School of Mathematical Science, Huaqiao University, 362021 Fujian, China.
Chaos. 2012 Jun;22(2):023121. doi: 10.1063/1.4711008.
This paper investigates the impact of edge-adding number m and edge-adding distance d on both synchronizability and average path length of NW small-world networks generated from ring networks via random edge-adding. It is found that the synchronizability of the network as a function of the distance d is fluctuant and there exist some d that have almost no impact on the synchronizability and may only scarcely shorten the average path length of the network. Numerical simulations on a network of Lorenz oscillators confirm the above results. This phenomenon shows that the contributions of randomly added edges to both the synchronizability and the average path length are not uniform nor monotone in building an NW small-world network with equal-distance edge additions, implying that only if appropriately adding edges when building up the NW small-word network can help enhance the synchronizability and/or reduce the average path length of the resultant network. Finally, it is shown that this NW small-world network has worse synchronizability and longer average path length, when compared with the conventional NW small-world network, with random-distance edge additions. This may be due to the fact that with equal-distance edge additions, there is only one shortcut distance for better information exchange among nodes and for shortening the average path length, while with random-distance edge additions, there exist many different distances for doing so.
本文研究了通过随机添加边从环网生成 NW 小世界网络时，边缘添加数 m 和边缘添加距离 d 对网络同步能力和平均路径长度的影响。研究发现，网络的同步能力随距离 d 的变化是波动的，存在一些 d 对同步能力几乎没有影响，可能只是略微缩短了网络的平均路径长度。洛伦兹振子网络上的数值模拟验证了上述结果。这一现象表明，在等距添加边构建 NW 小世界网络时，随机添加边对同步能力和平均路径长度的贡献既不均匀也不单调，这意味着只有在构建 NW 小世界网络时适当地添加边才能有助于提高网络的同步能力和/或降低网络的平均路径长度。最后，与随机距离边缘添加的传统 NW 小世界网络相比，本文构建的 NW 小世界网络具有更差的同步能力和更长的平均路径长度。这可能是因为在等距边缘添加的情况下，只有一个短路径距离可以更好地在节点之间交换信息并缩短平均路径长度，而在随机距离边缘添加的情况下，存在许多不同的距离可以做到这一点。
