Bounds of memory strength for power-law series

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 nontrivial bounds on the memory strength (first-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 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.

Figure 1

(a) Theoretical bounds (black solid lines) compared with simulated sample mean (dots) of $M\left(t_{{\theta }_{max}}\right)$ and $M\left(t_{{\theta }_{min}}\right)$ under different sequence lengths $n={10}^3,{10}^4,{10}^6$, and ${10}^8$. Each dot is averaged over 1000 independent draws of the sequence $\left\{t_i\right\}$ and the error bar denotes one standard deviation. Finite-size scaling analysis of the (b) maximum and (c) minimum values of $M$ as $n$ increases, under $\alpha =1.6$, 3.0, and 6.0. Mean values approach their corresponding theoretical bounds (dashed lines) as $n$ increases.

Figure 2

Memory strength and fitted $\alpha$ for power-law-distributed interevent time series produced by (a) posting to Twitter, (b) rating on MovieLens, (c) phoning by Orange's Mobile Phone, and (d) browsing on Taobao. Each circle corresponds to a user. Theoretical bounds are drawn by blue solid lines.

Figure 3

$M\left(t_{{\theta }_{min}}\right)$ simulated for different $n$, where $\left\{t_i\right\}$ are sampled from a Gaussian distribution. The dot represents sample mean from 1000 independent draws and error bars represent one standard deviation.

Figure 4

Simulated $M\left(t_{{\theta }_{max}}\right)$ and $M\left(t_{{\theta }_{min}}\right)$ for various distributions also with nontrivial memory strength constraints: (a) Gamma distribution $p\left(x\right)\propto {(x/\lambda )}^{k-1}{e}^{-x/\lambda }$ with shape parameter $k$, (b) Student's $t$-distribution $p\left(x\right)\propto {(1+x^2/\nu )}^{-(\nu +1)/2}$ with degree of freedom $\nu$, (c) Weibull distribution $p\left(x\right)\propto {(x/\lambda )}^{k-1}{e}^{-{(x/\lambda )}^k}$ with shape parameter $k$, and (d) log-normal distribution $p\left(x\right)\propto x^{-1}{e}^{-\frac{{(lnx-\mu )}^2}{2{\sigma }^2}}$ with scale parameter $\sigma$. Each dot is averaged over 1000 independent draws with series length $n=1000$. Error bars represent one standard deviation.