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., $D_{t}\sim|\mathcal{T}_{t}|^{\ln3/\ln5}$ where $|\mathcal{T}_{t}|$ represents vertex number. In addition, we also 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 analytic formula for quantity $\langle\mathcal{H}_{t}\rangle$ as a function of vertex number $\mathcal{T}_{t}$ has a power-law form, i.e., $\langle\mathcal{H}_{t}\rangle\sim|\mathcal{T}_{t}|^{1+\ln3/\ln5}$ , a characteristic that is rarely encountered in the realm of scale-free tree networks. Lastly, we execute extensive experimental simulations, and find that empirical analysis is in strong agreement with theoretical results