CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, People's Republic of China.
Department of Computer Science, Duke University, Durham, North Carolina 27708, USA.
Phys Rev E. 2017 May;95(5-1):052314. doi: 10.1103/PhysRevE.95.052314. Epub 2017 May 19.
Many time series produced by complex systems are empirically found to follow power-law distributions with different exponents α. By permuting the independently drawn samples from a power-law distribution, we present nontrivial bounds on the memory strength (first-order autocorrelation) as a function of α, which are markedly different from the ordinary ±1 bounds for Gaussian or uniform distributions. When 1<α≤3, as α grows bigger, the upper bound increases from 0 to +1 while the lower bound remains 0; when α>3, the upper bound remains +1 while the lower bound descends below 0. Theoretical bounds agree well with numerical simulations. Based on the posts on Twitter, ratings of MovieLens, calling records of the mobile operator Orange, and the browsing behavior of Taobao, we find that empirical power-law-distributed data produced by human activities obey such constraints. The present findings explain some observed constraints in bursty time series and scale-free networks and challenge the validity of measures such as autocorrelation and assortativity coefficient in heterogeneous systems.
许多复杂系统产生的时间序列在经验上被发现遵循具有不同指数 α 的幂律分布。通过对来自幂律分布的独立抽样进行排列，我们给出了作为 α 的函数的记忆强度（一阶自相关）的非平凡边界，这与高斯或均匀分布的普通 ±1 边界明显不同。当 1<α≤3 时，随着 α 的增大，上界从 0 增加到 +1，而下界保持为 0；当 α>3 时，上界保持为 +1，而下界下降到 0 以下。理论边界与数值模拟吻合得很好。基于 Twitter 上的帖子、MovieLens 的评分、Orange 移动运营商的通话记录以及 Taobao 的浏览行为，我们发现人类活动产生的经验幂律分布数据遵守这些约束。目前的发现解释了一些观察到的突发时间序列和无标度网络中的约束，并挑战了自相关和聚类系数等度量在异构系统中的有效性。
