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 G is navigable when an agent is able to efficiently reach any target node in 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 {\em 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 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)

Branching processes reveal influential nodes in social networks

Branching processes are discrete-time stochastic processes which have been largely employed to model and simulate information diffusion processes over large online social networks such as Twitter and Reddit. Here we show that a variant of the branching process model enables the prediction of the popularity of user-generated content and thus can serve as a method for ranking search results or suggestions displayed to users. The proposed branching-process variant is able to evaluate the importance of an agent in a social network and, thus we propose a novel centrality index, called the Stochastic Potential Gain (SPG). The SPG is the first centrality index which combines the knowledge of the network topology with a dynamic process taking place on it which we call a graph-driven branching process. SPG generalises a range of popular network centrality metrics such as Katz’ and Subgraph. We formulate a Monte Carlo algorithm (called MCPG) to compute the SPG and prove that it is convergent and correct. Experiments on two real datasets drawn from Facebook and GitHub demonstrate that MCPG traverses only a small fraction of nodes to produce its result, thus making the Stochastic Potential Gain an appealing option to compute node centrality measure for Online social networks

A general centrality framework-based on node navigability

Centrality metrics are a popular tool in Network Science to identify important nodes within a graph. We introduce the Potential Gain as a centrality measure that unifies many walk-based centrality metrics in graphs and captures the notion of node navigability, interpreted as the property of being reachable from anywhere else (in the graph) through short walks. Two instances of the Potential Gain (called the Geometric and the Exponential Potential Gain) are presented and we describe scalable algorithms for computing them on large graphs. We also give a proof of the relationship between the new measures and established centralities. The Geometric potential gain of a node can thus be characterized as the product of its Degree centrality by its Katz centrality scores. At the same time, the exponential potential gain of a node is proved to be the product of Degree centrality by its Communicability index. These formal results connect potential gain to both the "popularity" and "similarity" properties that are captured by the above centralities