Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices

Abstract

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 $q$. 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, while the small-world phenomenon arises. Finally, the dynamical processes of intentional attacks and collective synchronization are studied and the comparisons between HLs and Barab{ási}-Albert (BA) networks as well as SWHLs are shown. 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.

26 pages, 8 figures

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• Self-similar disk packings as model spatial scale-free networks
• A deterministic small-world network created by edge iterations
• Recursive graphs with small-world scale-free properties
• Distance-d covering problems in scale-free networks with degree correlations
• Better Synchronizability Predicted by Crossed Double Cycle
• Integer Networks

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