Sharpening the definition of centrality

Since Web 2.0, online content has become social. The popularization of Internet access added to the ease of producing online content have brought social communities to go online. This penetration of social assets in the modern internet has a wide impact, which naturally led computer scientists to analyze these online social networks. The promise of this research is for instance, from a network point of view, to discover who will exchange which information, and when. These three pieces of information are obviously central in network design. Among the available tools to dig and leverage information out of complex interaction structures, probably the most interesting one is the centrality notion. Originally defined to capture “the importance of an actor in a network”[4], it has since found many application in various fields, such as management, physics, or biology. In fact, centrality appears to be a much more fine-grain tool to understand the inner structure of social networks than other information such as degree distribution. Recently, Newmann even used it to study one of the core elements of online social networks: communities [9] However, the apparently simple notion of centrality hides many different concepts. Despite its success, centrality lacks a proper definition, and may more be referred to as an abstract family of measures rather than to a single notion. There is unfortunately no generic definition of a centrality as a set of mathematical properties, as there is no absolute definition of the “importance” of an actor. So for each subjective definition of importance, an ad-hoc centrality definition has been provided. In this paper, we prospect for a more precise definition of centrality. We believe that the mathematical usefulness of this tool could greatly benefit from such definition.