Scale-free tree network with an ultra-large diameter

Abstract

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 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 γ = 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 T t have an ultra-large diameter D t , i.e., D t ∼ |T t | ln 3/ ln 5 where |T t | represents vertex number. In addition, we also study random walks on tree networks T t and derive exact solution to mean hitting time H t . The results suggest that analytic formula for quantity H t as a function of vertex number T t has a power-law form, i.e., H t ∼ |T t | 1+ln 3/ ln 5 , 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.

Mean Hitting Time on Recursive Growth Tree Network

“…( 5), which implies that we finish the proof of Theorem I.5. It should be pointed out that another proof of Theorem I.5 has been reported in the prior work [47]. Interested reader is encouraged to refer to [47] for more details.…”

“…Equivalently speaking, we first insert two new vertices on each edge in the initial graph G(V, E) and then connect k u new vertices to each existing vertex u with degree k u of graph G(V, E). As shown in the prior work [47], this kind of operations have been used to create a family of networked models with interesting properties including fractal feature. In addition, stochastic versions are also generated via introducing randomness into the process of constructing graphs with respect to primitive operations.…”