Time reversibility from visibility graphs of nonstationary processes

Visibility algorithms are a family of methods to map time series into networks, with the aim of describing the structure of time series and their underlying dynamical properties in graph-theoretical terms. Here we explore some properties of both natural and horizontal visibility graphs associated to several nonstationary processes, and we pay particular attention to their capacity to assess time irreversibility. Nonstationary signals are (infinitely) irreversible by definition (independently of whether the process is Markovian or producing entropy at a positive rate), and thus the link between entropy production and time series irreversibility has only been explored in nonequilibrium stationary states. Here we show that the visibility formalism naturally induces a new working definition of time irreversibility, which allows us to quantify several degrees of irreversibility for stationary and nonstationary series, yielding finite values that can be used to efficiently assess the presence of memory and off-equilibrium dynamics in nonstationary processes without the need to differentiate or detrend them. We provide rigorous results complemented by extensive numerical simulations on several classes of stochastic processes.

Figure 1

Sample time series of an unbiased random walk, as canonical example of a nonstationary process. By definition, the process is (infinitely) time irreversible, although if we remove the $Y$ axis, then it is impossible to know in which direction time is flowing, as both pictures (forward and backward) are equally likely.

Figure 2

White noise: VG. (a) Semilog plot of the in and out degree distributions of the natural visibility graph associated to a time series of $2^{15}$ i.i.d. uniformly random uncorrelated variables $\sim U[0,1]$. Both distributions are identical up to finite-size effects fluctuations, suggesting that the underlying process is VG reversible. (b) Log-log plot of the irreversibility measure $D_{kld}\left(\text{in}\right|\left|\text{out}\right)$ as a function of the series size $n$ (each dot is an average over ten realizations). This measure vanishes asymptotically as $1/n$, showing that finite irreversibility values for finite size are due to statistical fluctuations that vanish asymptotically.

Figure 3

Dissipative chaos: VG. (a) Semilog plot of the in and out degree distributions of the natural visibility graph associated to a time series of $2^{15}$ data generated from a fully chaotic logistic map $x(t+1)=4x\left(t\right)[1-x(t\left)\right]$. Distributions are clearly different, suggesting that the underlying process, although stationary, is VG irreversible. (b) Irreversibility measure $D_{kld}\left(\text{in}\right|\left|\text{out}\right)$ as a function of the series size $n$ (each dot is an average over ten realizations). This measure converges to a finite value with series size, confirming that the process yields a positive irreversibility measure.

Figure 4

Nonstationary unbiased (memoryless) additive random walk: VG. (a) Log-log plot of the in and out degree distributions of the natural visibility graph associated to an unbiased random walk of $2^{17}$ steps generated from $x(t+1)=x\left(t\right)+\xi$, where $\xi \sim U[-0.5,0.5]$. Both distributions are identical up to finite-size effect fluctuations, suggesting that the underlying process is VG reversible. The distributions follow a power-law tail $k^{-2}$, something that can be heuristically justified according to scaling laws (see the text). (b) Log-log plot of the irreversibility measure $D_{kld}\left(\text{in}\right|\left|\text{out}\right)$ as a function of the series size $n$ (each dot is an average over ten realizations). This measure vanishes asymptotically as $n^{-1/3}$, suggesting that, albeit being a nonstationary process, it is VG reversible.

Figure 5

Nonstationary unbiased (memoryless) additive random walk: HVG. (a) Semilog plot of the in and out degree distributions of the horizontal visibility graph associated to an unbiased random walk of $2^{17}$ steps generated from $x(t+1)=x\left(t\right)+\xi$, where $\xi \sim U[-0.5,0.5]$. Both distributions are identical up to finite-size effect fluctuations, suggesting that the underlying process is HVG reversible. (b) Log-log plot of the irreversibility measure $D_{kld}\left(\text{in}\right|\left|\text{out}\right)$ as a function of the series size $n$ (each dot is an average over ten realizations). This measure vanishes asymptotically as $n^{-1}$, certifying that, albeit being a nonstationary process, it is HVG reversible.

Figure 6

Nonstationary (memoryless) additive random walk with drift: HVG. (a) Semilog plot of the in and out degree distributions of the horizontal visibility graph associated to an unbiased random walk of $2^{17}$ steps generated from $x(t+1)=x\left(t\right)+\xi$, where $\xi \sim U[-0.4,0.6],\langle \xi \rangle =0.1$. Distributions are different, suggesting that the process is HVG irreversible. (b) Log-log plot of the irreversibility measure $D_{kld}\left(\text{in}\right|\left|\text{out}\right)$ as a function of the series size $n$ (each dot is an average over ten realizations, and error bars denote the standard deviation). This measure converges asymptotically to a finite value, certifying that the process is HVG irreversible.

Figure 7

Non-Markovian additive random walk with memory: VG. (a) Semilog plot of the in and out degree distributions of the natural visibility graph associated to a biased random walk (see the text) of $2^{17}$ steps with delay $\tau =6$ and reset rate $r=0.3$. Both distributions are similar, suggesting that the onset of memory effects are not effectively captured by VG reversibility, and although these are slightly different than for the baseline random walk, no major qualitative differences are observed. (b) Log-log plot of the irreversibility measure $D_{kld}\left(\text{in}\right|\left|\text{out}\right)$ as a function of the series size $n$ (each dot is an average over ten realizations). This measure vanishes asymptotically with series size as slowly as $n^{-1/3}$, so finite-size values can still be used for comparison with other models.

Figure 8

Non-Markovian additive random walk with memory: HVG. (a) Semilog plot of the in and out degree distributions of the horizontal visibility graph associated to a biased random walk (see the text) of $2^{18}$ steps with delay $\tau =6$ and reset rate $r=0.3$. Both distributions are clearly different, suggesting that the onset of memory effects are effectively captured by the HVG. This seems to be a unique property of HVGs (as VGs fail to accurately capture this trait). (b) Irreversibility measure $D_{kld}\left(\text{in}\right|\left|\text{out}\right)$ as a function of the series size $n$ (each dot is an average over ten realizations). This measure converges with increasing series size to a finite, non-null value, certifying that the process is HVG irreversible. As the unbiased (memoryless) random walk [Eq. (7)] is in turn HVG reversible, we conclude that the source of irreversibility captured in this process is only due to memory effects, as nonstationarities are filtered out.

Figure 9

(a) Log-linear plot of a sample time series generated through the process $x(t+1)=\xi x\left(t\right),\text{ln}\xi \sim U[-0.5,0.5]$. This multiplicative process is additive in logarithmic space, hence in a log-linear plot, the time series looks similar to an additive random walk with no drift, as $\text{ln}\xi$ is uniformly distributed in a symmetric interval so $\langle \text{ln}\xi \rangle =0$. Irreversibility measures on $x\left(t\right)$ are depicted in Fig. 10. (b) Log-log plot of the in and out degree distributions for two different realizations of $2^{15}$ data of a multiplicative random walk $x(t+1)=\xi x\left(t\right)$, where $\text{ln}\xi \sim U[-0.5,0.5]$. The curves have a power-law decay with a fairly stable exponent $k^{-0.75}$, followed by wildly fluctuating tails. These are related to the presence of extreme events in the series, which are exponentially rare but exponentially large, and dominate the tails [26].

Figure 10

Multiplicative random walk with uniformly distributed log returns. (a) Linear-log plot of the irreversibility measure $D_{kld}\left(\text{in}\right|\left|\text{out}\right)$ as a function of the series size $n$ (each dot is an average over 100 realizations and error bars account for $\pm \sigma )$ computed from the VG associated to a multiplicative random walk $x(t+1)=\xi x\left(t\right),\text{ln}\xi \sim U[-0.5,0.5]$. The measure converges to a finite value, so the process is VG irreversible. (b) Log-log plot of the same measure computed from the HVG. The measure decays with series size $n$, so the process is HVG reversible.

Figure 11

Log-log plot of the degree distribution of a VG associated to the first $2^n$ data (where, from top to bottom, $n=14,16,18)$ of a time series generated via the MRW with symmetric multiplicative noise $\xi \sim U[0.9,1.1]$, which is qualitatively similar to an additive random walk with negative drift in logarithmic space. There is a power-law contribution at small degrees (associated to the fast time scale) and a steady wave-fluctuating part associated to the envelope whose extension increases with series size (see the text for details).

Figure 12

(a) Log-linear plot of a sample time series generated through the process $x(t+1)=\xi x\left(t\right),\xi \sim U[0.9,1.1]$, so in logarithmic space the process shares similarities with an additive random walk with a negative drift, as $\langle \text{ln}\xi \rangle <0$. Irreversibility measures on $x\left(t\right)$ are depicted in Fig. 13. (b) Log-log plot of the in and out degree distributions for two different realizations of $2^{15}$ data of a multiplicative random walk $x(t+1)=\xi x\left(t\right)$, where $\xi \sim U[0.9,1.1]$.

Figure 13

Multiplicative random walk with drift. (a) Log-linear plot of the irreversibility measure $D_{kld}\left(\text{in}\right|\left|\text{out}\right)$ as a function of the series size $n$ (each dot is an average over 100 realizations and error bars account for $\pm \sigma )$ computed from the VG associated to a multiplicative random walk $x(t+1)=\xi x\left(t\right),\xi \in U[0.9,1.1]$, which induces an additive random walk with drift in logarithmic space. The measure diverges logarithmically with series size, hence the process is infinitely VG irreversible. (b) Log-log plot of the same measure computed from the HVG. The measure converges to a finite, non-null value with series size $n$, hence the process is HVG irreversible.