Potential gain as a centrality measure

TL;DR

This paper introduces the potential gain, a new centrality measure combining popularity and similarity, to explain why some nodes are more easily reachable in navigable graphs, supported by fast algorithms and theoretical proof.

Contribution

It proposes the potential gain as a novel composite centrality metric, integrating degree and Katz centrality, with formal proof and efficient computation methods.

Findings

Potential gain is equivalent to degree centrality times Katz centrality.

Two variants of potential gain are defined: geometric and exponential.

Algorithms for fast computation of potential gain are presented.

Abstract

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).