Kawasaki F, Yakubo K
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Sep;82(3 Pt 2):036113. doi: 10.1103/PhysRevE.82.036113. Epub 2010 Sep 24.
The fractal and the small-world properties of complex networks are systematically studied both in the box-covering (BC) and the cluster-growing (CG) measurements. We elucidate that complex networks possessing the fractal (small-world) nature in the BC measurement are always fractal (small world) even in the CG measurement and vice versa, while the fractal dimensions d{B} by the BC measurement and d{C} by the CG measurement are generally different. This implies that two structural properties of networks, fractality and small worldness, cannot coexist in the same length scale. These properties can, however, crossover from one to the other by varying the length scale. We show that the crossover behavior in a network near the percolation transition appears both in the BC and CG measurements and is scaled by a unique characteristic length ξ.
我们在盒覆盖（BC）和簇生长（CG）测量中系统地研究了复杂网络的分形和小世界特性。我们阐明，在BC测量中具有分形（小世界）性质的复杂网络，即使在CG测量中也总是分形（小世界）的，反之亦然，而通过BC测量得到的分形维数(d_{B})和通过CG测量得到的分形维数(d_{C})通常是不同的。这意味着网络的两种结构性质，分形性和小世界性，不能在相同的长度尺度上共存。然而，通过改变长度尺度，这些性质可以从一种转变为另一种。我们表明，在渗流转变附近的网络中的交叉行为在BC和CG测量中都出现，并且由一个独特的特征长度(\xi)进行标度。
