Zhang Zhongzhi, Zhang Yichao, Zhou Shuigeng, Yin Ming, Guan Jihong
Department of Computer Science and Technology, Tongji University, 4800 Cao'an Road, Shanghai 201804, China.
J Math Phys. 2009 Mar;50(3):033514. doi: 10.1063/1.3094757. Epub 2009 Mar 30.
Various real-life networks exhibit degree correlations and heterogeneous structure, with the latter being characterized by power-law degree distribution , where the degree exponent describes the extent of heterogeneity. In this paper, we study analytically the average path length (APL) of and random walks (RWs) on a family of deterministic networks, recursive scale-free trees (RSFTs), with negative degree correlations and various , with an aim to explore the impacts of structure heterogeneity on the APL and RWs. We show that the degree exponent has no effect on the APL of RSFTs: In the full range of , behaves as a logarithmic scaling with the number of network nodes (i.e., ), which is in sharp contrast to the well-known double logarithmic scaling previously obtained for uncorrelated scale-free networks with . In addition, we present that some scaling efficiency exponents of random walks are reliant on the degree exponent .
各种现实生活中的网络都呈现出度相关性和异质结构，后者的特征是幂律度分布，其中度指数描述了异质程度。在本文中，我们通过分析研究了一类具有负度相关性和不同度指数的确定性网络——递归无标度树（RSFT）上的平均路径长度（APL）和随机游走（RW），旨在探索结构异质性对APL和RW的影响。我们表明，度指数对RSFT的APL没有影响：在整个度指数范围内，APL与网络节点数量呈对数缩放关系（即），这与之前在度指数为的无关联无标度网络中得到的著名双对数缩放关系形成鲜明对比。此外，我们还表明，随机游走的一些缩放效率指数依赖于度指数。
