Bounds of memory strength for power-law series

Abstract

Many time series produced by complex systems are empirically found to follow power-law distributions with different exponents $\alpha$. By permuting the independently drawn samples from a power-law distribution, we present non-trivial bounds on the memory strength (1st-order autocorrelation) as a function of $\alpha$, which are markedly different from the ordinary $\pm 1$ bounds for Gaussian or uniform distributions. When $1<\alpha \le 3$, as $\alpha$ grows bigger, the upper bound increases from 0 to +1 while the lower bound remains 0; when $\alpha >3$, 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 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 like autocorrelation and assortativity coefficient in heterogeneous systems.

10 pages, 4 figures (revised)

References in corpus (20)

• Power-law distributions in empirical data
• Epidemic processes in complex networks
• Synchronization in complex networks
• Evolutionary games on graphs
• Small But Slow World: How Network Topology and Burstiness Slow Down Spreading
• On the Frequency of Severe Terrorist Events
• Role of Activity in Human Dynamics
• Human dynamics revealed through Web analytics
• Impact of memory on human dynamics
• Spreading Dynamics Following Bursty Human Activity Patterns
• Human Activity in the Web
• Modeling Human Dynamics with Adaptive Interest
• Structural constraints in complex networks
• Heavy-tailed statistics in short-message communication
• Unfolding large-scale online collaborative human dynamics
• Testing a priority-based queue model with Linux command histories
• Scaling behavior of online human activity
• Lower bound of assortativity coefficient in scale-free networks
• Network Topology of an Experimental Futures Exchange
• Dynamic Patterns of Academic Forum Activities

Cited by in corpus (5)

• Misinformation spreading on correlated multiplex networks
• Correlated bursts in temporal networks slow down spreading
• Copula-based algorithm for generating bursty time series
• Containing rumors spreading on correlated multiplex networks
• Analytically solvable autocorrelation function for weakly correlated interevent times