Potential gain as a centrality measure

Navigability is a distinctive features of graphs associated with artificial or natural systems whose primary goal is the transportation of information or goods. We say that a graph $\mathcal{G}$ is navigable when an agent is able to efficiently reach any target node in $\mathcal{G}$ by means of local routing decisions. In a social network navigability translates to the ability of reaching an individual through personal contacts. Graph navigability is well-studied, but a fundamental question is still open: why are some individuals more likely than others to be reached via short, friend-of-a-friend, communication chains? In this article we answer the question above by proposing a novel centrality metric called the potential gain, which, in an informal sense, quantifies the easiness at which a target node can be reached. We define two variants of the potential gain, called the geometric and the exponential potential gain, and present fast algorithms to compute them. The geometric and the potential gain are the first instances of a novel class of composite centrality metrics, i.e., centrality metrics which combine the popularity of a node in $\mathcal{G}$ with its similarity to all other nodes. As shown in previous studies, popularity and similarity are two main criteria which regulate the way humans seek for information in large networks such as Wikipedia. We give a formal proof that the potential gain of a node is always equivalent to the product of its degree centrality (which captures popularity) and its Katz centrality (which captures similarity).