最小直径无标度图的结构性质与平均攻丝时间
Structural properties and average tapping time on scale-free graphs with smallest diameter
摘要：
本文提出了一类图$G^{\star}(m,t)$，并首先研究了它的一些结构性质，如平均度等。结果表明：（1）图$G^{\star}(m,t)$具有密度特征，这是因为其平均度与时间步长$t$成正比，但在大图尺寸范围内不是常数;（2）图$G^{\star}(m,t)$服从指数等于$2$的幂律分布，这在以往的无标度模型中是很少见的，（3）图$G^{\star}(m,t)$具有超小直径和高聚类系数的小世界性质;（4）图$G^{\star}(m,t)$具有皮尔逊相关系数小于零的离散结构。此外，我们考虑了所提出的图$G^{\star}(m,t)$上的捕获问题，发现它们都具有更优的捕获效率，其平均捕获时间都达到了理论上的下限，这是在现有的无标度模型中很少观察到的现象.我们进行了大量的模拟，与我们的理论分析是一致的。
英文摘要：
In this paper, we propose a class of graphs $G^{\star}(m,t)$ and first study some structural properties, such as, average degree, on them. The results show that (1) graphs $G^{\star}(m,t)$ have density feature because of their average degrees proportional to time step $t$ not to a constant in the large graph size limit, (2) graphs $G^{\star}(m,t)$ obey the power-law distribution with exponent equal to $2$, which is rarely found in most previous scale-free models, (3) graphs $G^{\star}(m,t)$ display small-world property in terms of ultra-small diameter and higher clustering coefficient, and (4) graphs $G^{\star}(m,t)$ possess disassortative structure with respect to Pearson correlation coefficient smaller than zero. In addition, we consider the trapping problem on the proposed graphs $G^{\star}(m,t)$ and then find that they all have more optimal trapping efficiency by means of their own average trapping time achieving the theoretical lower bound, a phenomenon that is seldom observed in existing scale-free models. We conduct extensive simulations that are consistent with our theoretical analysis.
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