Non-Ergodic Complexity Management

Non-Ergodic Complexity Management Nicola Piccinini 1 , David Lambert 1 , Bruce J. West 2 , Mauro Bologna 3 , Paolo Grigolini

arXiv:1511.08140v1 [nlin.AO] 25 Nov 2015

1

1

Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427 2 Information Science Directorate, Army Research Office, Research Triangle Park, North Carolina, 27709, USA and 3 Instituto de Alta Investigation, Universidad de Tarapac´ a, Casilla 6-D, Arica, Chile The principle of complexity management (PCM) determines the response of systems, characterized by long-time correlation, to stimuli produced by other systems with similar anomalous properties. The 1/f -noise paradigm is widely adopted to illustrate this condition and applies to a broad range of phenomena, from physics to neurophysiology to sociology. Herein we show that 1/f noise is an ideal condition at the border between phenomena having infinite memory, with stationary, but very slow autocorrelation functions, and those with just as slow, but non-stationary autocorrelation functions, generated by renewal events. This Letter presents a theoretical approach that allows investigators to properly deal with complexity management on both sides of this border region. We show that as an effect of non-stationarity, the observation of the response to such stimuli may lead to random behavior, hiding the correlation between driven and driving systems, and we offer prescriptions that reveal the correlations in the entire domain of 1/f noise.

The transfer of information from one complex network to another is one of the most important issues in the emerging field of complex networks. Network studies clearly indicate that the probability density functions (PDFs) for network properties are inverse power-law and that the power-law index is a measure of network complexity. The correlation between two complex networks with a large, although finite number of units, may be hidden by non-stationary fluctuations, thereby making the detection of transferred information difficult, if not impossible, with the traditional observation of single realizations. It is compelling to design techniques of analysis appropriate to address the perennial out-of-equilibrium condition that is shared by the dynamics of many complex networks. Traditional methods of non-equilibrium statistical physics have not been successful in addressing the question of information transfer. For example, in studying the response of non-stationary networks to harmonic perturbations it was determined by many authors, see Ref. [1] and references therein, that linear response theory (LRT) was “dead”. An assessment of this premature death was made by Aquino et al. [2, 3] resulting in a generalization of LRT that was successfully applied to the question of information transfer. In their discussion these latter authors focused on the intimate connection between neural organization and information theory, as well as the production of 1/f noise. Psychologists interpret the generation of 1/f noise as a manifestation of cognition [4, 5], although no psychologically well founded model for the origin of 1/f noise yet exists [6]. Experimental observation of brain dynamics either by monitoring EEG activity [7] or through actigraphy [8] confirm that the awake condition of the brain is a source of 1/f noise [9]. The theoretical approaches to 1/f noise are based on the Wiener-Khinchin theorem, and consequently on the stationary assumption [10]. We refer to this form of 1/f noise as due to infinite memory fluctuations. To understand the reason for the name “infinite memory”, con-

sider the spectrum S(f ), given by the the Fourier transform of the stationary autocorrelation function Φξ (t) = hξ(t1 )ξ(t2 )i, with t = |t1 − t2 |, the signal under study is ξ(t) and the brackets denote an average over an ensemble PDF. In the case limt→∞ Φξ (t) ∝ 1/tδ a Tauberian theorem establishes that the spectrum S(f ) ∝ f −β , with β ≡ 1 − δ. If δ < 1, the equilibrium autocorrelation function Φξ (t) is not integrable, and the fluctuation ξ(t2 ) has memory of the earlier fluctuation ξ(t1 ), no matter how large the time interval t is between the two fluctuations. In this infinite memory case we have β < 1. The last decade has witnessed increasing attention devoted to the breakdown of the stationary condition and to the 1/f noise generated by the occurrence of non-Poisson renewal events, which reflects that breakdown [11, 12]. Further interesting non-stationary effects on 1/f noise have been more recently found [13–15]. Herein we limit ourselves to discuss the case in which the time interval between two consecutive renewal events is given by the waiting-time PDF: ψ(t) = (µ − 1)

T µ−1 , (t + T )µ

(1)

with µ > 1. Note that this analytical expression can be derived from an idealized [16] version of the Manneville map [17], originally created to study the extended time regions between two consecutive turbulent events. For this reason we call the time region between two consecutive events laminar. The corresponding cumulative probability reads  Ψ(t) =

T t+T

µ−1 .

(2)

The time series ξ(t) is realized according the ballistic prescription [18]. We fill the laminar regions with either ξ = 1 or ξ = −1, according to a coin tossing prescription and, as a consequence, if the first laminar region is filled,

2 for example with ξ = 1, the function Ψ(t) can be interpreted as a survival probability of this value, when we average over infinitely many realizations. In this case the occurrence of an event at time t would have the effect of annihilating the mean value of ξ. In the case µ < 2, the mean waiting time is infinite and yields a perennial nonequilibrium condition, as is clearly shown by the formula [19] R(t) ∝

1 , t2−µ

(3)

where R(t) is the number of events per unit of time following the preparation of an ensemble of realizations sharing the condition of an event occurring at time t = 0. This non-equilibrium condition destroys the equivalence between ensemble averages and time averages, and is usually referred to as the nonergodic condition. In this nonergodic case a proper extension of the Wiener-Khintchine theorem [12] yields the spectrum S(f ) ∝

1 L2−µ

1 , fβ

(4)

where L denotes the length of the time series under study and β ≡ 3 − µ > 1. The condition µ < 2 makes the spectrum intensity decrease upon increase of the time length L, with the decreasing factor 1/L2−µ being a direct consequence of the cascade of events given by Eq. (3). In the regime µ > 2, S(f ) ∝ 1/f β , with β = 3 − µ < 1. In summary, the ideal case β = 1 is the border between the infinite memory region β < 1 and the nonergodic region β > 1. This important property has been overlooked in the 1/f noise literature, as a consequence of a misleading interpretation of flicker noise. The signal ξ(t) for flicker noise is characterized by an exponential autocorrelation function Φξ (t) = exp(−γt), and is thus stationary and ergodic. It is straightforward to show that when γ  f we obtain the spectrum S(f ) ∝ f −2 . If we wrongly compare this to Eq. (4), we would get β = 2, thus µ = 1, namely a condition where the events are extremely rare. However Eq. (4) refers to nonergodic processes, while flicker noise has an exponential autocorrelation function. This implies that it belongs to the region µ > 3. In fact, the value µ = 2 is the border between the ergodic (µ > 2) and nonergodic (µ < 2) regions. The region 2 < µ < 3 is the L´evy walk domain [20]. In order to enter the the Gauss basin of attraction [21] we have to move to µ > 3. This Letter is limited to region 1 < µ < 3, so flicker noise is irrelevant for our discussion. Notice that Ψ(τ ) of Eq. (2) is a brand-new autocorrelation function that, as an effect of infinite aging [22, 23], becomes the equilibrium autocorrelation function  µ−2 T Φξ (t) = , (5) t+T thereby making the region 2 < µ < 3 ergodic. Moving from µ > 2 to µ < 2 corresponds to moving from the

ergodic to the nonergodic situation. A diffusion process x˙ = ξ generated by µ < 2 is ballistic. In fact, using analytical arguments it is shown [18] that such a diffusion process is described by a Lamperti PDF [24]. Using the Rt time average of ξ(t) given by ξ = 0 dt0 ξ(t0 )/t = x/t allows us to write the Lamperti PDF:   2 α−1 1 − ξ sin(πα) 2   Π(ξ) = , π 1 − ξ
2α + 1 + ξ
2α + 1 − ξ 2 α cos(πα) (6) with α = µ − 1. This Lamperti PDF is a clear manifestation of ergodicity breaking since it favors 1 and −1, while ensemble averages would yield vanishing values. The main purpose of this Letter is to provide a theoretical and statistical perspective to study the response of a network S to the stimulus produced by a network P , when both networks are generators of 1/f noise. An attractive application of the theory illustrated herein is the response of the brain to music, since music is found to be described by means of 1/f noise spectral processes [25]. There is significant interest in the Mozart effect [26], an enhancement of neurophysiological activity associated with listening to Mozart’s music, which generated substantial debate in the literature that we cannot pursue here. We limit ourselves to note that the latest contributions to this research are based on observations of EEG response to music [27]. To study the response of network S to network P we perform a numerical experiment, find the most convenient way to establish the response in the nonergodic case and obtain an analytical theoretical expression to explain the results of the numerical experiment. The experiment is done with a pair of time series, ξP (t) and ξS (t), both being realized according to the ballistic prescription [18]. The sequence ξP (t) exerts a weak influence on the sequence ξS (t) as follows. If ξS = 1 and is embedded in a laminar region of the perturbing sequence, with ξP = 1, then the perturbed sequence is influenced to remain in that state for a more extended time by assigning to its parameter T the value T+ = T (1 + ). If ξS = 1 and is embedded in a laminar region of the perturbing sequence with ξP = −1, then the perturbed sequence is forced to make its sojourn in this state shorter by assigning to its parameter T the value T− = T (1 − ). Note that this form of perturbation is a natural consequence of the analytic form of the waiting-time PDF given by Eq. (1), with only two parameters, T and µ. The parameter µ quantifies network complexity and is a collective property that can be forced to change only in response to a very strong perturbation. A noninvasive perturbation, therefore, is expected to only change T . This is in line with the prescriptions of the dynamical approach to linear response theory (LRT) which was used in [28] to design a generalized LRT [2, 3] that led to remarkably good agreement with experimental observation. It has to be stressed that the dimensionless parameter  must satisfy the condition  < 1 and that, according

3 where both the perturbed and the perturbing networks are nonergodic. Note that for µP = 1, Φ∞ coincides with the exact prediction of Eqs. (7) and (8). When only a single time series is available and ensemble averages can no longer be used, the original proof of the PCM [2, 3] is no longer applicable. For instance, this is the case when assessing the response of a patient to music, since each brain is unique. Thus, we have to find a way to establish the statistical response using a time average. To connect with the ensemble average of the earlier work [2, 3] we evaluate the time average crosscorrelation function Z tl +TW 1 C(tl , TW ) ≡ dtξS (t)ξP (t), (9) TW tl

FIG. 1: (Color online) The cross-correlation Φ∞ ≡ limt→∞ hξS (t)ξP (t)i / for two complex networks S and P calculated with an ensemble PDF. The asymptotic cross-correlation function is graphed as a function of the inverse power-law index µS of the responding network S and of the power index µP of the stimulating network P. This figure is derived from the earlier work of Ref. [2]

to the exact calculation in Ref. [29] lim

t→∞

hξS (t)i = (µS − 1), 

(7)

hξS (t)i =1 

(8)

for µS < 2 and lim

t→∞

for 2 < µS < 3. To properly appreciate the importance of the prescription afforded by this Letter to address the issue of complexity management without using ensemble averages, it is convenient first to discuss Figure 1, which illustrates the Principle of Complexity Management (PCM) with a cross-correlation cube which was derived earlier [2, 3] using ensemble averages. This landscape records the crosscorrelation function Φ∞ ≡ limt→∞ hξS (t)ξP (t)i /. The four quadrants are distinguished by the power-law indices of the stimulating and responding networks. Recalling that β = 3 − µ, we have: quadrant I, βS > 1, βP > 1; quadrant II, βS > 1, βP < 1; quadrant III, βS < 1, βP > 1; quadrant IV , βS < 1, βP < 1. We see that network S does not respond to the stimulus in quadrant II, indicating that a nonergodic complex network is asymptotically insensitive to the influence of an ergodic stimulus; whereas quadrant III indicates a complete asymptotic enslavement of the ergodic responding network to the nonergodic stimulating network. However, the most interesting quadrant might be quadrant I,

where tl is the initial position of the time window of size TW and we move it along the sequences ξS (t) and ξP (t), after preparing the system at time t = 0. The two time series (ξS (t), ξP (t)) have length L, and by making L as large as possible we establish a connection with the ensemble average autocorrelation function limt→∞ hξS (t)ξP (t)i. Due to the non-equilibrium nature of this process, the distribution of C 0 s is expected to be sharp in the ergodic case, while it should be broader in the nonergodic case. We evaluate the center of gravity (COG) of the PDF of the cross-correlation function and assess how much it departs from the vanishing mean value that corresponds to a lack of response. This is illustrated in Figure 2 and is the main result of this Letter. Note that these plots are based on observing only a single realization of a time series. In the left panel of Fig. 2 we see that, as expected, there is good agreement with the earlier results depicted in Figure 1 in all the quadrants, except the crucial quadrant I. In this quadrant it is evident that, in spite of making averages on different positions of the moving window, the coupling of the two networks changes from correlated to anti-correlated, with small changes of the index pair (µS , µP ). As an aside, we observe that these fluctuations can be suppressed to recover the results of Figure 1 by supplementing these time averages with additional averages over infinitely many realizations. This can be done in numerical simulations. In real-world applications, however, this is not possible, as we have already noticed in the particular case of a patient listening to music. Therefore, this is not a viable method for treating experimental data. The problem of strongly fluctuating averages is solved by adopting the following prescription that is a cornerstone of the results of this Letter. We locate the left hand side of the window on an event of the perturbed or perturbing network. This has the effect of producing the right panel of Fig. 2. A benefit of this prescription is that the random fluctuations, between positive and negative correlations depicted in the right panel in quadrant I are eliminated and regular behavior emerges. Another important result of this Letter is that the entire region µS < µP of quadrant I shows no sign of response to

4

FIG. 2: (Color online) COG as a function of the inverse power-law index µS of the responding network S and of the power index µP of the stimulating network P. Left panel: The time sequence of length L is divided into L/TW intervals of length TW . The cross-correlation function of Eq. (9) is evaluated for each interval and the landscape is obtained plotting the mean of the resulting distribution of values. Right panel: The times ti of Eq. (9) are the times of event occurrence and the landscape is obtained by plotting the mean of the resulting distribution of values. The exact prediction of Eq. (14) is shown by the two red lines.

perturbation, as in quadrant II. Let us now discuss how to relate these numerical results to an exact analytical prediction. The probabilities WS and WP of finding S and P events at time t after the preparation are given by WS (t) =

RS (t) RS (t) + RP (t)

(10)

WP (t) =

RP (t) , RS (t) + RP (t)

(11)

and

respectively, with R(t) given by Eq. (3) along with µ indexed by the network to which it corresponds. It is straightforward to prove that for t → ∞ we obtain WS = 1 and WP = 0, if µS > µR and WS = 0 and WP = 1 if µS < µR . Thanks to the prescription of locating the beginning of the moving window at the occurrence of an event, we have the following. When µS < µR , events of the perturbing system happen in long laminar regions of the perturbed system. This results in an unperturbed Lamperti distribution (the COG of which is 0). When µS > µR events of the perturbed system happen in long laminar regions of the perturbing system, thereby turning the problem of evaluating the correlation between S and P into that of observing the response of S to a constant perturbation. This observation brings us back to the theory of Ref. [18] that when properly adapted to establish the pertur-

bation of the parameter T yields   2 α−1 1−ξ sin(πα) 2   Π(ξ) = , π 1 − ξ
2α η + 1 + ξ
2α 1 + 1 − ξ 2 α cos(πα) η

(12) where the perturbation strength defines the asymmetry parameter α  1− . (13) η≡ 1+ This exact prediction coincides with the recent result of Akimoto [30]. Thus, supplementing the straightforward algebra used by Akimoto with our arguments leading to no response if µP < µS , we obtain for the center of gravity of the perturbed Lamperti distribution the exact prediction  µS −1 1 − 1− 1+ B(µS , µP ) =  (14)  µS −1 θ(µS − µP ), 1 + 1− 1+ where θ is the Heaviside theta function. It is straightforward to prove that for  → 0, the COG coincides with the ensemble average prediction of Eq. (7), which does not have any contribution of order higher than the first in . The time average, on the contrary, yields contributions of higher order in  that are exactly evaluated through the analytical expression of Eq. (14). The right panel of Fig. 2 shows a good agreement between the numerical results and the exact prediction of Eq. (14), shown in red. Note that the exact theory of Eq. (14) is not limited to µS < 2

5 but extends over the whole interval 1 ≤ µS ≤ 3. According to [31], it is possible to useh Feller [19] theory i to prove

to real experimental data, so as to assess, for instance, the response of the brain to noninvasive stimuli, in terms of renewal events. These crucial events do indeed exist in the brain, as shown by [9, 29]. In the literature there is an increasing interest for criticality as well as for intelligence-induced criticality [32– 37]. The theory and practical rules to detect correlation in the nonergodic case, afforded by this Letter, are expected to contribute to the advance of this field of research. acknowledgment NP, DL and GP thank Welch for financial support through Grant No. B-1577 and ARO for financial support through Grant W911NF-15-1-0245.

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µ−2

1 1 + T3−µ tµ−2 , which corthat in this case R(t) = (µ−2) T responds to a very slow regression to equilibrium for µ slightly larger than 2. In the case µ > 2 equilibrium exists, in contrast with µ < 2, where the out of equilibrium condition is perennial. Although slow regression to equilibrium is shown to generate a multi-scaling diffusion process [31], in the long-time limit it is expected to be ergodic. In this case, the asymptotic cross-correlation function limt→∞ hξS (t)ξP (t)i of Ref. [2] should coincide, as it does, with the exact time average presented herein. This Letter presents guidelines for applying the PCM