Prominence and Control: The Weighted Rich-Club Effect

Abstract

Complex systems are often characterized by large-scale hierarchical organizations. Whether the prominent elements, at the top of the hierarchy, share and control resources or avoid one another lies at the heart of a system's global organization and functioning. Inspired by network perspectives, we propose a new general framework for studying the tendency of prominent elements to form clubs with exclusive control over the majority of a system's resources. We explore associations between prominence and control in the fields of transportation, scientific collaboration, and online communication.PACS numbers: 89.75. Hc,89.40.Dd Research has long documented the abundance of systems characterized by heterogeneous distribution of resources among their elements [1,2]. Back in 1897, Pareto noticed the social and economic disparity among people in different societies and countries [1]. This empirical regularity inspired the 80−20 rule of thumb stating that only a select minority (20%) of elements in many real-world settings are responsible for the vast majority (80%) of the observed outcome. While recent studies have examined the tendency of prominent elements to establish connections among themselves [3], how they leverage on their connections to gain and maintain control over resources circulating in a system still remains largely unexplored. In particular, do they collude and choose to exchange a disproportionately large amount of resources among themselves rather than with others? Or does competition prevent them from deepening the connections they have with one another? To answer these questions, we need to test for the tendency of prominent elements to engage in stronger or weaker interactions among themselves than expected by chance. We call this tendency the weighted rich-club effect. In this Letter, we adopt the framework of network theory -where the elements of the system are seen as nodes and the links among the elements represent interactions [4,5,6,7,8] -and provide a novel method to properly assess this tendency.Previous work has focused on highly connected nodes and the degree to which they preferentially interact among themselves [3]. This feature is known as the richclub phenomenon [3,9], a metaphor that alludes to the tendency of the highly connected nodes (i.e., the rich nodes) to establish more links among themselves than randomly expected. Evidence of the phenomenon has been reported for scientific collaboration networks [3], transportation networks [3] and inter-bank networks [10]. Conversely, research has shown that highly connected routers on the Internet tend not to be connected with one another [3], whereas the pattern of interactions among proteins has been found to depend on the particular organism under consideration [3,11]. Although uncovering interesting structural aspects of the systems, these studies are limited in that they only detect whether or not links among prominent nodes are present. In so doing, they neglect a crucial piece of information encoded in the weight of links, which is...

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