Chimeras on a social-type network

We consider a social-type network of coupled phase oscillators. Such a network consists of an active core of mutually interacting elements, and of a flock of passive units, which follow the driving from the active elements, but otherwise are not interacting. We consider a ring geometry with a long-range coupling, where active oscillators form a fluctuating chimera pattern. We show that the passive elements are strongly correlated. This is explained by negative transversal Lyapunov exponents.


I. INTRODUCTION
Since their discovery about 20 years ago by Kuramoto and Battogtokh [1], chimera patterns attracted large interest in studies of complex systems. Chimera is an example of a symmetry breaking in a homogeneous system of coupled oscillators: together with a homogeneous fully synchronous state there exist non-homogeneous states where some oscillators are synchronized and some not. In spatially extended systems chimera appears as a localized pattern of asynchrony [2][3][4][5][6][7][8][9][10][11][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][14][15][16][17].
The basic model of Kuramoto and Battogtokh is a one-dimensional ring of phase oscillators with non-local coupling. Each oscillator is coupled to all others in a symmetric bidirectional way; the strength of coupling depends on the distance on the ring. There are two typical setups for this distance dependence: exponential as in [1] (or its modification taking into account spatial periodicity [10]), or cos-shape coupling [2]. In both cases, chimera lives on a symmetric weighted bidirectional network. This paper aims to generalize the basic setting 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 with two types of nodes: (i) active nodes that force other nodes and potentially are also forced by them (i.e., active nodes have outgoing links); (ii) passive nodes that are driven by active nodes but do not influence them (i.e., passive nodes have only in-going links). We illustrate this in Fig. 1. The name "social-type" is picked because separation into active and passive nodes is similar to the separation of social networks into "influencers" and "followers". The latter participants get input from the former ones, but not vice versa. In physics, there are several prominent models of such type. In a restricted many-body problem in celestial mechanics, one considers several heavy bodies that interact and move according to the gravitational forces they produce. Additionally, light bodies move in the gravitational field created by the heavy ones but do not produce gravitational forces themselves (in fact, these forces are neglected in this setup). Another situation is modeling of two-dimensional turbulence 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 do not contribute to it. Below we construct the STN by taking a symmetric Kuramoto-Battogtokh network, and equipping it with additional passive oscillators. We will mainly consider a situation where the number of passive units is much larger that the number of active ones. The model will be introduced in Section II. In Section III we will illustrate the dynamics of passive units, and in Section IV will perform its statistical evaluation.
One can see that this coupling implements an STN: while active oscillators are mutually coupled, passive ones just follow the force from the active ones.
In previous literature, several coupling kernels G(·) has been explored. Kuramoto and Battogtokh [1] used an exponential kernel, Abrams and Strogatz [2] used a cos-shaped one.
We will follow the latter option, and set G as Parameters A = 3/4 and α = π/2 − 0.05 are fixed throughout the paper.
Nontrivial properties in the social-type network (1), (2) can be expected if the number of active oscillators N is not too large. Indeed, in the thermodynamic limit N → ∞ the field created by active oscillators is stationary (in a certain rotating reference frame), and the dynamics of passive oscillators in this field is trivial. In contradistinction, for relatively small N there are significant finite-size fluctuations, which, as we will see, lead to nontrivial effects. On the other hand, it is known that chimera in a very small population is a transient state [20]. Below in this paper we choose N = 32; for the parameters chosen chimera in Eqs. (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 distance between the states of neighboring active oscillators D k = | sin( ϕ k+1 −ϕ k 2 )|. This quantity is close 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 phases here 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 following features: 1. First we mention that the passive elements which have exactly the same positions as the active ones, attain the same state. This is due to the fact that although initial conditions are different, these pairs are driven by exactly the same field, and the conditional Lyapunov exponents are negative (see a detailed discussion of Lyapunov exponents below), so that active and passive oscillators synchronize.

A. Cross-Correlations
To characterize the level of regularity of passive units, we calculate the cross-correlation between the phases. Here, as has been shown in Ref. [18], it is important to use a proper observable. Indeed, because the rotations of passive phases are not free, their distribution is not uniform -this can be clearly seen in Fig. 3, where the phases in the disordered domain are concentrated around the value ϑ ≈ 2.5. In Ref. [18], where the Kuramoto model on a STN was treated, the transformation from the inhomogeneous phase ϑ to a homogeneous observable θ was performed using the local instantaneous complex order parameter z = e iϑ loc by virtue of the Möbius transform .
In the chimera setup of this paper, we cannot properly define a local complex order parameter due to strong finite-size fluctuations. Instead, we use transformation (4) with the global order parameter of active oscillators After the transformation (4) is performed, the cross-correlation between passive oscillators is calculated according to where the averaging is performed over all the pairs of passive phases and over a long time interval. The latter has been chosen long enough that every oscillator was both in regular and 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 = m M tends to zero, what corresponds to the mentioned above continuity of the phase profiles. At large ∆y the correlations are low; this is the advantage of using the "cleansed" observable θ instead of the original phase ϑ, for the latter the cross-correlations do not drop below 0.4.

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 what from which the transversal LEs (they depend on the position y m ), can be expressed as Calculated in this way transversal LEs are shown in Fig. 5. They are all negative, with the minimum at the central position between the active units.
The interpretation of the transversal LEs is as follows. If there are two passive units at exactly the same position on the ring but with different initial conditions, then they will eventually approach each other and synchronize. Quantity λ t gives the average rate of this exponential approach. In particular, if a passive unit is at the same position as an active one, they will synchronize with the average rate λ t (0). The result of this synchronization has been already discussed in Section III.
Negative transversal LEs explain also correlations of neighboring passive units (Fig. 4).
is the difference in the forcing. One can roughly estimate ∆ϑ ≈ ∆h/|λ t |, i.e. neighboring passive units nearly synchronize for small ∆y. This picture is however, not exact, as the discussion 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, such satellites are perfectly synchronized to the host [18] (similar to the restricted many-body problem in gravitational dynamics, where light particles in a vicinity of a heavy body do not leave 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 are at least three hosts which are detached from the satellites (these are isolated red dots at x ≈ 0.075, 0.74, 0.97).
Such a behavior is not covered by the simple relation (8). The reason is in the fluctuations of the transversal LE, not taken into account in relation (8). Such fluctuations may generally lead to so-called modulational intermittency [21], and this happens also here. On average the transversal LE is negative, but the values of quantity (7)

V. 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 interconnections, which drives another, passive network. Moreover, we assume that there no interconnections in the passive subnetwork, so that it consists of individual driven elements. 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 oscillators. The role of this number is twofold: first, it leads to fluctuations of the force driving passive elements, and second, it leads to weak turbulence of the active oscillators which restores ergodicity in the system. Let us briefly discuss, how the effects change for large active population sizes N . In this case chimera will move so slowly that the time where ergodicity establishes is not available. Thus, one should distinguish passive oscillators in the synchronous and the asynchronous domains. Even larger effect on the dynamics of passive elements 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 LEs in the asynchronous domain. The correlations, which are due to negative LEs, disappear in this limit, and can be expected to be very weak for large population sizes N .