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

Abstract

Here, we propose a class of scale-free networked graphs G(t; m) with some intriguing properties, which can not be simultaneously held by all the theoretical models with power-law degree distribution in the existing literature, including (i) average degrees k of all the generated graphs are no longer a constant in the limit of large graph size, implying that they are not sparse but dense, (ii) power-law parameters γ of these models are precisely calculated equal to 2, as well (iii) their diameters D are all an 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 coefficient based on original graphs using some reasonable approaches, for instance, randomly adding some new edges under the premise of keeping the three important properties above unchanged. In addition, we study trapping problem on graphs G(t; m) and then obtain closedform solutions HT t to mean hitting time. As opposed to other models, our results show an unexpected phenomenon that HT t is approximately close to the logarithm of order of graphs G(t; m) however not to the order itself. From the theoretical point of view, these networked graphs considered here can be thought of as counterexamples for most of the published models obeying power-law distribution in current study. graphs on dynamics taking place on networks themselves, for instance, the mean hitting time for trapping problem, synchronization in networks, epidemic spread [5]- [7]. In this paper, we not only propose a family of networked models G(t; m) and discuss some commonly used topological measures for understanding models G(t; m) in much detail, but also consider the trapping problem on the proposed models G(t; m) and final derive the closed-form solution to mean hitting time.The main concern in the previous research of theoretical networked models is to focus on constructing models which have scale-free and small-world characters as described above. However, an overwhelming number of models are sparse, implying that the average degree of networks will tend to a constant in the limit of large graph size. This is because a great deal of real-life networks are found to display sparsity feature. By contrast, current studies in some areas turns out the existence of dense networks [8]. To describe these such networks, some available networked models have be proposed [9] and analytically investigated in some principled manners, including mean-field theory and master equation. Yet, most of them are stochastic. To our knowledge, almost no deterministic models with both density structure and scale-free feature are built in the past. Although there exist some disadvantages inherited by the latter in comparison with the former, the deterministic structure of model allows us to precisely derive the solutions to some quantities of great interest, such as clustering coefficient, degree distribution, average path length. To some extent, determining s...

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