小世界网络特征向量的多重分形分析
Multifractal analysis of eigenvectors of smallworld networks
摘要：
现实世界中的许多复杂系统都具有小世界拓扑结构，其特征在于节点的高度聚集性和短路径长度，而高度聚集性驱动网络的局部化，而短路径长度支持网络特征向量的离域化。利用多重分形技术，研究了由Watts-Strogatz算法构造的小世界网络的邻接矩阵的特征向量的局部化性质。我们发现特征值谱的中心部分具有强多重分形特征，而谱的尾部具有Dq->1。在小世界过渡开始之前，随机连接的增加导致特征向量局部化的增强，而在开始之后，特征向量的局部化逐渐减少。我们证实了在局域-离域转变过程中关联维数的急剧变化
英文摘要：
Many real-world complex systems have small-world topology characterized by the high clustering of nodes and short path lengths.It is well-known that higher clustering drives localization while shorter path length supports delocalization of the eigenvectors of networks. Using multifractals technique, we investigate localization properties of the eigenvectors of the adjacency matrices of small-world networks constructed using Watts-Strogatz algorithm. We find that the central part of the eigenvalue spectrum is characterized by strong multifractality whereas the tail part of the spectrum have Dq->1. Before the onset of the small-world transition, an increase in the random connections leads to an enhancement in the eigenvectors localization, whereas just after the onset, the eigenvectors show a gradual decrease in the localization. We have verified an existence of sharp change in the correlation dimension at the localization-delocalization transition
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