Complexity Matching and Requisite Variety

Korosh Mahmoodi 1,*, Bruce J. West2, Paolo Grigolini3

1 Department of Social and Decision Sciences, Carnegie MellonUniversity, Pittsburgh, PA 15213, USA2 Information Science Directorate, Army Research Office, ResearchTriangle Park, NC 27708, USA3 Center for Nonlinear Science, University of North Texas, P.O. Box311427, Denton, TX 76203, USA

* corresponding author, koroshm@andrew.cmu.edu

Abstract

Complexity matching characterizes the role of information in interactionsbetween systems and can be traced back to the 1957 Introduction to Cyber-netics by Ross Ashby. We argue that complexity can be expressed in termsof crucial events, which are generated by the processes of spontaneous self-organization. Complex processes, ranging from biological to sociological,must satisfy the homeodynamic condition and host crucial events that haverecently been shown to drive the information transport between complexsystems. We adopt a phenomenological approach, based on the subordina-tion to periodicity that makes it possible to combine homeodynamics andself-organization induced crucial events. The complexity of crucial eventsis defined by the waiting- time probability density function (PDF) of theintervals between consecutive crucial events, which have an inverse powerlaw (IPL) PDF ψ(τ) ∝ 1/(τ)µ with 1 < µ < 3. We show that the ac-tion of crucial events has an effect compatible with the shared notion ofcomplexity-induced entropy reduction, while making the synchronizationbetween systems sharing the same complexity different from chaos syn-chronization. We establish the coupling between two temporally complexsystems using a phenomenological approach inspired by models of swarmcognition and prove that complexity matching, namely sharing the sameIPL index µ, facilitates the transport of information, generating perfectsynchronization. This new form of complexity matching is expected to con-tribute significantly to progress in understanding and improving biofeed-back therapies.

Author summary

This paper is devoted to the control of complex dynamical systems, in-spired by real processes of biological and sociological interest. The conceptof complexity we adopt focuses on the assumption that the processes of self-organization generate intermittent fluctuations and that the time intervalbetween consecutive fluctuations is described by an IPL PDF making the

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second moment of these time intervals diverge. These fluctuations are iden-tified as crucial events and are responsible for the ergodicity breaking thatis widely revealed by the experimental observation of biological dynam-ics. We argue that the information transport from one to another complexsystem is ruled by these crucial events and propose an efficient theoreti-cal prescription leading to qualitative agreement with experimental results,shedding light into the processes of social learning. The theory developedherein is expected to have important medical applications, such as improv-ing biofeedback techniques, heart-brain communication and a significantcontribution to understanding how emotions balance cognition.

1 INTRODUCTION

The mathematician Norbert Wiener observed [1] in a 1948 ‘popular’ lec-ture that the complex networks in the social and life sciences appear tobehave differently from, but not in contradiction to, the laws in the physi-cal sciences. He makes the point that the force laws and therefore controlof social phenomena do not necessarily follow from changes in energy, butrather can be dominated by changes in entropy (information). We referto this honorifically as the Wiener Rule (WR), since it was presented inthe form of a statement and left unproven. That lecture appeared thesame year he introduced the new science of Cybernetics [2] to the scientificcommunity in which his interest in control and communication within andbetween animals and machines was made clear.

The economic webs of global finance and stock markets; the socialmeshes of governments and terrorist organizations; the transportation net-works of planes and highways; the ecowebs of food networks and speciesdiversity; the physical wicker of the Internet; the bionet of gene regulation,and so on, have all been modeled using the nascent science of networks.As these networks in which we are immersed become increasingly com-plex a number of apparently universal properties begin to emerge. One ofthese properties is a version of the WR having to do with how efficientlyinteracting complex networks exchange information with one another.

West et al. [3], among many others, noted that complexity often arisesin a network when the power spectrum Sp(f) takes on an IPL shape:

Sp(f) ∝ 1

fα, (1)

with the IPL index α in the interval 0.5 ≤ α ≤ 1.5. In fact, this 1/f−variability is taken by many scientists to be the signature of complexityand appears in a vast array of phenomena including the human brain [4],body movements [5], music [6–8], physiology [9], genomics [10], and soci-ology [11]. They [3] then used the IPL index as a measure of a network’scomplexity and reviewed the literature arguing that two complex interact-ing networks exchange information most efficiently when the IPL indicesof the two networks match. The hypothesis of the complexity managementeffect form of the WR was proven by Aquino et al. [12, 13] using averagesover PDFs and more generally using time averages [14].

In less than a decade after the initial formulation of cybernetics, RossAshby, captured in his introduction to the subject [15], the difficulty ofregulating biological systems and that “the main cause of difficulty is thevariety in the disturbances that must be regulated against”. This insight-ful observation, which was subsequently extended to complex networks in

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general, led to the conclusion that it is possible to regulate them if theregulators share the same requisite variety (complexity) as the systems be-ing regulated. Herein we refer to Ashby’s requisite variety with the morerecent term complexity matching effect (CME) [3]. The term complexitymatching has been widely used in the recent past to include such informa-tion exchange activities as side-by-side walking [16], ergometer rowing [17],syncopated finger tapping [18], dyadic conversation [19], and interpersonelcoordination [20,21]. These synchronizations are today’s realizations of theregulation of the brain, in conformity with the observations of Ashby.

Crucial events and the brain: For the purposes of the presentpaper it is important to stress that there exists further research directedtoward the foundations of social learning [22–25] that is even more closelyconnected to Ashby’s challenge of the regulator and the regulated sharinga common level of complexity. In fact, this social learning research aims atevaluating the transfer of information from the brain of one player to thatof another by way of the interaction the two players established throughtheir avatars [23], which is to say, it is a virtual reality experiment cap-turing the social cognition shared by a pair of humans. The results areexciting in that the trajectories of pairs of players turn out to be signif-icantly synchronized. But even more important than synchronization isthe fact that the trajectories of the two avatars have a universal structurebased on the shared EEGs of the paired human brains.

Herein we provide a theoretical rational for the universal structure rep-resenting the brains of the pairs of interacting individuals, based on theCME. More broadly the theory can be adapted to the communication, orinformation transfer, between the heart and the brain [26] of a single indi-vidual. The key to this understanding is the existence of crucial events. Ina system as complex as the human brain [27] there is experimental evidencefor the existence of crucial events, which, for our purposes here, can be in-terpreted as organization rearrangements, or renewal failures. The timeinterval between consecutive crucial events is described by a waiting-timeIPL PDF

ψ(τ) ∝ 1/tµ with 1 < µ < 3. (2)

The crucial events generate ergodicity breaking and are widely studied toreveal fundamental biological statistical properties [28].

The transfer of information between interacting systems has been ad-dressed using different theoretical tools, examples of which include: chaossyncronization [29], self-organization [30], and resonance [31]. However,none of these theoretical approaches have to date explained the experi-mental results that exist for the correlation between the dynamics of twodistinct physiological systems [32]. Herein we relate this correlation to theoccurrence of crucial events, which are responsible for the generation of1/f− variability with an IPL spectrum having an IPL index 3 − µ andfor the results of a number of psychological experiments including thoseof Correll [33]. The experimental data imply that activating cognition hasthe effect of making the IPL index µ < 3 cross the barrier between theLevy and Gauss basins of attraction, namely making µ > 3 [34]. This isin line with Heidegger’s phenomenology [35].

The crossing of a basin’s boundary is a manifestation of the signifi-cant effect of violating the linear response condition, according to whicha perturbation should be sufficiently weak that it does not affect a sys-tem’s dynamic complexity [36]. The experimental observation obliged us

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to go beyond the linear response theory adopted in earlier works in orderto explain the transfer of information from one complex system to anotherand to formally prove the WR. This information transfer was accomplishedthrough the matching of the IPL index of the crucial events PDF of theregulator with the IPL index of the crucial events PDF of the system be-ing regulated [12,14]. This is consistent with the general idea of the CME,with the main limitation being that the perturbation intensity is sufficientlysmall that it is possible to observe the influence of the perturbing systemon the perturbed system through ensemble averages, namely a mean overmany realizations [12, 13], or through time averages, if we know the oc-currence time of crucial events [14]. But, the complexity matching theoryof this paper allows us to establish the transfer of information from onecomplex network to another at the level of single realizations, for examplematching between the movements of two Tango dancers.

Note that the 1/f-variability of the spectrum is a necessary, but not asufficient, condition to have maximum information exchange between twocomplex networks. This is where the present theory deviates from the earlyform of cybernetics. The present theory requires the existence of crucialevents.

Homeodynamics: Another important property of biological pro-cesses is homeodynamics [37], which seems to be in conflict with home-ostasis as understood and advocated by Ashby. Lloyd et al. [37] invoke theexistence of bifurcation points to explain the transition from homeostasisto homeodynamics. This transition, moving away from Ashby’s emphasison the fundamental role of homeostasis, has been studied by Ikegami andSuzuki [38] and by Oka et al. [39], who coined the term dynamic home-ostasis. They used Ashby’s cybernetics to deepen the concept of self andto establish if the behavior of the Internet is similar to that of the humanbrain.

Turalska et al. [40], based on the direct use of the dynamics of twocomplex networks, studied the case when a small fraction of the units ofthe regulated system perceive the mean field of the regulating system. Atcriticality the choice made by these units is interpreted as swarm intelli-gence [41], and, in the case of the Decision Making Model (DMM) adoptedin [40] is associated with the index µ = 1.5. Synchronization is observedin [40] when both systems are in the critical condition µ1 = µ2 = 1.5 and itis destroyed if one system is critical and the other is sub-critical, or vicev-ersa. This suggests that maximal synchronization is realized when bothsystems are at criticality, namely, they share the same IPL index µ.

The present theory covers the complexity matching between networkswith different complexity indices µ1 6= µ2. Also, the present theory issupplemented by homeodynamics, which had not been considered before.This theory should not be confused with the unrelated phenomenon ofchaos synchronization. In fact, the intent of the present approach is toestablish the proper theoretical framework to explain, for instance, brain-heart communication. Here the heart is considered to be a complex, butnot chaotic, system, in accordance with a growing consensus that the heartdynamics are not chaotic [42].

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2 METHOD

To address Ashby’s challenge [15] we adopt the perspective of subordina-tion theory [43]. This theoretical perspective is closely connected to theContinuous Time Random Walk (CTRW) [44], which is known to gener-ate anomalous diffusion. We use this viewpoint to establish an approachto explaining the experimental results showing the remarkable oscillatorysynchronization between different areas of the brain [32].

It has to be stressed that a natural choice may rest on the use of Ku-ramoto’s model [45]. In fact, the model of Kuramoto affords a simpleparadigm to explain synchronization of rotators, as explained in the excel-lent review paper of Ref. [46]. We think that this popular model can beproperly generalized to replace the adoption of its control parameter withthe same self-control of Refs. [47–49] that is shown to generate critical-ity. The theory of these papers, called Self-Organized-Temporal Crtitical-ity (SOTC), makes the control parameter spontaneously evolve towards acondition of fluctuation around a critical value that, interpreted as a fluc-tuating temperature may lead to physical effects similar to those of thetemperature gradients studied in Ref. [50], which affords again a naturalway to explain the synchronization of rotators. However, this direction,which is left as a subject of future investigation, would make more difficultfor us to explain the important role of crucial events for the realization ofsynchronization. For this reason in this paper we adopt the subordinationto a periodic motion of Ref. [51], which is a generalization of CTRW [44],based, indeed, on activating the action of crucial events.

Hereby we describe subordination to periodic and regular rotation. Thistheory, supplemented by the intelligence necessary to realize CTRW leadsus to the central algorithmic prescription of Eq. (8) ) and Eq. (9): there isno progressive phase shift between the clocks if there is no crucial event anda shift if there is. This is the central idea we use to supplement CTRWwith the intelligence realizing the spontaneous synchronization that theKuramoto models generates with a proper choice of the control parameter[46].

Consider a clock, whose discrete hand motion is punctuated by ticksand the time interval between consecutive ticks is, ∆t = 1, by assumption.At any tick the angle θ of the clock hand increases by 2π/T , where Tis the number of ticks necessary to make a complete rotation of 2π. Weimplement subordination theory by selecting for the time interval betweenconsecutive ticks a value τ/ < τ > where τ is picked from an IPL waiting -time PDF ψ(τ) with a complexity index µ > 2. This is a way of embeddingcrucial events within the periodic process. Notice that in the Poisson limitµ → ∞ the resulting rotation becomes virtually indistinguishable fromthose of the non-subordinated clock. Note further, that when µ > 2, themean waiting time 〈τ〉 is finite. As a consequence, if T is the informationabout the frequency Ω = 2π/T , this information is not completely lost inthe subordinated time series.

During the dynamical process the signal frequency fluctuates around Ωand the average frequency is changed into an effective value

Ωeff = (µ− 2)Ω. (3)

This formula can be easily explained. In fact, µ = 3 is the border betweentwo distinct statistical regions, the Levy and the waiting-time PDF of theGaussian region µ > 3 where both the first and second moment of ψ(τ)

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Figure 1: The spectrum S(f) of subordinations to the regular clock motion.(a) Ω = 0.06283, µ = 2.1 (black curve), µ = 2.9 (red curve). (b) µ = 2.1,Ω = 0.06283 (black curve), Ω = 0.0006283 (red curve).

are finite, and the average of the fluctuating frequencies is identical to Ω.In the region µ < 2 the process is non-ergodic, the first moment < τ > isdivergent and the direct indications of homeodynamics vanish. The condi-tion 2 < µ < 3 is compatible with the emergence of a stationary correlationfunction, in the long-time limit, with µ replaced by µ− 1. Thus, using theresult of earlier work [51] we obtain for the equilibrium correlation functionexponentially damped regular oscillations. At the end of this oscillatoryprocess, an IPL tail proportional to 1/tµ−1is obtained. Using a Tauberiantheorem explains why the power spectrum S(f) becomes proportional to1/f3−µ for f → 0. In summary, in a log-log representation, we obtain acurve with different slopes: β = 3−µ, to the left of the frequency-generatedbump, and β = 2, to its right. The slope β = 2 is a consequence of theexponentially damped oscillations.

These predictions are confirmed in The spectrum S(f) of subordina-tions to the regular clock motion. (a) Ω = 0.06283, µ = 2.1 (blackcurve), µ = 2.9 (red curve). (b) µ = 2.1, Ω = 0.06283 (black curve),Ω = 0.0006283 (red curve). , which illustrates the result of a numericalapproach to the subordination to a periodic clock with frequency Ω. In theabsence of subordination the projection on the abscissa axis would gener-ate x(t) = Re(eiΩt). When we apply subordination to this regular motion,we interpret the time evolution of x(t) as the result of a cooperative inter-action between many oscillators. The IPL index µ quantifies the temporalcomplexity, spontaneously realized as an effect of oscillator-oscillator inter-actions. The spectrum S(f) of subordinations to the regular clock motion.(a) Ω = 0.06283, µ = 2.1 (black curve), µ = 2.9 (red curve). (b) µ = 2.1,

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Ω = 0.06283 (black curve), Ω = 0.0006283 (red curve). shows that as aneffect of periodicity, the low-frequency region where the ideal 1/f noise isexpected to emerge at µ = 2, can be strongly reduced giving more roomto the region of noise 1/f2 (see panel b).

To make network-1 (S1) drive network-2 (S2) we have to generalize theswarm intelligence prescription adopted in earlier work [40,41,47–49]. Thisgeneralization is necessary because the earlier work was limited to matchingof two identical networks at their criticality, and also was based on theassumption that the single units of the complex networks, in the absenceof interaction, undergo dichotomous fluctuations without the periodicityimposed here. In the absence of periodicity, the mean field x(t) of thecomplex network can be written as x(t) = (U(t)−D(t))/N , where U(t) isthe number of individual in the state “Up”, x > 0, and D(t) is the numberof individuals “Down”, x < 0. In the case considered herein the number ofunits in a network N = U(t) + D(t) is constant. Using this notation (seeSection 3) we show that network-2 under influence of network-1 changesas:

∆x2 ∝ K(t), (4)

whereK(t) ≡ 2(x1(t)− x2(t)). (5)

To properly take periodicity into account note that the mean field in S1

given by x1(t) has the functional form

x1(t) = cos(Ω1n1(t)). (6)

The mean field in S2 has the same periodic functional form, up to a time-dependent phase,

x2(t) = cos(Ω2n2(t) + Φ(t)). (7)

The phase Φ(t) is a consequence of the fact that the units of S2 try to com-pensate for the effects produced by the two independent self-organizationprocesses. The number of ticks of the S1 clock, n1(t), due to the occurrenceof crucial events, becomes increasingly different from the number of ticksof the S2 clock, n2(t). The units of S2 try to imitate the choices made bythe units of S1. This is modeled by adjusting the phase Φ(t) of Eq. (7).The phase change is proportional to K(t) and to the derivative of x2(t)with respect to t. Thus, we obtain the central algorithmic prescription ofthis paper:

Φ(t+ 1) = Φ(t), (8)

if at t+ 1 no crucial event occurs, and

Φ(t+ 1) = Φ(t)− r1K(t)sin (Ω2n2(t) + Φ(t)) , (9)

if at t+1 a crucial event occurs. Note that the real positive number r1 < 1,defines the proportionality factor left open by Eq. (4), or, equivalently,defines the strength of the perturbation that S1 exerts on S2.

S1 (blue curve) drives S2 (red curve). Two systems are identical: µ =2.2, Ω = 0.063, r1 = 0.05. The connection (one directional) is realizedusing Eq. (4) and Eq. (14). illustrates the significant synchronizationbetween the driven and the driving system obtained for µ = 2.2, close tothe values of the crucial events of the brain dynamics [27]. This result alsocan be used to explain the experimental observation of the synchronizationof two people walking together [16] (see Section 3).

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Figure 2: S1 (blue curve) drives S2 (red curve). Two systems are identical:µ = 2.2, Ω = 0.063, r1 = 0.05. The connection (one directional) is realizedusing Eq. (4) and Eq. (14).

The top panel of The spectrums of subordinations. (a) Black curve(S1): µ = 2.1,Ω = 0.063; red curve (S2): µ = 2.9,Ω = 0.063; blue curve:S2 after being connected (one directional) to S1 with r1 = 0.1. (b) Blackcurve (S1): µ = 2.9, Ω = 0.0063; red curve (S2): µ = 2.1, Ω = 0.063;blue curve: S2 after being connected (one directional) to S1 with r1 = 0.1.shows that S2, with µ2 = 2.9, is very close to the Gaussian border andadopts the higher complexity of S1 with µ1 = 2.1, namely the complexityof a network very close to the ideal condition, µ = 2, to realize 1/f noise.

In the bottom panel of The spectrums of subordinations. (a) Blackcurve (S1): µ = 2.1,Ω = 0.063; red curve (S2): µ = 2.9,Ω = 0.063; bluecurve: S2 after being connected (one directional) to S1 with r1 = 0.1. (b)Black curve (S1): µ = 2.9, Ω = 0.0063; red curve (S2): µ = 2.1, Ω = 0.063;blue curve: S2 after being connected (one directional) to S1 with r1 = 0.1.we see that a driving network very close to the Gaussian border does notmake the driven network less complex, but it does succeed in forcing it toadopt the regulator’s periodicity. Here we have to stress that the perturbingnetwork is quite different from the external fluctuation that was originallyadopted to mimic the effort generated by a difficult task [33, 34]. In thislatter case, according to Heidegger’s phenomenology [35] the transitionfrom ready-to-hand to unready-to-hand makes the IPL index µ depart fromthe 1/f -noise condition µ = 2 [33, 34] so as to reach the Gaussian borderµ = 3 and to go beyond it. Here the perturbation is characterized byan intense periodicity and while it does not change the complexity of theperturbed network very much, it does transfer its own periodicity.

The theory developed herein may shed light on the crucial role of coop-eration. Recent psychological research on collective intelligence [52] showsthat a cooperative interaction between the members of a group may im-prove the global intelligence of a group. To realize a condition that is closeto that of Ref. [52] we study the case where S1 is influenced by S2 in thesame way S2 is influenced by S1. To make this extension we have to in-troduce the new parameter r2, which defines the intensity of the influenceof S2 on S1. As a result of this mutual interaction, we have µ1 → µ′1 andµ2 → µ′2. When µ1 < µ2 we expect

µ1 < µ′1 < µ′2 < µ2. (10)

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Figure 3: The spectrums of subordinations. (a) Black curve (S1): µ =2.1,Ω = 0.063; red curve (S2): µ = 2.9,Ω = 0.063; blue curve: S2 afterbeing connected (one directional) to S1 with r1 = 0.1. (b) Black curve(S1): µ = 2.9, Ω = 0.0063; red curve (S2): µ = 2.1, Ω = 0.063; blue curve:S2 after being connected (one directional) to S1 with r1 = 0.1.

The spectrums of subordinations. Black curve (S1): µ1 = 2.2, Ω1 = 0.063;red curve (S2): µ2 = 2.9, Ω2 = 0.063. Blue and pink curves are the spec-trums of S1 and S2 after being connected (bidirectional) with r1 = r2 = 0.1respectively. shows that µ′1 ≈ µ1, thereby suggesting that the system withhigher complexity does not perceive its interaction with the other systemas a difficult task, which would force it to increase its own µ [33,34], whilethe less complex system has a sense of relief. We interpret this result as animportant property that should be the subject of psychological experimentssimilar to that of Ref. [52] to shed light on the mechanisms facilitating thecontrolled exchange of information in the teaching and learning processes.The theory underlying complexity matching and therefore requisite vari-ety, makes it possible to go beyond the limitation of the earlier work oncomplexity management, as illustrated in Section 3.

The term “intelligent” that we are using herein is equivalent to assessinga network to be as close as possible to the ideal condition µ = 2, corre-sponding to the ideal 1/f noise. The spectrums of subordinations. Blackcurve (S1): µ1 = 2.2, Ω1 = 0.063; red curve (S2): µ2 = 2.9, Ω2 = 0.063.

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Figure 4: The spectrums of subordinations. Black curve (S1): µ1 = 2.2,Ω1 = 0.063; red curve (S2): µ2 = 2.9, Ω2 = 0.063. Blue and pink curvesare the spectrums of S1 and S2 after being connected (bidirectional) withr1 = r2 = 0.1 respectively.

Blue and pink curves are the spectrums of S1 and S2 after being connected(bidirectional) with r1 = r2 = 0.1 respectively. shows that as a resultinteraction, the complexity of S1 remains virtually unchanged while thatof S2 significantly increases thereby generating a mean value of µ that iscloser to the ideal condition of full intelligence µ = 2.

Note that in the same sense two very intelligent networks are the brainand heart that when healthy share the property of a µ being close to 2. Theargument presented herein therefore provides a rationale for (an explana-tion of) the synchronization between the heart and brain time series [26]showing that the concept of resonance, based on tuning the frequency ofthe stimulus to that of the network being perturbed, may not be appropri-ate for complex biological networks. Resonance is more appropriate for aphysical network, where the tuning has been adopted over the years for thetransport of energy not information. The widely used therapies resting onbiofeedback [53], are the subject of appraisal [54] and the present resultsmay contribute to making therapeutic progress by establishing their properuse.

3 Supporting Information

This section affords an example of the application of the theory developedherein to the analysis of experimental data. We focus on the close con-nection between Fig. 2 of this paper and Fig. 3 of Ref. [16]. For reader’sconvenience we illustrate this connection with the help of Experimentalwalking synchronization. These results have been derived with permis-sion from Ref. [16]. The top panel shows two distinct walking trajectories.These are two human subjects trying to walk together. The bottom panelshows the same trajectories so as to emphasize their synchronization. .

This figure is the result of the real experiment of Ref. [16] and it shouldbe compared to the qualitatively similar Time difference between the eventsof two identical systems connected back to back, µ1 = µ2 = 2.2, Ω1 = Ω2 =

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Figure 5: Experimental walking synchronization. These results have beenderived with permission from Ref. [16]. The top panel shows two distinctwalking trajectories. These are two human subjects trying to walk together.The bottom panel shows the same trajectories so as to emphasize theirsynchronization.

0.062, r1 = r2 = 0.1. obtained with the theory of this paper.

Figure 6: Time difference between the events of two identical systemsconnected back to back, µ1 = µ2 = 2.2, Ω1 = Ω2 = 0.062, r1 = r2 = 0.1.

We obtain Time difference between the events of two identical systemsconnected back to back, µ1 = µ2 = 2.2, Ω1 = Ω2 = 0.062, r1 = r2 = 0.1.using Eqs. (2-5) of the text and hereby we afford details on how to derivethese important equations. We use numerical results of the same kind asthose illustrated in Fig. 2 properly modified to connect the two trajectoriesback to back. To make the qualitative similarity with the results of theexperiment [16] more evident we adopt the same prescription as that usedby Delignieres and his co-worker and interpret the time interval betweenconsecutive crossings of the origin, x = 0, of Fig. 2 as the time duration ofa stride. We evaluate the mean stride duration and for both the driven andthe driving, for any stride we plot the deviation from mean value. Time

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difference between the events of two identical systems connected back toback, µ1 = µ2 = 2.2, Ω1 = Ω2 = 0.062, r1 = r2 = 0.1. illustrates the resultof this procedure, which can also be used to explain the synchronizationbetween heart and brain [26].

3.1 Group intelligence

Although subordination theory does not explicitly depend on the interac-tion between different units with their own periodicity, S2 is driven by S1

with a prescription inspired to create a swarm intelligence [47–49]. At agiven time the units of the driven systems look at the driving system andaccording to its state increase or decrease the phase of the driven systemas described in the Methods section.

The single individuals of the complex network may have only the value1, cooperation, or −1, defection. We introduce the angle θ to take period-icity into account and interpret cosθ as the ratio of the difference betweenthe number of cooperators and the number of defectors to the total numberof units. Thus the majority of cooperators corresponds to 0 < θ < π andthe majority of defectors corresponds to π < θ < 2π. We express that thechange in S2 because of interaction with S1 as

∆x2 ∝ p(D2 → U2)− p(U2 → D2), (11)

where the probability of making a transition from the state down to thestate up in S2 is given by

p(D2 → U2) =D2

U2 +D2

U1

U1 +D1. (12)

The form of Eq. (12) is due to the fact that this probability is the productof the probability of finding a unit in the driven system in the down stateby the probability of finding a unit in the driving system in the up state.Using the same arguments we find

p(U2 → D2) =U2

U2 +D2

D1

U1 +D1. (13)

Let us plug Eq. (12) and Eq. (13) into Eq.(11). We obtain

K(t) ≡ (1− x2(t))(1 + x1(t))− (1 + x2(t))(1− x1(t)), (14)

which is the important prescription of Eq. (5).

3.2 Walking together

To facilitate appreciation of the similarity between the complexity match-ing prescription of this paper and the walking synchronization of Ref. [16],we invite the readers to look at the experimental results of Experimentalwalking synchronization. These results have been derived with permis-sion from Ref. [16]. The top panel shows two distinct walking trajectories.These are two human subjects trying to walk together. The bottom panelshows the same trajectories so as to emphasize their synchronization. .The real data are not available to us, and we use surrogate data instead.These surrogate data are derived from the numerical results adopted toget Time difference between the events of two identical systems connected

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back to back, µ1 = µ2 = 2.2, Ω1 = Ω2 = 0.062, r1 = r2 = 0.1. , withtwo major adjustmentsm, as earlier explained, of focusing on the obser-vation of the origin crossing. The remarkably good qualitative agreementbetween Time difference between the events of two identical systems con-nected back to back, µ1 = µ2 = 2.2, Ω1 = Ω2 = 0.062, r1 = r2 = 0.1. andExperimental walking synchronization. These results have been derivedwith permission from Ref. [16]. The top panel shows two distinct walkingtrajectories. These are two human subjects trying to walk together. Thebottom panel shows the same trajectories so as to emphasize their syn-chronization. proves the efficiency of the complexity matching approachof this paper.

3.3 Beyond Complexity Management

Complexity management is difficult to observe, since it is based on en-semble averages, thereby requiring the average over many identical real-izations [13]. In the case of experimental signals of physiological interest,for instance time series relating to brain dynamics, taking the ensembleaverage is not possible. Recently a procedure was proposed [14] to convertan individual time series into many independent sequences, so as to againhave recourse to an average over many realizations. This procedure, how-ever, requires knowledge of time occurrence of crucial events. The theorydeveloped herein makes it possible to evaluate the correlation between thedriving and the driven networks using only one realization. We stress thatwhile complexity management [13] does not affect the power index µ of theinteracting complex networks, the present theory, as shown by The spec-trums of subordinations. (a) Black curve (S1): µ = 2.1,Ω = 0.063; redcurve (S2): µ = 2.9,Ω = 0.063; blue curve: S2 after being connected (onedirectional) to S1 with r1 = 0.1. (b) Black curve (S1): µ = 2.9, Ω = 0.0063;red curve (S2): µ = 2.1, Ω = 0.063; blue curve: S2 after being connected(one directional) to S1 with r1 = 0.1., affords important information onhow the cooperative interaction makes the unperturbed values of µ change.

In Dependence of Cmax (as a measure for complexity matching) onthe periodicity of the drive and driven systems. µ1 = µ2 = 2.8. r1 =0.1. the maximum value of the cross correlation function, Cmax, betweenthe driving and the driven. This figure shows that a significantly largefrequency mismatch strongly reduces the intensity of synchronization.

Dependence of Cmax on the complexity index of the drive and drivennetworks. T1 = T2 = 50, r1 = 0.1. shows the effect of changing µ1 and µ2

on Cmax, while keeping the frequencies Ω identical.

4 Complexity, Information and Conclusions

In the recent literature on self-organization, see, for example, Gershensonand Fernandez [55], emergence of complexity is interpreted as correspond-ing to information reduction, while emphasizing Ashby’s concept of home-ostasis. Variety increases with a complex system performing multitaskactions and decreases with a complex system focusing on a single task [56].More recent work confirms this property in sociological systems [57] while itis well known that it holds true for physiological processes [58,59]. The hy-pothesis of self-organization has been known and used in biology for nearlyhalf a century [60,61] (see also Chapter 5 of Eigen’s important book [62]).

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Figure 7: Dependence of Cmax (as a measure for complexity matching) onthe periodicity of the drive and driven systems. µ1 = µ2 = 2.8. r1 = 0.1.

Figure 8: Dependence of Cmax on the complexity index of the drive anddriven networks. T1 = T2 = 50, r1 = 0.1.

The theoretical approach to the complexity matching presented hereinmay afford a unifying view accounting for the main properties emphasizedby this literature while adapting the homeostasis perspective of Ashby [15]to the concept of homeodynamics. Consider the four distinct propertiesstressed in the current literature: Information reduction, requisite variety,multitasks, homeostasis.

Information reduction: The entropic approach used to deal with cru-cial events is the Kolmogorov-Sinai (KS) entropy hKS [63], which is welldescribed by the formula

hKS = z(2− z)ln2, (15)

where z ≡ µµ−1 . Eq.(15) indicates that the KS entropy vanishes at z = 2

and it remains equal to 0 in the whole infinite interval 2 < z <∞ (µ < 2).Allegrini et al. [64] noticed that z = 1, corresponding to µ = ∞, is thecondition of total randomness, namely, the case where an infinitely largeamount of information is necessary to control the system. The condition

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z = 1.5, corresponding to µ = 3, makes the sequence of crucial eventscompressibile, namely, it reduces the amount of information necessary tocontrol the system, and finally the KS entropy vanishes when µ < 2. Thisis a region characterized by the diverging value of < τ >.

The recent generalization to the mechanism of self-organized criticalitygiven by SOTC [47] generates crucial events with µ < 2, and, albeit a formof self-organization yielding values of µ in the interval 2 < µ < 3 is not yetknown, we make the plausible conjecture that complex processes that areexperimentally proven to generate crucial events in this interval as well as inthe interval 1 < µ < 2, are the result of a process of self-organization. Thecondition z > 2 (µ < 2) is where Korabel and Barkai [65] had to modify Eq.(15) leaving this expression unchanged for 1 < z < 2 and making it increasefrom the vanishing value with z > 2. Actually KS entropy is a Lyapunovcoefficient and Korabel and Barkai defined the Lyapunov coefficient forz > 2, by comparing the departure between two trajectories moving fromvery close initial conditions to tµ−1, rather than to t, as correctly done forz < 2. This means that the region z > 2 (µ < 2) is not fully deterministic,but the amount of information necessary to control the system is drasticallyreduced.

Requisite Variety : Ivanov et al. [66] noticed that the healthy heart-beats have a variability that makes it impossible to adopt the conventionalmethod of analysis of anomalous scaling based on the stationary assump-tion. Consequently they made the assumption of a scaling fluctuation thatled them to adopt a multi-fractal approach. Their proposal turned outto be very successful and was adopted to distinguish healthy heartbeatsfrom heart failure heartbeats [66]. Allegrini et al. [67] examined the samepatients studied in [66] using the crucial events defined herein and foundthat healthy patients have a µ very close to the value µ = 2, which makesthe KS entropy vanish. They also conjecture that a self-organization gen-erating crucial events may also be the generator of multi-fractality. Thisconjecture has been fully confirmed by the recent work of Ref. [68, 69]

Of remarkable importance for the requisite variety issue is the work byStruzik et al [59], emphasizing the transition from 1/f noise to 1/f2 noiseas a manifestation of variability suppression. Healthy heart physiology isbased on the balance between the conflicting action of the sympathetic andparasympathetic nervous systems, thereby resulting in the ideal 1/f noisefor healthy individuals and in the 1/f2 noise for pathological individuals.This condition is examined herein with the help of Fig. 1. The SOTC ofRef. [47] yields µ < 2. We examined the case of µ moving in the interval2 < µ < 3 using subordination to regular oscillatory motion, a phenomeno-logical way of combining crucial events and periodicity. We believe thatSOTC can be extended to this condition, and hope that future work mayrealize this important goal. We see that for f → 0, the IPL spectrumSp(f) ∝ 1/fβ , with β = 3− µ. The ideal condition of 1/f noise is realizedwhen µ = 2. The transition from 1/f noise to white noise is realized byincreasing µ from the ideal value µ = 2 to the value µ = 3 and beyond.In the presence of periodicity, though, the 1/f noise region can also beaffected by moving the periodicity peak from the right to the left, in sucha way as to make the 1/f2 noise become the predominant contribution tothe spectrum in accordance with the experimental observation of Ref. [59].

Monotasking versus Multitasking : The theoretical perspective adoptedherein affords an efficient way to approach this problem, while suggestingan interesting approach to cognition. In a recent paper Gershenson [70]

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addresses the important issue of the connection between cognition andinformation and writes: “Just like Buddhist philosophy, information theoryand current cognitive science are pointing towards a worldview not centeredon objective phenomena (studied traditionally by physics), but centeredon information, which can represent object, subject, and action within thesame formalism”. The readers can find another attempt at making progresson the cognition issue in the paper [71]. These authors rest on Buddhism tocreate a bridge between consciousness as a phenomenon in the operationalarchitectonics of brain organization and quantum mechanics. The earlierwork of these authors led to the discovery of rapid transition processes(RTP) that have been studied in [72], and found to be crucial events withµ > 2, but very close to µ = 2. For this reason, we are convinced that thetheoretical approach adopted herein may help to build the bridge betweenWest and East that the authors of [70] and [71] are trying to establish. Itis very encouraging to notice that Tuladhar et al [73] have recently foundthat meditation has the surprising effect of enhancing heartbeat coherencegenerating effects qualitatively similar to those illustrated in Fig. 1 ofthis paper, thereby leading us to interpret meditation as a mental processreducing variety to help the realization of specific tasks.

Homeodynamics: The important results established herein are basedon the action of crucial events, which are a manifestation of temporalcomplexity, thereby explaining why we replace Ashby’s homeostasis withhomeo-dynamics.

Finally, we conclude this paper stressing that the surprisingly accuratesynchronization of the walking together process ought not to be confusedwith either chaos synchronization or resonance. In fact, chaos synchro-nization requires finite Lyapunov coefficients and resonance requires fre-quency tuning. Complex systems with µ very close to the ideal conditionµ = 2, where the traditional Lyapunov coefficient vanishes, have the affectof transferring their temporal complexity to systems with higher values ofµ. The numerical results show, that, although communication through fre-quencies still exists (bottom panel of Fig. 4), the action of crucial events ismore important for the transfer of intelligence. Our theoretical approachis based on the essential role of crucial events. The crucial events with µbecoming closer to µ = 2 are generators of multifractality, as pointed outin Refs. [68] and [69]. Thus, our prediction that the walker with µ closeto 2 attracts the µ of the walker close to the Gaussian region µ = 3 canbe interpreted as a transmission of multifractality from the healthy to thesick walker in a surprising agreement with the recent experimental resultof [74].

Acknowledgments The authors thanks Dr. Herbert Jelineck drawingour attention to Ref. [59], PG thanks ARO for financial support of thiswork through grant W911NF1901.

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