Tore Opsahl

Thesis: 3 Prominence and Control: The Weighted Rich-club Effect

As described in the previous chapters, there is a growing sense of urgency within the scientific community to develop measures for weighted networks. Most efforts have been directed towards the generalisation of binary network measures (e.g. Barrat et al., 2004; Freeman et al., 1991; Newman, 2001c; Opsahl and Panzarasa, 2009); however, weighted networks can contain patterns and regularities that do not exist in binary networks. For example, the location of the strongest ties in a network is only possible to identify if the ties are weighted. Due to the increase in the number of collected weighted networks, the development of methods to detect new features of these networks is imperative.

This Chapter aims to study whether a subset of nodes exchange among themselves the strongest ties in the network. To select a subset of nodes, we rely on previous studies have shown that the elements of a wide range of systems, ranging from technological to economic and social ones, are organised into hierarchies (Pareto, 1897; Price, 1965; Simon, 1955; Zipf, 1935). For example, Pareto (1897) argued that 80% of the wealth within a society is controlled by 20% of the population. This hetrogeniety is also found for a variety of properties in a wide range of networks (Barabasi and Albert, 1999; Barrat et al., 2004; Pastor-Satorras and Vespignani, 2004). For example, Barabasi and Albert (1999) found that the vast majority of links on the Internet point towards a relatively small subset of webpages. Studying the nature of the interactions between the nodes at the top of the hierarchies (the prominent nodes) can provide useful insights into the network’s organisation and functioning. In fact, scholars have already embarked on this avenue of investigation. In particular, Colizza et al. (2006) tested the tendency of highly connected nodes to form tighter interconnected groups than randomly expected. This property is known as the topological rich-club phenomenon (Colizza et al., 2006; Zhou and Mondragon, 2004). By allowing us to discover patterns of interactions (or their absence) at the top of the hierarchy, the rich-club phenomenon helps highlight organisational principles in the network. This approach, however, is limited by the binary nature of ties on which it draws, whereas a wealth of information is contained within the strength of ties (Barrat et al., 2004; Reagans and McEvily, 2003).

Given the relevance of the strong ties in many processes (see Section 1.1 for more details), this chapter proposes a new general measure aimed at evaluating whether, and the extent to which, strong ties occur among prominent nodes. Unlike the topological rich-club assessment which focused on the highly connected nodes, we do not limit the analysis to solely this group of nodes and define the prominent nodes as the ones that rank at the highest levels according to any ordering property in the network. This flexibility reflects the empirical findings that a wide range of ordering properties are heterogeneously distributed in networks (Pastor-Satorras and Vespignani, 2004; Serrano et al., 2007; Zlatic et al., 2008)..

The analysis is undertaken within a two-fold framework. First, by focusing on an ordering property, a subset of prominent nodes is selected. Second, the preference of these nodes to direct their efforts towards one another, by forging ties that are stronger than randomly expected, is examined. By shifting attention from the binary structure to tie strength, this method thus extends previous research on the rich-club phenomenon, and provides a general framework for detecting non-trivial patterns of interaction among the prominent nodes of a weighted network.