Distance dependent competitive interactions in a frustrated network of mobile agents

Complex network [1, 2] is the unifying paradigm behind many scientific disciplines ranging from physics, mathematics to computer science and biology. Besides the study related to local and global statistical properties of complex networks [3], researchers are equally interested in paying attention to the dynamics of their interacting units [4]. Among those collective dynamics, a widely studied phenomenon is that of synchronization [5, 6, 7, 8, 9, 10, 11, 12] of coupled oscillators arranged over the constituents of complex networks. Although, in recent times, there have been attempts in examining synchronizability of time-varying networks with different underlying structural arrangements, in most of these studies of synchronization, the nodes of the complex networks are spatially static. But, in real world networks, the individuals (nodes) move around and exchange information within close proximity. Such nodes in the contextual literature are well known as mobile agents [13, 14, 15, 16, 17, 18, 19, 20, 21] referring to those nodes whose movement highly influence the systems’ collective dynamics. But, in general, the states of those oscillators situated on top of those agents do not affect the agents’ mobility. The studies on the process of synchronization in mobile agents [22, 23, 24, 25, 26, 27, 28] have, so far, primarily concentrated on those particular cases when the mobile agents interact with each other only when they are closed enough in space. But, in all of these studies, the network architecture, i.e.,  how the oscillators are connected with each other, depends only on the attractive interaction. So, a basic query arises that how is the flow of information gets affected when different types of interactions are present in the movement. For instance, an attempt to answer this question can be made while incorporating repulsive coupling with the attractive one in such a networked system. Such co-existing dynamical interactions can be observed in a variety of systems and have been studied previously in different network set-ups yielding diverse collective states [29, 30, 31, 32, 33].

Synergy of attractive-repulsive coupling can lead to several interesting dynamical behaviors. One of such example is the emergence of solitary states in multiplex networks [34] possessing positive inter-layer and negative intra-layer interactions. Earlier, Maistrenko et al. [35] found a variety of stationary states in networks of globally coupled identical oscillators with attractive and repulsive interactions. Sathiyadevi et al. [36, 37] found diverse chimera states due to the introduction of nonlocal repulsive coupling together with an attractive coupling in a network of coupled oscillators. Also, interplay between “conformists” and “contrarians” leads to traveling wave along with the stationary states [38]. The effect of competitive interactions on the synchronization manifold among globally coupled identical Van der Pol oscillators through their velocities is rigorously addressed in ref. [39]. Even, the presence of a tiny fraction of repulsive couplings is found to enhance the synchronization of nonidentical dynamical units that are attractively coupled in a small-world network [40]. Origination of dragon-king-like extreme events as a result of the coexistence of excitatory and inhibitory chemical synaptic couplings is reported in the ref. [41]. Besides emergent dynamical patterns, co-presence of attractive and repulsive couplings carry high significance to several biological [42], physical [43], ecological [44] and even social scenarios [45].

Recently, Nag Chowdhury et al. [46] proposed an universal $0-\pi$ rule to identify the bipartiteness of networks from the anti-phase synchronization states. This article describes such a system of attractively coupled oscillators with negative links as frustrated networks [47]. Such frustrated systems under the influence of attractive-repulsive co-existing interactions may induce unanticipated phenomenon, like extreme events [48]. Most of the existing articles [49, 50, 41, 51, 52] on extreme events in dynamical systems and networks define such an event as rare event. Mathematically, a rare event is defined on a probability space $\chi$ equipped with a $\sigma$-algebra $\mathbb{B}$ of events such that $P(A)$ is small, where $P$ is the probability measure designed to quantify the probability of the occurrence of any event $A\in\mathbb{B}$. But, extreme events can be rare events [53, 54, 55], or they can often be frequent in time and space [56, 57, 58, 59]. Examples of such intermittent frequent extreme events include population size of Paris [60], the bubbling of share market [61] before a crash and many more.

Along with its small occurrences, extreme event has another threat in the form of large impact. Financial and commodity market crashes, tsunamis, hurricanes, floods, epidemic disease spread, global warming-related changes in climate and weather, warfare and related forms of violent conflict, asteroid impacts, solar flares, acts of terrorism, industrial accidents, 8.0+ Richter magnitude earthquake are few examples of such large impact events. In dynamical systems, researchers are recently starting to measure the impact of such events in terms of the amount of variation from the central tendency of that observable [49, 41, 51, 62, 63, 64]. So, if $\psi$ is the observable defined on the state space $U$ to $\mathbb{R}$, then the extreme events are members of the set $\{u\in U:\psi(u)>H_{S}\}$, where $H_{S}=m+d\sigma$ with $d$ is the non-zero integer. Here, $m$ and $\sigma$ are respectively the mean value and the standard deviation of all the peak values in a time series of $u$. But, if $H_{S}$ is too large, there will be very few values to model the tail of the distribution correctly as the variance is likely to be large due to only very extreme observations remaining. On the other hand, a low threshold value of $H_{S}$ will include too many values giving a high bias. Recently, $m+8\sigma$ is found to appropriate indicating extreme event threshold for Weibull distribution [65].

Inspired by these observations, we consider a frustrated network of limit cycle oscillators, where each node of the network is a mobile agent. Earlier all studies [66, 22, 23, 25, 27, 28, 65] related to mobile agents, considered only limited local interactions. Instead, we consider here a global network of mobile agents with co-existing switching interactions. Depending solely on the relative distance between the agents, attractive or repulsive interaction is activated. Specifically, we choose attractive interaction between those agents which are staying apart from each other, while we consider repulsive coupling for the agents that are sufficiently close. We consider two types of coupling schemes, namely symmetry-breaking and symmetry-preserving couplings. Under this set up, we explore the effect of coupling strengths which reveal quite interesting phenomena like cluster synchronization [67], complete synchronization, inhomogeneous small oscillation, extreme events etc. We also analytically derive the critical coupling strength for achieving complete synchronization using time-average Laplacian matrix in absence of repulsive interaction and numerically verify the results.

We consider $N$ mobile agents which are moving independently on a two-dimensional ($2$D) physical plane $S=[-g,g]\times[-g,g]$. Initially, the agents are randomly distributed on $S$. The $i$-th mobile agent can move in any direction with velocity $\big{(}v\cos\theta_{i}(t),v\sin\theta_{i}(t)\textbf{})$ (any kind of collision among the agents are not allowed), where $v$ is the uniform moving velocity and $\theta_{i}$ is drawn randomly from $[0,2\pi]$. Higher values of velocity $v$ implies that the agents move the whole physical space and which increases the possibilities of interactions. Thus, if $\big{(}p_{i}(t),q_{i}(t)\big{)}$ is the position of the $i$-th agent at time $t$, then motion updating process will be maintained by the following relation,

If at some time $t$, $p_{i}(t),q_{i}(t)$ exceed $|g|$, then we re-generate a new $\theta_{i}(t)$ in $[0,2\pi]$ such that $-g\leq p_{i}(t),q_{i}(t)\leq g$. We define the distance between any two agents by the standard Euclidean metric as

Next, we write the the dynamical equations characterizing the time-evolution of each agent ($i=1,2,\cdots,N$) in the networked system as follows,

where $\mathbf{x}_{i}\in\mathbb{R}^{n}$ is the state variable and $F(\mathbf{x_{i}})$ is the corresponding vector field of the $i$-th agent. Here $H(\mathbf{x_{i}},\mathbf{x_{j}})$ is chosen as the diffusive type coupling function, $k_{A}>0$ and $k_{R}<0$ respectively correspond to the attractive and repulsive coupling strengths. Moreover, $B$ and $C$ are the distance dependent adjacency matrices associated to the attractive and repulsive interactions such that

and

where $\alpha$ is the parameter lying within $[0,2\sqrt{2}g]$. Here $2\sqrt{2}g$ is the maximum possible distance between any two agents lying in the $2$-dimensional plane $[-g,g]\times[-g,g]$. At a particular instance, the agents whose Euclidean distance is less than $\alpha$, repel to each other and repulsive interactions arise between them. Attractive coupling occurs between the agents when the distance between them is greater than $\alpha$.

To illustrate this, we draw a schematic diagram with $N=3$ mobile agents in the $2$-dimensional plane $[-g,g]\times[-g,g]$ in Fig. (1). At a particular time $t$, we plot the position of these three agents in Fig. (1) where $g=10$ and $\alpha=10$ are considered. For these choices of $\alpha$ and $g$, the adjacency matrices corresponding to attractive and repulsive interactions at that specific time instant look like

$B=\left(\begin{array}[]{ccccc}0&1&1\\ 1&0&0\\ 1&0&0\end{array}\right)$ and $C=\left(\begin{array}[]{ccccc}0&0&0\\ 0&0&1\\ 0&1&0\end{array}\right)$

Now, we consider fix physical plane with $g=10$ and fix the parameter $\alpha=10.$ Each agents moving within the physical plane $[-g,g]\times[-g,g]$ using the rule given in (1) with fixed velocity $v=2.0$. Now we see the variation of degree distributions of the time-varying network. For this, we calculate the number of attractive and repulsive neighbors depending on the critical distance $\alpha$. In Fig. 2, the degree distributions of moving agents for attractive and repulsive interactions are shown. We randomly select an agent which is moving with uniform velocity $v=2.0$. We then find that the number of attractive neighbors are varying within the interval $[5,92]$ (cf. Fig. 2(a)) and the number of repulsive neighbors of that agent is varying within $[7,94]$ (cf. Fig. 2(a)). At any particular time-step, the sum of the number of attractive neighbors and the number of repulsive neighbors is $N-1$ for any $i$-th agent. This scenario is well expressed by the lower panel of Figs. 2(c, d).

Now, on top of each mobile agent, a two-dimensional Stuart-Landau (SL) oscillator is placed where the state dynamics of each limit cycle oscillator is represented by

where $\omega_{i}=\omega=3.0$ is the identical intrinsic frequency for each $i=1,2,\cdots,N$.

We consider two types of diffusive interactions, namely symmetry-breaking coupling $H(\mathbf{x_{i}},\mathbf{x_{j}})=({x_{j}}-{x_{i}},0)^{T}$ and symmetry-preserving coupling $H(\mathbf{x_{i}},\mathbf{x_{j}})=({x_{j}}-{x_{i}},{y_{j}}-{y_{i}})^{T}$. The interesting and important part of this distance dependent coupling functions is for $\alpha=0,$ the Eq. (3) becomes globally coupled network with attractive coupling only. In this case, movement of the agents does not effect the attractive adjacency matrix $B$ and the network becomes static. But, for $\alpha>0,$ depending upon the values of attractive and repulsive coupling strengths $k_{A}$ and $k_{R}$, two different cases can be implemented:

1. 1.

   $k_{A}\neq 0$ and $k_{R}=0:$ In this case, the mobile agents will interact attractively when the relative distance $d_{ij}$ is greater than $\alpha$ and remain disconnected with each other when they are closed to each other and relative distance $d_{ij}$ is less than equal to $\alpha$. There is no repulsion among the agents. Here, synchronization is the most desired state for a suitable value of $k_{A}.$

2. 2.

   $k_{A}\neq 0$ and $k_{R}\neq 0:$ In this case, interplay between attractive and repulsive interactions is considered. During the spatial movement of the agents, they experience both types of interaction among them. Here, different states may arise, like inhomogeneous small oscillation, synchronization, extreme event and other intermittent states.

In the following section, we will explore the dynamics of the time-varying network (3) with two types of diffusive couplings and under above mentioned distinct coupling conditions. Our main emphasis will be to identify the parameter space by varying the two coupling strengths $k_{A}$ and $k_{R}$ with fixed values of other network parameters $g=\alpha=10.0$ and $v=2.0$ related to the movement of the agents.

For numerical simulations, we integrate Eq. (3) using fifth order Runge-Kutta-Fehlberg method with a integration time step $\delta t=0.01$. At each integrating time step, $\theta_{i}(t)$ and thus $v_{i}(t)$ is updated according to rule given in (1) for each $i$-th mobile agent ($i=1,2,\cdots,N$). Hence, both the matrices $B$ and $C$ change at each integrating time step leading to fast switching approximation [68]. All simulations are done for fixed initial conditions

unless stated otherwise ¹¹1 Inhomogeneous small oscillation depends crucially on the initial conditions, and the basin of attraction of this state is small. If the initial conditions are not chosen suitably from the basin of attraction, then the trajectories may converge to a different periodic attractor. However, other observed dynamical states can be reached from any random initial conditions starting from $[-1,1]\times[-1,1]$.. In this Sec. (3), the symmetry-breaking coupling $H(\mathbf{x_{i}},\mathbf{x_{j}})=({x_{j}}-{x_{i}},0)^{T}$ is considered.

In this section, we deal with the scenario in which the local dynamical units, i.e., the SL-oscillators are coupled through both the variables so that the coupling becomes (rotational) symmetry-preserving. Our main emphasis will be to examine how this change affects the collective dynamics of the networked system. We depict the dynamical outcomes through Fig. 9 in which the phase diagram in the $k_{R}-k_{A}$ parameter plane is portrayed. As can be seen from the figure, the presence of only attractive coupling readily induces complete synchronization in the system. Moreover, in contrast to our earlier observation for symmetry-breaking coupling (cf. Fig. 8), this synchrony sustains as $k_{A}$ increases. This is mainly because now the coupling function does not allow to break the symmetry yielding inhomogeneous small oscillation. On the other hand, switching the repulsive coupling strength $k_{R}$ on, one observes occasional jumps in the error dynamics (for suitable strengths of couplings) that indicates the appearance of intermittent states. More importantly, further increment of $k_{R}$ leads to the intermittent states that satisfy the criteria of being considered as extreme events. This result demonstrates that our observations of complete synchronization or the extreme events on the considered network set-up is not limited to a specific choice of symmetry-breaking coupling function.

In this study, we have considered Stuart-Landau oscillators on top of mobile agents moving in a finite region of two-dimensional plane. Due to the spatial movement of the agents, the relative distance between any two agents always varies with time. Using this spatial distance between the agents, two competing interactions are introduced. Whenever the agents stay inside a closed ball of radius $\alpha$, the oscillators are repulsively coupled bidirectionally through diffusive linear coupling. On the contrary, whenever the mobile agents lie beyond the closed communication ball of radius $\alpha$, they are coupled through attractive coupling. Under this set-up of the networked system, we have encountered diverse collective behaviors and provided rigorous investigation of each of them.

Firstly, we have addressed the effect of sole attractive interaction among the dynamical agents. In order to accomplish this, we first set the repulsive strength to zero for which complete synchronization is found for $\alpha=10$. This result is quite interesting as for a static time-independent network, complete synchronization is not possible. Mobility of the agents play the decisive role to converge all the attractors to a single one. Analytically, the critical attractive coupling strength for synchrony is calculated, which exhibit excellent agreement with the numerical results. To investigate further the sole influence of attractive coupling, the repulsive matrix is set to be a null matrix by keeping $\alpha=0$ fixed. This different strategy leads to complete synchronization for suitable choice of $k_{A}$ and cluster synchronization beyond a certain value of $k_{A}$.

Next we dealt with our prime issue of coexisting attraction and repulsion. The coexistence of such competing interactions can demolish the synchronized behavior of the whole system. Under suitable choices of both coupling strength, inhomogeneous small oscillation is perceived. In this state, each oscillator maintains chaotic behavior though we set initially each oscillator into limit cycle regime. Further reduction of repulsive strength helps to regain their periodic behavior through the path of attractor switching. Moreover, further lessening leaves the error dynamics intermittent. This irregular away journey of error trajectory from the synchronization manifold gives rise to infrequent large deviated events. Using few characterizations, we confirmed these states as extreme events. The mean return interval of these extreme events is approximated using time series analogs of Kac’s lemma in relation (12). This relation indicates the dependency of average return interval $R_{H_{S}}$ on the threshold $H_{S}$. The larger $R_{H_{S}}$ clearly corresponds to larger $H_{S}$ and vice-versa. The route for generating such large amplitude events is also addressed in terms of on-off intermittency. An upper bound for the probability of occurrence of extreme events is calculated in relation (10) depending on the choice of $H_{S}$.

Further, to elucidate the mechanism behind inhomogeneous small oscillation of those oscillators, vector coupling is introduced. This coupling helps to preserve the rotational symmetry of SL-oscillators. Although, this vector coupling yields behavior like complete synchrony and extreme events, but it eliminates the appearance of such small oscillation, which is occurred during symmetry-breaking coupling.

We have been able to portray the whole scenario of all these emerging collective behaviors in a two-dimensional parameter space of the competing coupling strengths. This parameter space manifests the fact that by tuning any one of the coupling strength, one can avoid such devastating extreme events. This fact can be used as one of the influential strategy for controlling extreme events. Such controlled usage of antibiotics [87, 88] already found to be helpful in avoiding worsening medical cost and mortality especially for life-threatening bacteria infections.

In this article, we analyzed the interplay of positive and negative interactions in coupled identical limit cycle oscillators, by considering SL system on top of each mobile agent. An important direction of future generalization may include networks of chaotic oscillators, which is where our approach might make a difference. The chaotic dynamics [89, 90, 91, 92] are expected to reveal an even wider spectrum of interesting dynamical states. This remains the core interesting avenue for future work.

SNC and DG were supported by Department of Science and Technology, Government of India (Project No. EMR/2016/001039). SNC would also like to thank Physics and Applied Mathematics Unit of Indian Statistical Institute, Kolkata for their support during the pandemic COVID-19.