The Relation Between Local and Global Influence of Individuals in Scale-Free Networks

Abstract

Large-degree nodes in scale-free networks are normally responsible for large cascades of epidemics. However, recent research shows small-degree nodes can also produce large-scale epidemics in the real world. In this letter, we investigate the relation between local and global influence of individuals in scale-free network in order to theoretically explain this real-world phenomenon. The local influence of an individual corresponds to the node degree, and the global influence of an individual reflects the expected number of individuals directly or indirectly influenced by this individual in epidemics. We formalize the later as the novel epidemic betweenness concept, to mathematically estimate the global influence of individuals. Our analysis shows that the global influence follows power-law distributions in scale-free networks. We also observe that the average global influence of individuals is power-law to the degree of nodes, which well explains the reason why large-degree nodes are more likely to produce large cascades of epidemics. In addition, we discover that some smalldegree nodes also possess large global influence in terms of epidemics betweenness. This well explains the counter-intuitive phenomenon in recent research.