The behavior of rich-club coefficient in scale-free networks

The rich-club phenomenon, which provides information about the association between nodes, is a useful method to study the hierarchy structure of networks. In this work, we explore the behavior of rich-club coefficient (RCC) in scale-free networks, and find that the degree-based RCC is a power function of degree centrality with power exponent $\beta$ and the betweenness-based RCC is a linear function of betweenness centrality with slope $\theta$. Moreover, we calculate the value of RCC in a BA network by deleting nodes and obtain a general expression for RCC as a function of node sequence. On this basis, the solution of RCC for centrality is also obtained, which shows that the curve of RCC is determined by the centrality distribution. In the numerical simulation, we observe: $\beta$ and $\gamma$ (the degree distribution exponent) increase together, $\theta$ increases with the average degree and decreases to convergence as $\gamma$ increases.