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.

翻译：无标度网络在多种复杂系统中普遍存在，这引发了关于此类网络模型的多项研究。本文中，我们提出了一类以迭代方式生成的生长树网络$\mathcal{T}_{t}$，并证明其具有无标度特性。与已发表的大多数具有无标度特征的树模型不同，我们的树网络具有幂律指数$\gamma=1+\ln5/\ln2$，该指数明显大于$3$。同时，由于模型$\mathcal{T}_{t}$具有超大直径$D_{t}$（即$D_{t}\sim|\mathcal{T}_{t}|^{\ln3/\ln5}$）和更大的平均最短路径长度$\langle\mathcal{W}_{t}\rangle$（即$\langle\mathcal{W}_{t}\rangle\sim|\mathcal{T}_{t}|^{\ln3/\ln5}$，其中$|\mathcal{T}_{t}|$表示顶点数），因此无法观察到“小世界”特性。接下来，我们确定皮尔逊相关系数，并验证网络$\mathcal{T}_{t}$呈现异配混合结构。此外，我们研究了树网络$\mathcal{T}_{t}$上的随机游走，并推导出平均到达时间$\langle\mathcal{H}_{t}\rangle$的精确解。结果表明，作为顶点数$|\mathcal{T}_{t}|$的函数的量$\langle\mathcal{H}_{t}\rangle$的解析公式呈现幂律形式，即$\langle\mathcal{H}_{t}\rangle\sim|\mathcal{T}_{t}|^{1+\ln3/\ln5}$。据此，我们进行了广泛的实验模拟，并证明实证分析与理论结果高度一致。最后，我们提供了扩展所提迭代方法的指南，以生成更具一般性的具有大直径的无标度树网络。