CChimeras on a social-type network
Arkady Pikovsky
Department of Physics and Astronomy,University of Potsdam, 14476 Potsdam-Golm, Germany andDepartment of Control Theory, Lobachevsky University of Nizhny Novgorod,Gagarin Avenue 23, 603950 Nizhny Novgorod, Russia (Dated: ...)
Abstract
We consider a social-type network of coupled phase oscillators. Such a network consists of anactive core of mutually interacting elements, and of a flock of passive units, which follow the drivingfrom the active elements, but otherwise are not interacting. We consider a ring geometry with along-range coupling, where active oscillators form a fluctuating chimera pattern. We show thatthe passive elements are strongly correlated. This is explained by negative transversal Lyapunovexponents. a r X i v : . [ n li n . PS ] N ov . INTRODUCTION Since their discovery about 20 years ago by Kuramoto and Battogtokh [1], chimera pat-terns attracted large interest in studies of complex systems. Chimera is an example of asymmetry breaking in a homogeneous system of coupled oscillators: together with a homo-geneous fully synchronous state there exist non-homogeneous states where some oscillatorsare synchronized and some not. In spatially extended systems chimera appears as a localizedpattern of asynchrony [2–12]. In globally coupled populations chimeras are also possible:they emerge not as spatial patterns, rather a group of asynchronous oscillators “detaches”from the synchronous cluster [13–17].The basic model of Kuramoto and Battogtokh is a one-dimensional ring of phase os-cillators with non-local coupling. Each oscillator is coupled to all others in a symmetricbidirectional way; the strength of coupling depends on the distance on the ring. There aretwo typical setups for this distance dependence: exponential as in [1] (or its modificationtaking into account spatial periodicity [10]), or cos -shape coupling [2]. In both cases, chimeralives on a symmetric weighted bidirectional network. This paper aims to generalize the basicsetting of the Kuramoto and Battogtokh to a social-type network (STN). Such a network,introduced in [18], deserves a detailed description. It is a weighted directional network withtwo types of nodes: (i) active nodes that force other nodes and potentially are also forcedby them (i.e., active nodes have outgoing links); (ii) passive nodes that are driven by activenodes but do not influence them (i.e., passive nodes have only in-going links). We illustratethis in Fig. 1. The name “social-type” is picked because separation into active and passivenodes is similar to the separation of social networks into “influencers” and “followers”. Thelatter participants get input from the former ones, but not vice versa. In physics, thereare several prominent models of such type. In a restricted many-body problem in celes-tial mechanics, one considers several heavy bodies that interact and move according to thegravitational forces they produce. Additionally, light bodies move in the gravitational fieldcreated by the heavy ones but do not produce gravitational forces themselves (in fact, theseforces are neglected in this setup). Another situation is modeling of two-dimensional tur-bulence by a motion of point vortices [19]. The vortices move as interacting fluid particles,while other particles, like passive tracers, follow the velocity field created by vortices but donot contribute to it. 2
IG. 1. Illustration of a social-type network. Central blue units are active, they interact with eachother, but do not get inputs from passive units (peripherial red ones). The passive units are drivenby the active ones, and do not interact with each other.
Below we construct the STN by taking a symmetric Kuramoto-Battogtokh network, andequipping it with additional passive oscillators. We will mainly consider a situation wherethe number of passive units is much larger that the number of active ones. The model willbe introduced in Section II. In Section III we will illustrate the dynamics of passive units,and in Section IV will perform its statistical evaluation.
II. BASIC MODEL
We consider a network consisting of N active phase oscillators ϕ n and M passive phaseoscillators ϑ m . Both are uniformly distributed in space on a ring [0 , x n = ( n − /N , n = 1 , . . . , N ; the coordinates of passive units are y m = ( m − /M , m = 1 , . . . , M . All oscillators have identical frequency (which we set tozero chosing the appropriate rotating reference frame), and are nonlocally coupled:˙ ϕ n = 1 N N (cid:88) k =1 G ( x k − x n ) sin( ϕ k − ϕ n − α ) , n = 1 , . . . , N , (1)˙ ϑ m = 1 N N (cid:88) k =1 G ( x k − y m ) sin( ϕ k − ϑ m − α ) , m = 1 , . . . , M . (2)One can see that this coupling implements an STN: while active oscillators are mutuallycoupled, passive ones just follow the force from the active ones.In previous literature, several coupling kernels G ( · ) has been explored. Kuramoto andBattogtokh [1] used an exponential kernel, Abrams and Strogatz [2] used a cos-shaped one.3e will follow the latter option, and set G as G ( x ) = 1 + A cos(2 πx ) . (3)Parameters A = 3 / α = π/ − .
05 are fixed throughout the paper.Nontrivial properties in the social-type network (1),(2) can be expected if the numberof active oscillators N is not too large. Indeed, in the thermodynamic limit N → ∞ thefield created by active oscillators is stationary (in a certain rotating reference frame), andthe dynamics of passive oscillators in this field is trivial. In contradistinction, for relativelysmall N there are significant finite-size fluctuations, which, as we will see, lead to nontrivialeffects. On the other hand, it is known that chimera in a very small population is a transientstate [20]. Below in this paper we choose N = 32; for the parameters chosen chimera inEqs. (1) is strongly fluctuating and has a long life time. III. VISUALIZATION OF CHIMERA
In Fig. 2 we illustrate the chimera state in the set of active units ϕ k . We show the distancebetween the states of neighboring active oscillators D k = | sin( ϕ k +1 − ϕ k ) | . This quantity isclose to zero if the phases ϕ k and ϕ k +1 are nearly equal, and is 1 if the phase difference is π . In Fig. 2 the black region corresponds to a coherent domain of chimera (all the phaseshere are nearly equal), and the rest with red/yellow colors is the disordered state.Next we illustrate what happens to passive oscillators in the regime depicted in Fig. 2.In Fig. 3 we show a snapshot of the states of active and passive oscillators. It has followingfeatures:1. First we mention that the passive elements which have exactly the same positions asthe active ones, attain the same state. This is due to the fact that although initialconditions are different, these pairs are driven by exactly the same field, and theconditional Lyapunov exponents are negative (see a detailed discussion of Lyapunovexponents below), so that active and passive oscillators synchronize.2. The active oscillators show typical for chimera domains where the phases are nearlyequal (here 0 . (cid:46) x (cid:46) .
0 500 1000time 0 0.5 1 s pa c e FIG. 2. Chimera in a set of 32 active units (1). Color coding shows the distances betweenneighboring units D k , as function of time. Black region corresponds to a synchronized domain,yellow-red irregular pattern to the desynchronized one. The position of the synchronized domainexperiences a random walk, so that the dynamics on the long time scale is ergodic - each oscillatorsparticipates in synchronous and asynchronous motions. pha s e space FIG. 3. A snapshot of an STN with N = 32 active units (large red filled circles) and M = 8192passive units (blue dots). close values of the phases. Visually this appears as a continuous profile of passivephases values. Of course, this profile cannot be exactly continuous because of phaseslips, which are also clearly visible in Fig. 3 (e.g., at x ≈ .
21 and at x ≈ . V. STATISTICAL PROPERTIESA. Cross-Correlations
To characterize the level of regularity of passive units, we calculate the cross-correlationbetween the phases. Here, as has been shown in Ref. [18], it is important to use a properobservable. Indeed, because the rotations of passive phases are not free, their distribution isnot uniform – this can be clearly seen in Fig. 3, where the phases in the disordered domainare concentrated around the value ϑ ≈ .
5. In Ref. [18], where the Kuramoto model on aSTN was treated, the transformation from the inhomogeneous phase ϑ to a homogeneousobservable θ was performed using the local instantaneous complex order parameter z = (cid:104) e iϑ (cid:105) loc by virtue of the M¨obius transformexp[ iθ ] = exp[ iϑ ] − z − z ∗ exp[ iϑ ] . (4)In the chimera setup of this paper, we cannot properly define a local complex order parameterdue to strong finite-size fluctuations. Instead, we use transformation (4) with the global orderparameter of active oscillators z = 1 N (cid:88) n e iϕ n . After the transformation (4) is performed, the cross-correlation between passive oscillatorsis calculated according to c (cid:16) mM (cid:17) = |(cid:104) e i ( θ k − θ k + m ) (cid:105)| (5)where the averaging is performed over all the pairs of passive phases and over a long timeinterval. The latter has been chosen long enough that every oscillator was both in regularand irregular domains. The correlation function (5) is shown in Fig. 4, for N = 32 and M = 8192. One can see that the correlation function tends to one as ∆ y = mM tends to zero,what corresponds to the mentioned above continuity of the phase profiles. At large ∆ y thecorrelations are low; this is the advantage of using the “cleansed” observable θ instead ofthe original phase ϑ , for the latter the cross-correlations do not drop below 0 . B. Lyapunov exponents
In the context of STNs, there is a twofold application of the Lyapunov exponents (LEs).Usual LEs can be defined for a set of active particles, some of them are positive what6 C o rr e l a t i on f un c t i on distance FIG. 4. Cross-correlations in a chimera regime with N = 32 active and M = 8192 passive units,calculated according to (5). corresponds to turbulent dynamics depicted in Fig. 2. For passive oscillators, the LEs havea meaning of transversal Lyapunov exponents. Indeed, because passive units do not act onother oscillators, the system (1),(2) is a skew one, and linearization of Eqs. (2) for passiveoscillators leads to a set of independent one-dimensional equations for perturbations˙ δϑ m = − δϑ m N N (cid:88) k =1 G ( x k − y m ) cos( ϕ k − ϑ m − α ) , (6)from which the transversal LEs (they depend on the position y m ), can be expressed as λ t ( y ) = − (cid:42) N N (cid:88) k =1 G ( x k − y m ) cos( ϕ k − ϑ m − α ) (cid:43) . (7)Calculated in this way transversal LEs are shown in Fig. 5. They are all negative, with theminimum at the central position between the active units.The interpretation of the transversal LEs is as follows. If there are two passive units atexactly the same position on the ring but with different initial conditions, then they willeventually approach each other and synchronize. Quantity λ t gives the average rate of thisexponential approach. In particular, if a passive unit is at the same position as an activeone, they will synchronize with the average rate λ t (0). The result of this synchronizationhas been already discussed in Section III.Negative transversal LEs explain also correlations of neighboring passive units (Fig. 4).Indeed, neighboring sites (distance ∆ y ) experience different forcing fields, therefore theycannot synchronize completely. Instead, one can write a model linear equation for thedifference of states of passive units∆ ϑ ≈ −| λ t | ∆ ϑ + ∆ h , (8)7 L E position FIG. 5. Transversal LEs vs position on the ring, for N = 32. Due to periodicity with 1 /N , onlythe interval − / N < y < / N around an active unit at y = 0 is shown. where ∆ h ≈ ∆ y N N (cid:88) k =1 ∂G ( x k − y ) ∂y sin( ϕ k − ϑ − α )is the difference in the forcing. One can roughly estimate ∆ ϑ ≈ ∆ h/ | λ t | , i.e. neighboringpassive units nearly synchronize for small ∆ y . This picture is however, not exact, as thediscussion in next section shows. C. Intermittency of satellites
Here we focus on passive units that are extremely close to the active ones. We call them“satellites”, and the corresponding active unit the “host”. In the Kuramoto model, suchsatellites are perfectly synchronized to the host [18] (similar to the restricted many-bodyproblem in gravitational dynamics, where light particles in a vicinity of a heavy body do notleave this vicinity). In the present chimera setup, we however observe a different behavior.An inspection of Fig. 3 shows that indeed in many cases the satellites are close to the hosts(these cases a represented by blue “lines” passing through red dots). However, there areat least three hosts which are detached from the satellites (these are isolated red dots at x ≈ . , . , . o r de r pa r a m e t e r , d i s t an c e time FIG. 6. Illustration of intermittency in the satellites’ dynamics. Red line: 1 − | z s | ; blue dashedline: distance D . Outside of the burst 1 − | z s | ≈ D ≈
0. The burst has three stages. (i) first | z s | decreases from one, but D remains small; here the satellites are spread around the host. (ii) Both1 − | z s | and D are large, satellites are spread away from host. (iii) | z s | ≈ D is large; satellitesform a small cluster away from the host. distances, and the satellites detach from the host. It may take a long transient time untilthey attach again. This process is indeed intermittent, as Fig. 6 illustrates. In this figurewe take L = 32 passive satellites ϑ k of an active host, which are spread in the vicinity ofsize − − ≤ ∆ y ≤ − . To characterize these satellites, we calculate their complex orderparameter (using cleansed phases θ ) z s = 1 L L (cid:88) k =1 e iθ k and depict in Fig. 6, as functions of time, the absolute value | z s | and the distance from thehost ϕ measured as D = | sin arg( z s ) − ϕ | . Most of the time | z s | ≈ D ≈
0, what meansthat all the satellites are in a small neighborhood of the host. However, there is a burstwhere the satellites spread ( | z s | is as small as 0 .
2) and detach from the host. At the finalstage of the burst, the satellites congregate ( | z s | ≈ D is large). This is a quite unusual state, which we attribute to the fact that thetransversal Lyapunov exponent is smaller in absolute value close to the host, as one can seefrom Fig. 5. There is quite a long time interval 1300 (cid:46) t (cid:46) . CONCLUSION In this paper we considered a special class of networks - social-type networks STNs.From the mathematical viewpoint, they are skew systems: one active network with inter-connections, which drives another, passive network. Moreover, we assume that there nointerconnections in the passive subnetwork, so that it consists of individual driven elements.Furthermore, it is natural to asume that the number of active elements is small, and thenumber of passive units is large. This configuration mimics what really is observed in thesocial networks like the facebook [22, 23]. We, however, consider the effects related to STNfor oscillatory systems. We have considered both active and passive oscillators forming asymmetric ring, with long-range interactions. Active oscillators form a chimera pattern,with a synchronous and an asynchronous domains on a ring. Our main focus was on thedynamics of passive units. We have demonstrated that they are rather correlated, what isexplained by negative transversal Lyapunov exponents. A remarkable intermittent dynamicsis demonstrated by passive units (satellites) which are very close to an active host. Mostof the time the satellites follow the host, but there are bursts where they detach and leavethe host to move for certain time alone; after that the satellites again attach to the host.Probably, such a behavior by followers could be observed in social networks as well.We stress here that essential for our analysis was a rather small number of active oscil-lators. The role of this number is twofold: first, it leads to fluctuations of the force drivingpassive elements, and second, it leads to weak turbulence of the active oscillators whichrestores ergodicity in the system. Let us briefly discuss, how the effects change for largeactive population sizes N . In this case chimera will move so slowly that the time whereergodicity establishes is not available. Thus, one should distinguish passive oscillators in thesynchronous and the asynchronous domains. Even larger effect on the dynamics of passiveelements is due to smallness of finite-size fluctuations. Indeed, in the thermodynamic limit N → ∞ the field acting on oscillators is stationary in the proper rotating reference frame.Thus, passive elements will have negative LEs in the synchronous domain, and vanishing LEsin the asynchronous domain. The correlations, which are due to negative LEs, disappear inthis limit, and can be expected to be very weak for large population sizes N .10 CKNOWLEDGMENTS
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