Power-Hop: A Pervasive Observation for Real Complex Networks
CARNEGIE-MELLON UNIV PITTSBURGH PA PITTSBURGH United States
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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 Cr 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 dimensionD2 on a point-set, we consider the hop-count r to be the underlying distance metric between two vertices of the network, and we examine the scaling of Cr with r. We find that this relationship follows a power-law in real networks within the range 2 r d, where d is the effective diameter of the network, that is, the 90-th percentile distance. We term this relationship as power-hop and the corresponding power-law exponent as power-hop exponent h. We provide theoretical justification for this pattern under successful existing network models, while we analyze a large set of real and synthetic network datasets and we show the pervasiveness of the power-hop.
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