2013 New J. Phys. 15 093013
(http://iopscience.iop.org/1367-2630/15/9/093013 )

李明，刘润然，贾春晓，汪秉宏

连接边与相依边的重叠对网络渗流的重要影响

近期对网络结构与级联故障的研究中，节点之间的相依性以相依边的形式被引入渗流模型。这种边的引入，使得网络更加脆弱。并且不同于经典网络渗流的连续相变，当相依边数量较多时，系统展现出不连续相变。我们这篇文章中抛开了相依边与连接边的独立假设，考虑了两种边的重叠效应，研究了其对整个网络的渗流过程的影响。研究发现，节点之间的相依并不总是使得相变不连续。在较高的重叠率下，即使网络中含有大量的相依边，渗流过程依然是连续相变。从而，系统可以展现出两种相变过程，当重叠率由低到高变化时，系统的相变过程从一阶相变转化成二阶相变。这也说明相依边数量并不是系统相变类型的唯一决定因素，相依节点之间的连接性也是一个重要因素。利用生成函数技术，我们解析得到的序参量，相变点，三相点等值都与模拟结果精确吻合。

Critical effects of overlapping of  connectivity and dependence links on percolation of networks

Ming Li¹,  Run-Ran Liu^2,4,  Chun-Xiao Jia² and

Bing-Hong Wang^1,3,4

¹  Departmentof ModernPhysics, Universityof Science and Technologyof China, Hefei 230026, People’s Republic of China

 ² Institute of Information Economy, Hangzhou Normal University, Hangzhou 310036,

People’s Republic of China

 ³ Complex System Research Center, University of Shanghai for Science and Technology and

Shanghai Academy of System Science, Shanghai 200093, People’s Republic of China

E-mail: runranliu@gmail.com and bhwang@ustc.edu.cn

New Journal of Physics 15 (2013) 093013 (9pp)

Received 27 April 2013
Published 10 September 2013
Online at http://www.njp.org/
doi:10.1088/1367-2630/15/9/093013

Abstract. In a recent work Parshani et al (2011 Proc. Natl Acad. Sci. USA 108 1007), dependence links have been introduced to the percolation model and used to study the robustness of the networks with such links, which shows that the networks are more vulnerable than the classical networks with only connectivity links. This model usually demonstratesafirst order transition, rather than the second  order transition found in classical network percolation. In this paper, considering the real situation that the interdependent nodes are  usually connected, we study the cascading dynamics of networks  when dependence links partially overlap with connectivity links. We find that the percolation transitions are not always sharpened by  making nodes interdependent. For a high fractionofoverlapping, the networkis robust for randomfailures, and the percolation transition is second order, while for a low fraction of overlapping,  the percolation process shows a first order transition. This work  demonstrates that the crossover between two types of transitions does not only depend on the densityof dependence links  but also on the overlapping fraction of connectivity and dependence links. Using generating function  techniques, we present exact solutions for the size of the giant  component and the critical point, which are in good agreement  with the simulations.

 NJP article download:

NJP 1367-2630_15_9_093013.pdf