Scale-free tree network with an ultra-large diameter

Scale-free networks are prevalently observed in a great variety of complex systems, which triggers various researches relevant to networked models of such type. In this work, we propose a family of growth tree networks $\mathcal{T}_{t}$, which turn out to be scale-free, in an iterative manner. As opposed to most of published tree models with scale-free feature, our tree networks have the power-law exponent $\gamma=1+\ln5/\ln2$ that is obviously larger than $3$. At the same time, "small-world" property can not be found particularly because models $\mathcal{T}_{t}$ have an ultra-large diameter $D_{t}$ (i.e., $D_{t}\sim|\mathcal{T}_{t}|^{\ln3/\ln5}$) and a greater average shortest path length $\langle\mathcal{W}_{t}\rangle$ (namely, $\langle\mathcal{W}_{t}\rangle\sim|\mathcal{T}_{t}|^{\ln3/\ln5}$) where $|\mathcal{T}_{t}|$ represents vertex number. Next, we determine Pearson correlation coefficient and verify that networks $\mathcal{T}_{t}$ display disassortative mixing structure. In addition, we study random walks on tree networks $\mathcal{T}_{t}$ and derive exact solution to mean hitting time $\langle\mathcal{H}_{t}\rangle$. The results suggest that the analytic formula for quantity $\langle\mathcal{H}_{t}\rangle$ as a function of vertex number $|\mathcal{T}_{t}|$ shows a power-law form, i.e., $\langle\mathcal{H}_{t}\rangle\sim|\mathcal{T}_{t}|^{1+\ln3/\ln5}$. Accordingly, we execute extensive experimental simulations, and demonstrate that empirical analysis is in strong agreement with theoretical results. Lastly, we provide a guide to extend the proposed iterative manner in order to generate more general scale-free tree networks with large diameter.