The Derivatives of Centrality and their Applications in Visualizing Social Networks

In this paper, we introduce the notion of derivatives of centrality metrics for graph visualizations. As centrality represents the prestige or importance of a node in a network, its derivative with respect to any other node represents the influencing power it has over that node. Therefore, derivatives tell us how much a given node influences the importance of another node, even if they are not directly connected. We study three different centrality metrics and show the different results when visualizing their derivatives for a number of social and other scale-free networks. We show that derivatives not only provide an analysis tool for social networks, and also help us simplify the layout of complex networks in a way that retains the main structural properties. Centrality derivatives also help to visually measure the centralization degree of a network and provide the necessary information for estimating other metrics, such as structural balance and uncertainty. Through a number of examples, we show the flexibility and generality of this approach, and a general mechanism for extending this to any centrality metric.