Structural Properties on 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 \(\gamma=1{ + }\ln 5/\ln 2\) 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., \(D_{t}\sim|\mathcal{T}_{t}|^{\ln 3/\ln 5}\)) and a greater average shortest path length \(\langle\mathcal{W}_{t}\rangle\) (namely, \(\langle\mathcal{W}_{t}\rangle\sim|\mathcal{T}_{t}|^{\ln 3/\ln 5}\)) 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., \(\langle\mathcal{H}_{t}\rangle\sim|\mathcal{T}_{t}|^{1+\ln 3/\ln 5}\). 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.