Gao Jian, Zhou Tao, Hu Yanqing
CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, China.
Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China.
Sci Rep. 2015 Oct 1;5:14662. doi: 10.1038/srep14662.
Bootstrap percolation is a general representation of some networked activation process, which has found applications in explaining many important social phenomena, such as the propagation of information. Inspired by some recent findings on spatial structure of online social networks, here we study bootstrap percolation on undirected spatial networks, with the probability density function of long-range links' lengths being a power law with tunable exponent. Setting the size of the giant active component as the order parameter, we find a parameter-dependent critical value for the power-law exponent, above which there is a double phase transition, mixed of a second-order phase transition and a hybrid phase transition with two varying critical points, otherwise there is only a second-order phase transition. We further find a parameter-independent critical value around -1, about which the two critical points for the double phase transition are almost constant. To our surprise, this critical value -1 is just equal or very close to the values of many real online social networks, including LiveJournal, HP Labs email network, Belgian mobile phone network, etc. This work helps us in better understanding the self-organization of spatial structure of online social networks, in terms of the effective function for information spreading.
引导渗流是对某些网络激活过程的一种通用表示，它已被用于解释许多重要的社会现象，比如信息传播。受近期关于在线社交网络空间结构的一些研究结果启发，我们在此研究无向空间网络上的引导渗流，其中远程链接长度的概率密度函数为具有可调指数的幂律。将巨型活跃组件的大小设为序参量，我们发现幂律指数存在一个依赖于参数的临界值，当该指数高于此临界值时，会出现双重相变，它是二阶相变和具有两个可变临界点的混合相变的混合，否则仅存在二阶相变。我们进一步发现一个约为 -1 的与参数无关的临界值，围绕该值，双重相变的两个临界点几乎是恒定的。令我们惊讶的是，这个临界值 -1 恰好等于或非常接近许多真实在线社交网络的值，包括LiveJournal、惠普实验室电子邮件网络、比利时移动电话网络等。这项工作有助于我们从信息传播的有效功能方面更好地理解在线社交网络空间结构的自组织。
