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15

Complex ordering in spin networks: Critical role of adaptation rate for dynamically

evolving interactions

Anand Pathak∗ and Sitabhra Sinha†

The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India.

(Dated: April 30, 2019)

Many complex systems can be represented as networks of dynamical elements whose states evolvein response to interactions with neighboring elements, noise and external stimuli. The collectivebehavior of such systems can exhibit remarkable ordering phenomena such as chimera order corre-sponding to coexistence of ordered and disordered regions. Often, the interactions in such systemscan also evolve over time responding to changes in the dynamical states of the elements. Linkadaptation inspired by Hebbian learning, the dominant paradigm for neuronal plasticity, has beenearlier shown to result in structural balance by removing any initial frustration in a system thatarises through conflicting interactions. Here we show that the rate of the adaptive dynamics for theinteractions is crucial in deciding the emergence of different ordering behavior (including chimera)and frustration in networks of Ising spins. In particular, we observe that small changes in the linkadaptation rate about a critical value result in the system exhibiting radically different energy land-scapes, viz., smooth landscape corresponding to balanced systems seen for fast learning, and ruggedlandscapes corresponding to frustrated systems seen for slow learning.

I. INTRODUCTION

Many natural and technological complex systems canbe described as networks connecting a large number ofelements whose states evolve in time [1, 2]. The collectivebehavior of a system resulting from interactions betweenits components can exhibit non-trivial features, includ-ing critical phenomena [3]. For instance, a system ofbinary-state elements (e.g., representing individuals hav-ing opposing opinions on an issue) connected through anetwork having modular organization can exhibit order-ing dynamics at very distinct time-scales [4] and undercertain circumstances, self-organize into locally alignedclusters that correspond to the communities of the net-work (i.e., subnetworks characterized by a significantlyhigher connection density compared to the overall den-sity of the network) [5]. In many situations, the linksof the network (representing the interactions) can alsoevolve over time as a result of the changes in the statesof the components that they connect. Such connectionsmay not only be characterized by weights (indicating thestrength of interaction) but also sign (representing thenature of the interaction). For instance, in the contextof a network of synaptically-connected model neurons,positive links may correspond to excitatory interactions(whereby activation of one element can result in activa-tion of other connected elements) while negative linkscan give rise to inhibition (i.e., activation of an elementtends to suppress subsequent activation of neighboringelements) [6].The occurrence of negative links can lead to the emer-

gence of frustration because of the existence of inconsis-tent relations within cycles in the network [7, 8]. A net-work is said to be structurally balanced if its positive and

∗ anandp@imsc.res.in† sitabhra@imsc.res.in

negative links are arranged such that frustration is ab-sent. Such a network can always be represented as twocommunities, with interactions within each communitybeing exclusively positive while those between communi-ties can only be negative [9]. This is of interest not onlyfor social systems in the context of which the conceptof balance was first introduced [10], but also for biologi-cal systems. For instance, it has been recently observedthat the resting human brain is organized into a pair ofdynamically anti-correlated subnetworks [11]. This sug-gests that the network responsible for this behavior maybe structurally balanced. This possibility is intriguing inview of the observation of non-trivial collective behav-ior in balanced networks of binary-state dynamical ele-ments, such as, the coexistence of ordered and disorderedregions referred to as “chimera” order, in the presence ofan external field [12]. The process by which networks canevolve to a balanced configuration has been explored byconsidering a link adaptation process inspired by Hebb’sprinciple, the basis behind neural plasticity [13]. Ac-cording to this mechanism, weights associated with eachlink change in proportion to the correlation in the ac-tivity of the connected elements. Therefore, any initialfrustration in the network can be removed by modifyingthe connection weights in accordance with the dynami-cal states of the elements. Systems undergoing such linkadaptation in the presence of environmental noise showhigh variability in the convergence time required to reachthe balanced state [14].

In this paper we consider how the rate of dynamicalevolution of interactions in accordance with the Hebbianadaptation principle affects the collective behavior of thenetwork. Starting from a fully connected network that isstructurally balanced and introducing noise and externalfield so that the system exhibits chimera order, we showthat different rates of adaptation can result in distinctoutcomes. Fast learning rates result in persistence ofthe chimera regime, although the balanced network now

2

comprises communities with asymmetric sizes, viz., theordered sub-network containing a much larger fractionof elements of the network. Slow learning, on the otherhand, results in a completely ordered state with the inter-actions becoming only of positive nature. On removingthe external field, fast learning results in retrieving a net-work that is similar to the original one, i.e., structurallybalanced with communities of almost equal size. How-ever, slow learning gives rise to a fully frustrated networkexhibiting disorder. We observe that there exists a criti-cal interval for values of the learning rate, around whichsmall changes result in the system converging to distinctfinal states. In the following sections we first discuss themodel and adaptation dynamics, followed by descriptionof the results of numerical simulations. We conclude witha brief summary of our results and a discussion of theirimplications.

II. THE MODEL

For the purpose of investigating how collective behav-ior of a complex system is affected by adaptive dynamicsof interactions, we use one of the well-known spin mod-els of statistical physics which are generic systems foranalyzing cooperative phenomena. In particular, we usethe binary-state Ising spin, the spin orientations (“up”or “down”) representing a pair of mutually exclusivechoices. The simplicity of the model makes it appli-cable to not just magnetic materials (in the context ofwhich it was originally proposed) but any system wherethe elements choose between the two competing statesbased on interactions with neighboring elements, noise(represented by thermal fluctuations characterized by atemperature T ) and external stimulus (often representedby a magnetic field H). The interactions Jij betweenany pair of spins (i, j) can be either positive (promot-ing connected spins to have the same state) or negative(promoting connected spins to have opposite states) innature. For instance, in neuronal networks, one can viewthe neurons as binary-state devices that are either fir-ing (active) or quiescent (inactive). Correspondingly, aspointed out earlier, the excitatory and inhibitory connec-tions between neurons can be represented by positive andnegative interactions, respectively. At a different scale,one can view genetic regulatory networks in a similarvein, with genes being in either of two states, viz., get-ting expressed (active) or not(inactive). In this setting,genes promoting the expression of other genes correspondto positive interaction, while inhibiting the expression ofother genes correspond to negative interactions. In a dif-ferent context, the interactions between Ising spins canalso be used to represent social intercourse [15]. Here,the spin states are considered to be analogous to twocompeting opinions, while the interactions may representthe nature of the relation between a pair of individuals- positive corresponding to affiliative and negative corre-sponding to antagonistic relations.

We consider a system of 2N globally coupled Isingspins arranged into two sub-populations (called modulesor communities) each having N spins, at a constant tem-perature T . For the simulations reported below we havechosen N = 100. The interaction between a pair of spinsbelonging to the same module is positive having strengthJ (> 0), while that between spins belonging to differentmodules is negative with strength −J ′ (where J ′ > 0). Inthe absence of an external field and thermal fluctuations(i.e., H = 0, T = 0), the two modules will be completelyordered in opposite orientations. When subjected to anexternal field of strength H , the energy of the system isdescribed by

E = −J∑

i,j,α

σiασjα − J ′∑

i,j,α,β

σiασjβ −H∑

i,α

σiα, (1)

where σiα = ±1 is the Ising spin on the i-th node(i, j = 1, 2, . . .N) in the α-th module (α, β = 1, 2). Asevery spin interacts with every other spin, a mean-fieldtreatment should describe the system accurately. Forconvenience we define a = J(N − 1)/2 and b = J ′N assystem parameters. The state of all spins in the systemare updated stochastically at discrete time-steps usingthe Metropolis Monte Carlo (MC) algorithm with tem-perature T expressed in units of kBT/J where kB is theBoltzmann constant. The initial state of the system wehave chosen is one exhibiting chimera order [12], to ob-tain which we use a = 1, b = H = 10 and kBT/J = 5.We also allow the possibility that the interaction

strengths can change over time through an adaptationprocess inspired by the principle of Hebbian learning, aclassical concept in the area of neural networks [13]. Col-loquially often described as “neurons that fire together,wire together”, in this mechanism the strength of synap-tic connection between a pair of neurons is incrementedwhen the two exhibit correlated activation. In Ref. [14],this idea was used to propose a link adaptation rule thatapplies to a system of spins coupled via positive, as wellas, negative interactions as:

Jij(t+ 1) = (1− ǫ)Jij(t) + ǫσi(t)σj(t), (2)

which is applied after every MC step. The adaptationrate, ǫ, decides the time-scale over which the interactionstrength changes relative to the spin dynamics. Startingwith an initially frustrated spin system, implementingthe above Jij dynamics in absence of thermal fluctua-tions (i.e., T = 0) results in the system converging to astructurally balanced state [14]. This can be intuitivelyunderstood in terms of changes in the energy landscapeover which the state of the spin system evolves. Frus-trated systems have rugged energy landscapes compris-ing a large number of local minima (a fact which is ex-ploited in neural network models of associative memorywhere these minima are used to embed desired patternsone wishes the network to “memorize” [16, 17]. Thus,beginning at any randomly chosen initial state of spinconfigurations, the spin dynamics drives the system into

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0 0.5 1 1.5 2

−1

−0.5

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m1

m2a b

FIG. 1. Typical time-evolution of the magnetizations per spin ofthe two modules, m1 and m2, in (a) chimera ordered state and (b)disordered state, shown for MC simulations with N = 100 at (a)kBT/J = 5 and (b) kBT/J = 8, respectively.

the nearest local energy minimum. The subsequent evo-lution of the Jijs converts this into the global minimumof the system. However, in the presence of noise (T > 0),fluctuations in the spin dynamics can prevent the systemfrom being trapped in any state for sufficiently long du-ration. Thus, the adaptation dynamics fails to alter theenergy landscape sufficiently so as to turn the configura-tion into the global minimum. Therefore, with increasingtemperature, an extremely long time may be required toreach structural balance.In this paper we use a modified form of the Hebb-

inspired adaptation rule for interaction strengths thatwas introduced in Ref. [14]. This is to take into accountthe asymmetry in the strengths of positive and negativeinteractions necessary for observing chimera order (typ-ically J ′ ≃ 10J) [12]. We change the adaptation rulesuch that the upper and lower bounds for the evolvinginteractions have the same values as that of the positiveand negative interactions (respectively) which give riseto the chimera ordered state. Thus after each MC step,interaction strengths are changed according to:

Jij(t+1) = Jij(1− ǫ)+ ǫ[(J +J ′)σiσj

2+

1

2(J −J ′)]. (3)

The state of the system at any time is characterizedby two quantities. One of these is the frustration as-sociated with the interactions, measured by calculatingthe fraction of frustrated triads (i.e., connected sets of3 spins with an odd number of negative interactions) inthe system:

F =∑

i6=j 6=k

JijJjkJkiNC3

, ∀JijJjkJki < 0, (4)

which varies between 0 (no frustration) and 1 (completelyfrustrated).The other quantity is an order parameter p = |m1−m2|

that is used to identify a chimera state in a spin systemwith two modules that have magnetizations m1 and m2,respectively. To numerically estimate the order parame-ter p, we are confronted with a few potential complica-tions. First, as the magnetizations of the two modules

−1.5 −1 −0.5 0 0.5 1 1.50

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m1 − m

2

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2

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6

8

m1 − m

2

Fre

quen

cy d

istr

ibut

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ba

× 103 × 103

FIG. 2. Frequency distribution of the quantity m1 − m2 shownwhen the spin system exhibits (a) chimera order and (b) disorder.The kernel smoothened distribution function is represented by athick curve. Results shown for MC simulations with N = 100 at(a) kBT/J = 5 and (b) kBT/J = 8, respectively.

are stochastically fluctuating (especially the one which isdisordered, i.e., having lower magnetization), we can usetheir average values. However, this gives rise to an addi-tional problem as the modules can frequently exchangetheir order/disorder status, especially when the modulesare of the same size [Fig. 1 (a)]. Thus, a method is re-quired to measure p that is unaffected by the modulesswitching their magnetizations, while being able to deter-mine if the time-averaged magnetizations of the two mod-ules are different (the signature of chimera order). For in-stance, a simple time average of the absolute value of dif-ferences between m1 and m2 will not give correct resultsas one obtains a finite value (because of stochastic fluctu-ations) even when the time averaged magnetizations aresame for both modules [Fig. 1 (b)]. Therefore, to esti-mate p we have used the following algorithm. First, thefrequency distribution ofm1−m2 is computed. In the ab-sence of chimera, the distribution is unimodal [Fig. 2(a)]which is approximated as a Gaussian function, while forchimera ordering the distribution is bimodal [Fig. 2(b)]which is approximated as a superposition of two Gaus-sian functions. For a unimodal distribution, the value ofm1−m2 corresponding to the peak is used to calculate theorder parameter p. For bimodal distributions, a weightedaverage of the values corresponding to the peaks is used.For example, if one of the peaks occurs at p1 with valueh1 while the other is at p2 with value h2, the order pa-rameter is calculated as p = (h1|p1| + h2|p2|)/(h1 + h2).In order to make the automated detection of the peak lo-cations accurate, the frequency distributions need to besmoothed of all fluctuations in the frequencies so that thelocal maxima at the peaks can be uniquely determined.For this purpose we have used a kernel smoothing tech-nique [18] that gives a single peak location for unimodaldistributions and locations for two peaks in the case ofbimodal distributions.

4

800 900 1000 1100 12000

50

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Mod

ule

size

s n

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Learning ONLearning OFF

FIG. 3. Time-evolution of sizes of the modules resulting from linkadaptation dynamics that is introduced at time t = 1000 MC steps(indicated by broken line). The ordered module expands in size(n1) while the disordered one shrinks. Learning rate ǫ = 0.1

III. RESULTS

Starting from a structurally balanced network com-prising two modules, we first introduce field and thermalfluctuations so as to drive the system into chimera order- i.e., one of the modules becomes ordered while the otheris disordered. Once this is achieved we allow adaptationdynamics of the interactions to take place. As the evolu-tion of the interactions are related to the degree of fluc-tuations in the spin states, these would occur mostly inthe disordered module where the spins are subject to thecompeting forces of negative interactions with the spinsbelonging to the ordered module and the influence ex-erted by the uniform external magnetic field which triesto align the spins in parallel with those of the orderedmodule. Thus, at any instant, a few spins in the disor-dered module will get aligned with the spins in the othermodule and consequently may develop positive interac-tions with the latter as a result of link adaptation. Thismeans that they will now no longer be part of the dis-ordered module but become part of the ordered module.As a result, the size of the modules would change overtime - the ordered module increasing at the expense ofthe disordered one. This has to be taken into accountwhen measuring system properties, such as, magnetiza-tion per spin of the two modules. In order to have aconsistent definition of module size that would be valideven when the system is no longer in structural balance,we heuristically measure it as follows. At each updateof the Jijs, we randomly choose a spin from the orderedmodule and consider all spins that have positive interac-tions with it to belong to the ordered module, with theremaining spins comprising the disordered module. Fol-lowing this process, we can follow the time-evolution ofmodular membership of individual spins, as well as, thatof the module sizes. However, this measure becomes lessuseful as the system becomes increasingly frustrated.

Fig. 3 illustrates the evolution of module sizes withtime following the introduction of link adaptation dy-namics in the system. We can now measure the mag-netizations per spin, m1 and m2, of the two modules

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FIG. 4. (top) Time-evolution of magnetizations m1 and m2 of thetwo modules in the network (having sizes n1 and n2, respectively)before and after link adaptation dynamics is introduced at time t =1000 MC steps (indicated by broken line). In absence of adaptation,magnetizations show the characteristic signature for chimera order.Once adaptation dynamics is operational, the ordered module withmagnetization m1 is seen to increase in size. (bottom) In absenceof learning, the network exhibits no frustration as it is structurallybalanced. Following the introduction of adaptation dynamics, thereis a transient increase in frustration, before the system once againachieves balance, but with modules of asymmetric sizes.

FIG. 5. Snapshots of the time-evolution of the interaction matrixJ after adaptation dynamics is initiated at t = 1000 MC steps.Note that the time interval shown here corresponds to the sameperiod over which the frustration in the system initially rises andthen decreases again as the system once more reaches a balancedstate, as indicated in Fig. 4.

by taking into account the modular membership of eachspin and the module sizes. Fig. 4 shows these order pa-rameters, as well as, the frustration in the network, asa function of time before and after the link adaptationdynamics is introduced. As expected, we observe that fol-lowing the start of link adaptation dynamics, the orderedmodule begins to grow in size, which is also observed inthe time-evolution of the interaction matrix (shown assnapshots at regular intervals in Fig. 5).

The time-evolution of the initial chimera-ordered net-work subjected to link adaptation dynamics with variouslearning rates ranging from ǫ = 0.1 to 10−4 are indicatedin Figs. 6-9. For fast learning rates (e.g., ǫ = 0.1), thesystem immediately converges to a balanced state char-acterized by a large ordered module (comprising about95% of all the elements) and a small disordered one. Byobserving the magnetization time-series of the two mod-ules in Fig. 6 it is easy to infer that the system still ex-hibits chimera order. We note that a rise in frustration

5

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Learning OFF Learning ON

FIG. 6. Time-evolution of module sizes (top panel), magnetiza-tion (middle panel) and frustration (bottom panel) for an initiallychimera ordered system in the presence of an external field beforeand after link adaptation dynamics with rate ǫ = 0.1 is introduced(the time at which the link adaptation is initiated is indicated bythe broken line). Following a brief transient rise in the frustration,the system again attains a balanced state although with asymmet-ric module sizes.

0

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Learning ONLearning OFF

FIG. 7. Time-evolution of module sizes (top panel), magnetiza-tion (middle panel) and frustration (bottom panel) for an initiallychimera ordered system in the presence of an external field beforeand after link adaptation dynamics with rate ǫ = 0.01 is introduced(the time at which the link adaptation is initiated is indicated bythe broken line). The system eventually becomes completely or-dered with all interactions becoming positive (resulting in mergingof the two modules into a single one).

is seen for a very brief duration during the initial periodimmediately following initiation of the link adaptationdynamics. For a slower learning rate, viz. ǫ = 0.01, theprocess of expansion in size of the ordered module is dif-ferent (Fig. 7). We notice that soon after the beginningof link adaptation, the chimera ordering is lost as thedisordered module becomes ordered. However, the mod-ules still retain their different identities as they are de-fined based on the sign of interactions of the spins withinthem (which should be positive) and that with spins inthe other module (which should be negative). For evenslower learning rates such as ǫ = 10−3 or 10−4 (Figs. 8-9) the duration over which the two modules exist whileboth being ordered is seen to increase. During this pe-riod the modular membership of the spins do not changesignificantly (apart from small fluctuations). Chimera or-dering is lost as the interaction strengths for spins in the

0

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Learning ONLearning OFF

FIG. 8. Time-evolution of module sizes (top panel), magnetiza-tion (middle panel) and frustration (bottom panel) for an initiallychimera ordered system in the presence of an external field beforeand after link adaptation dynamics with rate ǫ = 103 is introduced(the time at which the link adaptation is initiated is indicated bythe broken line). Initially, the two modules both become ordered.However, eventually the two modules merge into a single one as allinteractions become positive. The time of merging coincides withthe transient rise in frustration.

0

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n

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Learning ONLearning OFF

FIG. 9. Time-evolution of module sizes (top panel), magnetiza-tion (middle panel) and frustration (bottom panel) for an initiallychimera ordered system in the presence of an external field beforeand after link adaptation dynamics with rate ǫ = 104 is introduced(the time at which the link adaptation is initiated is indicated bythe broken line). We observe that while one of the modules remainsordered, the initially disordered modules is gradually also tendingtowards complete order. Eventually one expects the two modulesto merge into a single one.

disordered module becomes weaker as a result of frequentswitchings of their orientations. As a result the effect ofthe external field becomes dominant, which causes thespins in the disordered module to align with it. As spinsin both modules are now aligned parallel to each other,adaptation of their interactions with time would eventu-ally make all interactions positive, thereby merging thetwo modules into one. The time at which this happensis indicated by a transient rise in the frustration (Figs. 7and 8), occurring much later for slower rates of adapta-tion.

The different scenarios we observe at fast and slowlearning rates suggest that there is a competition betweentwo processes: if the modular membership of the spinscan change rapidly following initiation of link adaptation

6

FIG. 10. The region of chimera order in the (H, T ) parameterspace shown for different ratios of the sizes of the two modules.Chimera is indicated in terms of high values of the order parameter|m1 −m2| (represented by different colors).

so that the system adopts a new structurally balancedconfiguration, the interaction strengths of the spins in thedisordered module do not decrease significantly - therebyallowing the two modules to coexist with the chimera or-dering intact (although the modules now have very dif-ferent sizes). Note that the modular membership of thespins will change rapidly as they continually change theirinteraction strengths because of the fast adaptation rate.However, the average number of spins in each module willremain fairly steady. If the learning rate is slow, the inter-action strengths will not have time to change sufficientlyrapidly to allow the system to reach a new structurallybalanced state following the initiation of link adaptationdynamics. As a result the field dominates over the weak-ened interactions of the spins in the disordered module,thereby eliminating chimera order and eventually causingthe two modules to merge.

The existence of chimera order even in situations wherethe two modules have very different sizes is a novel ob-servation as earlier it had only been observed in a systemwith symmetric module sizes [12]. In order to see how theregion in the field-temperature parameter space where weobserve chimera ordering varies with asymmetry in themodule sizes, we show in Fig. 10 the dependence of theorder parameter |m1 −m2| on the strength of the exter-nal field (H), the scaled temperature (kBT/J) and theratio of sizes of the two modules (n1/n2) in the absenceof link adaptation dynamics. We observe that the regionin parameter space where we find chimera order (highervalues of |m1 −m2|) increases significantly as we go to-wards higher module size ratios (i.e., n1/n2) especiallyfor lower values of temperature.

We now consider the situation when the external fieldis withdrawn while allowing the link adaptation dynam-ics to continue. A difference is expected with the resultsshown in the earlier work [14] where the effect of linkadaptation on the evolution of a system subject to ther-mal fluctuations was investigated, as the learning rule isdifferent on account of the asymmetry in the strengthsallowed for negative and positive interactions. As theadaptation dynamics used here allows for much strongernegative interactions (as compared to positive), we would

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a

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FIG. 11. (a) Time-evolution of magnetization (top panel) andfrustration (bottom panel) for an initially chimera ordered systemwith link adaptation rate ǫ = 0.1, before and after an external fieldis withdrawn (the time at which the field is switched off is indi-cated by the broken line). We observe that the system becomescompletely frustrated on withdrawal of the field. (b) The corre-sponding time-evolution shown for link adaptation rate ǫ = 0.01.

expect the system to become more frustrated as negativeinteractions are much more likely to contribute frustratedtriads. This is indeed what is observed for slow learningrates, e.g., ǫ = 0.1 and ǫ = 0.01 (Fig. 11). Note thatin a frustrated system, one cannot define modules in ameaningful way, and thus the distinction between mod-ules indicated in the time-series for magnetization is anartifact of the measurement method. For both of theslow learning rates we observe that the system convergesto a fully frustrated state corresponding to the maximumvalue of the measure of frustration in the system (= 1).

For faster link adaptation rates, however, we observeunexpected behavior in the system. For a learning rateǫ = 0.13 [Fig. 12], which is only slightly higher thanǫ = 0.1 [shown in Fig. 11 (a)], the system initially be-comes fully frustrated after the external field is with-drawn. Surprisingly, after a period of ∼ 2000 MC stepsduring which the system remains frustrated, it suddenlyconverges to a balanced state. This intervening periodbetween the field being switched off and the system spon-taneously splitting into two modules having positive in-teractions among all elements within them (and corre-spondingly, negative interactions between elements be-longing to different modules) becomes even shorter withincreasing learning rates (e.g., see Fig. 13 for ǫ = 0.15).Thus, we can identify a critical value of around 0.13 forthe link adaptation rate ǫ, below which the system re-mains frustrated and above which the system convergesto a balanced state, on withdrawal of the external field.

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m1

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FIG. 12. Time-evolution of module sizes (top panel), magnetiza-tion (middle panel) and frustration (bottom panel) for an initiallychimera ordered system with link adaptation rate ǫ = 0.13, beforeand after an external field is withdrawn (the time at which thefield is switched off is indicated by the broken line). We observethat the system initially becomes completely frustrated on with-drawal of the field. However after about 2000 MC steps the systemsuddenly becomes balanced.

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n1

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FIG. 13. Time-evolution of module sizes (top panel), magnetiza-tion (middle panel) and frustration (bottom panel) for an initiallychimera ordered system with link adaptation rate ǫ = 0.15, beforeand after an external field is withdrawn (the time at which the fieldis switched off is indicated by the broken line). We observe thatthe system quickly becomes balanced after a brief transient rise infrustration following the withdrawal of the field.

Indeed our simulations show that at ǫ = 0.13 there isa wide variation in the time required for the system toconverge to balance after the field is switched off.In the previous study of structural balance [14], it had

been observed that the time required to converge to astructurally balanced state exhibits a bimodal distribu-tion for a range of temperatures. We observe similarfeatures in our model also - for instance, for a link adap-tation rate of ǫ = 0.13. For slightly slower learning rates(e.g., ǫ = 0.12), the convergence time increases signifi-cantly. In our simulations, many realizations did not con-verge to the balanced state within the time of observation(104 MC steps). For higher learning rates, the systemrapidly converges to balance. However, unlike the con-vergence to balance seen in the previous study [14], themodules in the structurally balanced state that our modelsystem reaches are of almost equal size (see Figs. 12 and13) independent of the initial state of the network (in-

cluding the initial distribution of interactions which wasseen to affect the nature of the balanced state that thesystem converges to in the earlier work). This may ap-pear surprising as starting from a balanced state, oneexpects that the network will remain in that state as theinteractions adapt so as to make the corresponding en-ergy minimum even deeper. However, what we observeis that, on withdrawing the field the balanced state char-acterized by asymmetric module sizes is destabilized andthe energy minimum moves in the configuration spaceeventually reaching the region corresponding to balancedstate with modules having equal size. This difference be-tween our results and that of the previous study suggeststhat the adaptation rule used here (which is biased infavor of negative interactions) is responsible for an in-triguing meta-dynamics of the energy landscape itself.We have also observed how the structure of the energy

landscape underlying our system evolves as the interac-tions change through the adaptation rule. In the modelstudied earlier [14], the frustration of the initially dis-ordered system is around 0.5. On initiating link adap-tation, this value decreases as the system tends towardsbalance. When frustration reaches a value around 0.49,we observe that the energy landscape becomes such thatstarting from any spin configuration one can reach thesame minimum energy spin arrangement. This suggeststhat a small decrease in the frustration can accompany aa very large change in the basin of attraction of an energyminimum (which now encompasses a significant fractionof the configuration space). We observe similar behav-ior in our model system where the adaptation occurs inthe presence and subsequent absence of an external field.For example, for an adaptation rate of ǫ = 0.13, whenthe field is withdrawn the system initially becomes com-pletely frustrated with the value of frustration measuredas around 0.97. The corresponding energy landscape hasa very large number of minima, each having very smallbasins of attraction. As the system evolves towards abalanced state, we note that even when the frustrationdecreases by a very small amount, e.g., to 0.93, the en-ergy landscape transforms to one having an extremelydeep energy minimum with a very large basin of attrac-tion. We believe that this radical transformation of theenergy landscape of the system at the initial stages ofapproach to balance points to features of landscape evo-lution that deserve further study.

IV. CONCLUSIONS

In this paper we have explored the behavior of a spinsystem that is subjected to an external field, while theinteractions adapt in response to the system dynamics.The results obtained through our simulations that is de-scribed above is summarized in Fig. 14. From an earlierstudy, we had known that introducing link adaptationinspired by the Hebbian principle in a frustrated sys-tem can result in it converging to a structurally balanced

8

FIG. 14. Schematic representation of our key results. Startingfrom the structurally balanced network (with the different com-munities showing opposite ordering) at the bottom left quadrant,we obtain chimera state on introducing an external field of req-uisite strength (top left quadrant). Allowing the interactions toadapt with different learning rates result in the network maintain-ing balance but with different outcomes for the ordering. Fast andintermediate learning rates retain chimera order, although the dis-ordered region is much diminished, while slow learning leads tocomplete order (top right quadrant). On withdrawing the field,the system subject to fast learning rate again converges to a struc-turally balanced network with oppositely ordered communities ofapproximately equal size, similar to the original network. Inter-mediate and slow learning, however, results in a fully frustratednetwork with complete disorder (bottom right quadrant).

state - corresponding to evolution of the underlying en-ergy landscape - that depends on the initial state of thesystem. Applying this adaptation dynamics to a systemexhibiting chimera order (as done in here) allows us toexplore how the dynamics of the energy landscape is af-fected by an external field, as well as, asymmetry in theadaptation rule regarding negative and positive interac-tions.

We observe that when system is adapting throughfaster learning rates, the chimera order is maintainedthroughout its evolution, with the ordered module in-creasing in size at the expense of the disordered module.For slower learning rates, first the interactions of the ele-ments belonging to the disordered module become weakwhich results in loss of chimera order. This is followed bythe system converging to a fully ordered state where al-most all the interactions are positive. The sizes of the twomodules defining the system where one observes chimeraorder can be varied to see how the ratio of module sizescan affect the subsequent evolution of the system. In-deed, we see that the region in the field-temperature pa-rameter space over which chimera order is seen increasesas the ratio increases from 1. When the external field iswithdrawn, the system returns to a structurally balancedstate with modules of similar sizes if the adaptation rateis high. However, for slower learning rates, the systembecomes completely frustrated.

We have also tried to explore how the structure of theenergy landscape of the system evolves during learning.

Starting from a chimera ordered state that is subjectedto adaptation rate and external field, we observe changesin the landscape after the field is switched off. Initiallythe system becomes completely frustrated. However, af-ter an interval (which decreases with increasing learningrate) the system shows a sudden large increase in thebasin of attraction of an energy minimum, although thefrustration of the system has decreased only negligibly.This surprisingly radical change in the landscape result-ing from relatively small degree of change in the interac-tion structure of the network is a question that needs tobe explored further.

It is of interest to consider the implications of the re-sults reported here for adaptation in biological systems.As connections in the brain evolve according to long-termpotentiation that embodies the Hebbian principle thathas inspired the link adaptation dynamics used in ourstudy, it is natural to expect that the brain may exhibitat least some of the features observed here. Indeed, asmentioned earlier, the experimental observation of twodynamically anti-correlated subnetworks in the restinghuman brain [11] strongly suggests that in the absenceof strong external stimulation the underlying network isstructurally balanced. On the other hand, when exposedto stimuli, correlated brain activity is indeed observed -although not encompassing the entire network. This is,to some extent, reminiscent of the chimera ordered statethat is seen in our model system. Thus, the brain may beseen as corresponding to a system subject to relativelyfast adaptation rate. However, the Hebbian principlecan apply to a much broader class of biological systems,e.g., gene regulation networks where the co-expression ofgenes has been suggested to result in their co-regulationover evolutionary time-scales [19]. As the adaptation inthis case is provided by natural selection, which is or-ders of magnitude slower than the learning process in thebrain mentioned earlier, it is probably not unreasonableto conclude that this can be seen as one corresponding toour model system subject to slow rate of adaptation. It iseasy to see that the frustrated system with a large num-ber of energy minima can be considered to be analogousto the cellular differentiation process that allows conver-gence to any one of a large number of possible cell fatesdependent upon initial conditions. Indeed this analogyhas been used earlier by Kauffman to motivate Booleannetwork models for explaining differentiation during bi-ological development [20]. Extending this analogy, onemay wonder whether the system has a state correspond-ing to the ordered state seen on exposing it to an exter-nal stimulus. As this state is unique and will be attainedby the system independent of all initial conditions, it istempting to suggest that induced pluripotency in cellsexposed to chemical stimuli [21] may be the biologicalanalogue. Therefore, it may be of interest to study thephenomena reported here in biologically realistic modelsof networks adapting at different time scales.

9

ACKNOWLEDGEMENTS

We thank Rajeev Singh, Shakti N. Menon and R.Janaki for helpful discussions. This work was supportedin part by IMSc Complex Systems Project (XII Plan)

funded by the Department of Atomic Energy, Govern-ment of India. The numerical work was carried out inmachines of the IMSc High-Performance Computing fa-cility, including “Satpura” which is partially funded byDST (Grant No. SR/NM/NS-44/2009).

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