Power-hop: A pervasive observation for real complex networks

Abstract

Complex networks have been shown to exhibit universal properties, with one of the most consistent patterns being the scale-free degree distribution, but are there regularities obeyed by the r-hop neighborhood in real networks? We answer this question by identifying another power-law pattern that describes the relationship between the fractions of node pairs C(r) within r hops and the hop count r. This scale-free distribution is pervasive and describes a large variety of networks, ranging from social and urban to technological and biological networks. In particular, inspired by the definition of the fractal correlation dimension D2 on a point-set, we consider the hop-count rto be the underlying distance metric between two vertices of the network, and we examine the scaling of C(r) with r. We find that this relationship follows a power-law in real networks within the range 2