Clustering in a hyperbolic model of complex networks

In this paper we consider the clustering coefficient, and function in a random graph model proposed by Krioukov et al. 2010. model, nodes are chosen randomly inside disk hyperbolic plane two connected if they at most certain distance from each other. It has been previously shown that various properties associated with complex networks, including power-law degree distribution, “short distances” non-vanishing coefficient. The is specified using three parameters: number of $n$, which think as going to infinity, $\alpha , \nu > 0$, constant. Roughly speaking, parameter $ controls power law exponent sequence $\nu average degree. Here show coefficient tends probability constant $\gamma give explicitly closed form expression terms special functions. This improves earlier work Gugelmann al., who proved remains bounded away zero high probability, but left open issue convergence limiting Similarly, able $c(k)$, over all vertices exactly $k$, limit (k)$ We extend last result also sequences $(k_{n})_{n}$ where $k_{n}$ grows $n$. Our results scales differently, $k$ grows, for different ranges $. More precisely, there exists constants $c_{\alpha ,\nu }$ depending on $, such $k \to \infty (k) \sim c_{\alpha } \cdot k^{2 - 4\alpha $\frac {1}{2} < \alpha \frac {3}{4}$, \log k^{-1}$ =\frac {3}{4}$ when {3}{4}$. These contradict claim stated should always scale $k^{-1}$ let grow.