bottom). Results obtained with the three null models are shown. We have omitted the confidence interval of the random networks for the sake of clarity.

B.2 Weighted rich-club effect in the Network Science collaboration network

Here we report the results of the weighted rich-club effect on the Network Science collaboration network (Newman, 2006). To motivate the choice of additional prominence parameters, Figures 8a and b showed the scientists with high degree and strength, respectively, working on networks (theory and experiments). As shown in Figure 22, the smaller collaboration network displays the same behaviour observed in the larger overall scientific collaboration network: a marked positive trend in the topological ordering, as opposed to a random behaviour when the intensity of collaborations is considered.

Figure 22: Weighted rich-club ordering among the most connected nodes in the Network Science collaboration network. The inset refers to the topological rich-club ordering.