## Thesis: 3.1 The topological rich-club effect

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

This work draws on, and extends, the topological measure of the rich-club phenomenon (Colizza et al., 2006; Zhou and Mondragon, 2004), that quantifies the extent to which the nodes with a high degree (i.e., the number of ties originating from a node, k) are connected with each other to a greater extent than randomly expected. In this measure, the prominent nodes are defined as the hubs that preside over many ties with other nodes. This choice was based on the discovery of skewed degree distributions (i.e., the probability that a given tie has degree k, $P(k)$) in a host of networks (Barabasi and Albert, 1999; Dorogovtsev and Mendes, 2003, pg. 80-81).

Formally, the topological rich-club coefficient is the proportion of ties connecting prominent nodes, with respect to the maximum possible number of ties among them. For the set of prominent nodes with degree larger than k, $N_{>k}$, the coefficient is defined for undirected networks as (Zhou and Mondragon, 2004) (equation 6):

$\phi(k)=\frac{2A_{>k}}{N_{>k}(N_{>k}-1)}$

where $A_{>k}$ represents the number of ties connecting the $N_{>k}$ prominent nodes. This equation is not enough to test whether highly connected nodes are connected with each other to a greater extent than randomly expected. Since the highly connected nodes have relatively many ties compared with the other nodes in the network, the likelihood that a tie is randomly located between them is higher than the likelihood of a tie in the overall network (Colizza et al., 2006). Therefore, in order to detect the non-random tendency towards the generation of rich-club structures, $\phi(k)$ measured on the observed network must be compared with the corresponding $\phi_{\mathrm{null}}(k)$ obtained from an appropriate null model. The null model is typically used as a benchmark to assess whether a property measured in a real-world network deviates from what would be observed by chance (Amaral and Guimera, 2006). For the topolgical rich-club measure, Colizza et al. (2006) proposed the following ratio (equation 7):

$\rho(k)=\frac{\phi(k)}{\phi_{\mathrm{null}}(k)}$

This ratio enables us to examine the extent to which the observed rich-club phenomenon diverges from what would be expected by chance.

A positive topological rich-club phenomenon has been found in networks of scientific collaborations among researchers (Colizza et al., 2006), in transportation networks (Colizza et al., 2006; Opsahl et al., 2008), in the Italian interbank network (De Masi et al., 2006), and in content-based networks (Balcan and Erzan, 2007). On the contrary, a negative tendency was found for the Internet, where highly connected routers are not typically connected with one another (Colizza et al., 2006). Biological networks, such as protein-protein interaction networks, do not show a consistent trend. Studies suggests that the trends are related to specific features of the organisms under study (Colizza et al., 2006; Guimera et al., 2007; Wuchty, 2007).