Scale-free networks with invariable diameter and density feature: Counterexamples.

Here, we propose a class of scale-free networks G(t;m) with intriguing properties, which cannot be simultaneously held by all the theoretical models with power-law degree distribution in the existing literature, including the following: (i) average degrees 〈k〉 of all the generated networks are no longer constant in the limit of large graph size, implying that they are not sparse but dense; (ii) power-law parameters γ of these networks are precisely calculated equal to 2; and (iii) their diameters D are all invariant in the growth process of models. While our models have deterministic structure with clustering coefficients equivalent to zero, we might be able to obtain various candidates with nonzero clustering coefficients based on original networks using reasonable approaches, for instance, randomly adding new edges under the premise of keeping the three important properties above unchanged. In addition, we study the trapping problem on networks G(t;m) and then obtain a closed-form solution to mean hitting time 〈H〉_{t}. As opposed to other previous models, our results show an unexpected phenomenon that the analytic value for 〈H〉_{t} is approximately close to the logarithm of the vertex number of networks G(t;m). From the theoretical point of view, these networked models considered here can be thought of as counterexamples for most of the published models obeying power-law distribution in current study.