反例：具有不变直径和密度特征的无标度网络图
Counterexample: scale-free networked graphs with invariable diameter and density feature
摘要：
本文提出了一类无标度网络$G(t;m)$，它具有一些有趣的性质，这些性质是现有文献中所有幂律度分布的理论模型都不能同时保持的，包括（i）在大图尺寸的限制下，所有生成网络的平均度$\langle k\rangle$不再是常数，这意味着它们不是稀疏的而是密集的，（ii）这些网络的幂律参数$γ$被精确计算为等于$2$，以及（iii）它们的直径$D$都是模型生长过程中的不变量。虽然我们的模型具有聚类系数等于零的确定性结构，但我们可以在保持上述三个重要性质不变的前提下，通过一些合理的方法，例如随机添加一些新的边，在原始网络的基础上获得具有非零聚类系数的各种候选网络。此外，我们还研究了网络$G(t;m)$上的陷阱问题，得到了平均命中时间$\langle \mathcal{H}\rangle_{t}$的封闭解。与以往的模型不同，我们的结果显示了一个意想不到的现象，即$\langle \mathcal{H}\rangle_{t}$的解析值近似于网络顶点数的对数$G(t;m)$ .从理论的角度来看，这些网络模型可以被认为是目前研究中大多数服从幂律分布的模型的反例。
英文摘要：
Here, we propose a class of scale-free networks $G(t;m)$ 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 $\langle k\rangle$ of all the generated networks 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 networks 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 networks 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 networks $G(t;m)$ and then obtain closed-form solution to mean hitting time $\langle \mathcal{H}\rangle_{t}$. As opposed to other previous models, our results show an unexpected phenomenon that the analytic value for $\langle \mathcal{H}\rangle_{t}$ is approximately close to the logarithm of 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
