Choi Woosik, Chae Huiseung, Yook Soon-Hyung, Kim Yup
Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Dec;88(6):060802. doi: 10.1103/PhysRevE.88.060802. Epub 2013 Dec 17.
Transport properties in random and scale-free (SF) networks are studied by analyzing the betweenness centrality (BC) distribution P(B) in the minimum spanning trees (MSTs) and infinite incipient percolation clusters (IIPCs) of the networks. It is found that P(B) in MSTs scales as P(B)∼B(-δ). The obtained values of δ are classified into two different categories, δ≃1.6 and δ≃2.0. Using the mapping between BC and the branch size of tree structures, it is proved that δ in MSTs which are close to critical trees is 1.6. In contrast, δ in MSTs which are supercritical trees is shown to be 2.0. We also find δ=1.5 in IIPCs, which is a natural result because IIPC is physically critical. Based on the results in MSTs, a physical reason why δ≥2 in the original networks is suggested.
通过分析网络的最小生成树（MST）和无限初渗簇（IIPC）中的中介中心性（BC）分布P(B)，研究了随机网络和无标度（SF）网络中的输运性质。研究发现，MST中的P(B)按P(B)∼B^(-δ)缩放。得到的δ值分为两类，δ≃1.6和δ≃2.0。利用BC与树结构分支大小之间的映射关系，证明了接近临界树的MST中的δ为1.6。相比之下，超临界树的MST中的δ为2.0。我们还在IIPC中发现δ = 1.5，这是一个自然的结果，因为IIPC在物理上是临界的。基于MST的结果，提出了原始网络中δ≥2的物理原因。
