Statistical Properties of Long-Range Interaction in Complex Networks

Abstract

We introduce a linking force between two nodes in a complex network; the force considered to be proportional to the degree of each node and the inverse square of the shortest path length between them. The importance of a node can be inferred from the resultant linking force (RLF), which is the sum of all linking forces acting on the node. To characterize the statistical properties of the RLF, we measure the RLF F(k) as a function of the degree k and find that it follows the power law F(k)∼k^α, with α≈1.3, for scale-free networks with the degree exponent ranging from 2 to 3 and the random network. It indicates that the exponent α is same irrespective of the structures of complex networks, which differ from the previous findings in which the most exponents depends on the structure of networks. We also find that the distribution of the RLF follows power law, with the exponent depending logarithmically on the degree exponent; additionally the mean RLF on all nodes also shows a power law relation with mean degree.