Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices
Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices
复制标题
层次格中的自相似性、小世界、无标度缩放、不协调性和鲁棒性
DOI：
10.1140/epjb/e2007-00107-6
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发表时间：
2007-04-01
影响因子：
1.6
通讯作者：
Zou, T.
中科院分区：
文献类型：
--
作者：
Zhang, Z.-Z.;Zhou, S.-G.;Zou, T.
本文首先从复杂网络的角度对一族层次格(HLS)的拓扑特征进行解析研究。我们得到了由参数q控制的HLS的一些基本性质：指数为γ=2+ln2/(Lnq)的无标度分布，零聚集系数，网格系数的幂定律行为，平均路径长度的指数增长(非小世界)，维度为Db=ln(2q)/(Ln2)的分形标度，以及无序性。我们的结果表明，无标度网络并不总是小世界的，并支持自相似无标度网络不是种类的猜想。其次，我们定义了一族确定性的图，称为小世界层次格(SWHL)。我们的构造保留了层次格的结构，包括它的度分布、分形结构、聚类系数，同时出现了小世界现象。最后，研究了故意攻击和集体同步的动态过程，并给出了HLS网络与BA网络以及SWHL网络的比较。我们发现，与非常脆弱的非分形BA网络相比，HLS和SWHL的自相似特性显著提高了此类网络对中枢目标损伤的健壮性，并且HLS的同步性比相应的SWHL和BA网络差。我们证明了无标度网络的度分布不足以刻画其可同步性，并且平均路径长度较小的网络并不总是更容易同步。
In this paper, firstly, we study analytically the topological features of a family of hierarchical lattices (HLs) from the view point of complex networks. We derive some basic properties of HLs controlled by a parameter q: scale-free degree distribution with exponent γ=2+ln 2/(ln q), null clustering coefficient, power-law behavior of grid coefficient, exponential growth of average path length (non-small-world), fractal scaling with dimension dB=ln (2q)/(ln 2), and disassortativity. Our results show that scale-free networks are not always small-world, and support the conjecture that self-similar scale-free networks are not assortative. Secondly, we define a deterministic family of graphs called small-world hierarchical lattices (SWHLs). Our construction preserves the structure of hierarchical lattices, including its degree distribution, fractal architecture, clustering coefficient, while the small-world phenomenon arises. Finally, the dynamical processes of intentional attacks and collective synchronization are studied and the comparisons between HLs and Barabási-Albert (BA) networks as well as SWHLs are shown. We find that the self-similar property of HLs and SWHLs significantly increases the robustness of such networks against targeted damage on hubs, as compared to the very vulnerable non fractal BA networks, and that HLs have poorer synchronizability than their counterparts SWHLs and BA networks. We show that degree distribution of scale-free networks does not suffice to characterize their synchronizability, and that networks with smaller average path length are not always easier to synchronize.