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Fluctuation of Dynamical Robustness in a Networked Oscillators System

In this work, we study the dynamical robustness in a system consisting of both active and inactive oscillators. We analytically show that the dynamical robustness of such system is determined by the cross link density between active and inactive subpopulations, which depends on the specific process of inactivation. It is the multi-valued dependence of the cross link density on the control parameter, i.e., the ratio of inactive oscillators in the system, that leads to the fluctuation of the critical points. We further investigate how different network topologies and inactivation strategies affect the fluctuation. Our results explain why the fluctuation is more obvious in heterogeneous networks than in homogeneous ones, and why the low-degree nodes are crucial in terms of dynamical robustness. The analytical results are supported by numerical verifications.


I. INTRODUCTION
Dynamical systems in nature and human labs usually comprise a large number of interacting individual elements, such as synchronizing fireflies 1 , neurons in human brain 2 , cardiac pacemaker cells 3 , power grids 4,5 , and Josephson junction arrays 6 , just to name a few, which can be modeled by networked oscillators 7 . One important issue in the study of such dynamical systems is the robustness, i.e., the ability to maintain basic structure and function under attacks or disfunctions [8][9][10][11][12] . It has been shown that the huge blackouts, which inevitably cause tremendous economic loss, are related to the cascading failure of power-grids 8 . Such structure robustness involving network connectedness has been intensively investigated in previous works 8- 10 .
On the other hand, networked systems typically carry dynamics, for example, the circadian rhythms of mammals, the synchronization of cardiac cells, and spatio-temporal patterns in brain 2,3 . Of equal importance is the dynamical robustness, i.e., the ability of a networked system to maintain its normal dynamical activity when the topology or the local dynamics of network components are subjected to changes.
In Ref. 13 , Daido et al. investigated the dynamical robustness of a globally coupled system which simultaneously consists of active and inactive oscillators. With the increase of the ratio of inactive oscillators, the dynamical activity in the system decreases until finally it totally vanishes at certain critical point. Such phenomenon is termed as aging transition which characterizes the dynamical robustness of such networked systems 11,13 . Recently, Tanaka et al. further studied this phenomenon in various complex network topologies 12 . Surprisingly, it is found that the low-degree nodes, rather than the hubs, play a dominant role in terms of dynamical robustness. This finding uncovered the great differences between dynamical robustness and structure robustness.
For such dynamical systems, we always expect the dynamical robustness of them could be controllable. While sometimes we wish a strong robust system to maintain its functionality when suffering attacks, sometimes we prefer a fragile system because it is harmful and we need to destroy it. Tanaka's work reported that coupled oscillators network with smaller mean degree appears more robust, which give us a hint that we could increase the dynamical robustness of the system by reducing connections between elements. However, in this paper, we report the fluctuation of dynamical robustness in the above mentioned networked system. We find that even the network topology and the coupling strength in the dynamical system are fixed, there is still a fluctuation for the critical point of aging transition measured by the ratio of inactive oscillators in the system, which characterizes the dynamical robustness of the system. Moreover, the fluctuation could be large enough that totally changes the robustness property of the system. So it is important to understand the fluctuation for the potential applications of dynamical robustness. In addition, this fluctuation occurs more obviously in heterogeneous networks than in homogeneous ones. By applying meanfield approximation and linear stability analysis, we show that the cross link density between subpopulations of active and inactive oscillators plays a dominant role. In principle, different inactivation realizations lead to different link densities, which finally result in fluctuations of the critical points. Based on this analysis, we can explain why the fluctuations in heterogeneous networks are severer than that in homogeneous networks. Our results are also helpful to understand the crucial role of the low-degree nodes in determining the dynamical robustness of networked oscillators reported in Ref. 12 . The rest of this paper is organized as follows: section II introduces the dynamical model; theoretical analyses and numerical verifications are presented in section III; finally, conclusions are presented in section IV.

II. MODEL
In this work, we investigate a dynamical model of networked oscillators. It has following features: (1) the dynamical system is described by coupled oscillators; (2) there are two types of oscillators in the system, i.e., active and inactive, representing oscillating state and non-oscillating state, respectively; (3) the coupling among oscillators forms a complex network. The general form of this model can be written as: (1) Here, j = 1, · · · , N is the index of oscillator or node. In the present work, we mainly choose the Stuart-Landau (SL) oscillators as the local dynamics. Specifically, the networked SL oscillators system can be described by the following coupled ODEs: where z j and Ω j are the complex amplitude and the inherent frequency of the jth SL oscillator, respectively. α j is the control parameter denoting the distance from a Hopf bifurcation point.
When α j > 0, the oscillator is a limit cycle with an amplitude √ α j . However, it settles down to a fixed point when α j < 0. Thus generally the oscillator is active when α j > 0 and inactive when α j < 0. The oscillator will lose its activity as its α value converts from positive to negative. This can be used to model the two distinct dynamical states of oscillators. We define parameter ρ as the ratio of inactive oscillators in the network. Reasonably, the global activity of the networked system can be characterized by the normalized order parameter Q, defined as As ρ increases to a critical value ρ c , the networked system will totally loses its global activity until Q = 0, i.e., an aging transition occurs 13 . Because the ratio ρ c is the largest ratio with which the dynamical system can maintain certain activity, it measures the dynamical robustness of this networked system. The larger the ρ c , the better the dynamical robustness.
For simplicity, we will not consider the case in which the frequencies of oscillators have a distribution. Throughout this paper, we set Besides SL oscillators, we also consider the following networked Rössler oscillators in this tify the global activity, the order parameter can

III. ANALYSIS AND RESULTS
Recently, in our study we find an interesting phenomenon, i.e., even for fixed network topol-

A. Linear stability analysis
We start from Eq. (2). When ρ = 0, i.e., the system contains only identical active oscillators.
In this case, it will easily achieve global synchronization as the coupling strength increases.
When we randomly choose some nodes and change the oscillators into inactive states, i.e., the proportion of inactive nodes ρ > 0, the system simultaneously contains both active and inactive oscillators. Such a process is called random inactivation. Throughout this paper, it is adopted by default unless otherwise stated. Usually, the non-trivial aging transition would only occur when the coupling strength is large enough 13 , so before the transition all the active oscillators already well synchronize into a cluster, and so does the inactive oscillators. This has been numerically verified and one such example is illustrated in Fig. 4. Although the synchronization is not rigorous, it is good enough for us to approximately reduce the high-dimensional system into two identical clusters of oscillators, denoted as S A and S I for active and inactive oscillators, respectively. We use A and I to represent the dynamics of active and inactive subpopulations, respectively, as treated in Ref. 13 . Then the dynamics of the networked system, i.e., Eq.
(2), can be written as: Here, K j means the degree of oscillator j, and µ j is the proportion of inactive oscillator among the neighbors of oscillator j. Summing the equations of all active and inactive oscillators, we obtaiṅ In the above equations, the summing terms at the RHS, i.e., j∈S A µ j K j and j∈S I (1 − µ j )K j , both represent the total number of cross links, denoted by L, between the subpopulations of active oscillators and inactive ones. Actually they are the same thing in different ways.
To avoid the influence of network size, we define the density of cross links as λ = L/N = j∈S A µ j K j /N = j∈S I (1 − µ j )K j /N. Then Eqs. (6) and (7) becomė From these equations, we immediately find that with β = 4σab 2 λ/[ab + σ(a + b)λ] 2 . For typical dense complex network, λ ≫ 1. Therefore, β is a small quantity. We apply Taylor expansion to √ 1 − β in Eq. (10) and keep the linear term. It finally becomes B. Analysis to fluctuation of critical point Eq. (11)  Naturally, one may want to know which inactivation process will give the maximal ρ c or the minimal ρ c , which directly limits the fluctuation scope of ρ c given the network topology.
Interestingly, this problem can be mapped into the ground state problem of anti-ferromagnetic Ising model or the MAX-CUT problem in combinatorial optimization [19][20][21] . It has been proved that the solution of these problems in general networks is a NP-Complete problem, i.e., we cannot find an effective algorithm to determinate it in polynomial time 19 . Since analytical method is not available to get the fluctuation scope of ρ c , we turn to numerical study. To this end, we numerically plot the possible curves λ(ρ) on the parameter panel of ρ-λ. The two crosspoints that define the largest range on the axis of ρ roughly gives the fluctuation range of ρ c . As shown in Fig. 5(b), it is found that λ(ρ) for a heterogeneous network has a much larger fluctuation area compared to a homogeneous network. Therefore, more obvious fluctuation of ρ c is observed in the former case. Physically, this result can be heuristically understood.
We know that the fluctuation of ρ c is caused by the multi-valued λ(ρ), which depends on specific inactivation processes. Obviously, an inactivation process is more sensitive to the topology in heterogeneous networks than in homogeneous ones because there are huge differences among the node degrees in the former case. This leads to the fluctuation of λ(ρ) to a greater degree, which finally contributes to the fluctuation of the critical point ρ c .
As one application of the above theory, we now consider the case of targeted inactivation rather than random activation in network. In Ref. 12 , It has been reported that under targeted inactivation the dynamical robustness of the system depends more on the low-degree nodes rather than the hubs. This important finding reveals the crucial role of the low-degree nodes in the context of dynamical robustness. Based on our analytical treatment, we can provide an explaination here. In our study, we apply three typical strategies of inactivation: (1) Inactivation goes from the node with the maximal degree to the one with the minimal degree; (2) Inactivation takes the inverse order of (1); (3) Random inactivation. Numerical results for both networked SL oscillators and networked Rössler oscillators are shown in Fig. 6. In both cases, ρ c under strategy 1 is always greater than that under strategy 2, regardless of the heterogeneity/homogeneity of the network topology. To understand the above results, we plot the curves λ(ρ) on the λ-ρ parameter plane. As shown in Fig. 7, all curves are unimodal with λ(0)=λ(1)=0. Considering the specific processes of the three strategies, it is not difficult to figure out that with the increase of ρ, curve 1 increases faster than curve 2 to the maximum, while the decrease from maximum to zero is just the opposite. Strategies 1 and 2 are two extreme cases, and strategy 3 is between them. As shown in Fig. 7, the crosspoints of curves λ(ρ) with the bifurcation curve defined by Eq. (10) determine that ρ 1 c > ρ 3 c > ρ 2 c . Therefore, our theory successfully explain why the low-degree nodes plays more important role than the hubs in terms of dynamical robustness.

C. Random inactivation
In the above analysis, the only limitation for the network topology is that it should be dense enough so that the cross link density λ ≫ 1, and there is no requirement for the strategy of inactivation. Thus the above result holds for general inactivation processes. In fact, in our study, we find for typical random inactivation, Eqs. (8) and (9) can take another forms, and the theoretical treatment can be simplified.
We notice that in principle for different oscillator j, K j and µ j are not necessarily the same. However, they must satisfy the constraint j K j /N = K and j µ j /N = ρ over all the nodes when the mean degree of the network is fixed. Let µ j = ρ + ξ j , where ξ j is the deviation of µ j from its mean value. Normally, a node with larger degree K j would have smaller deviation. In a dense network, it is reasonable to expect that |ξ j | ∝ 1/K j and µ j distributes symmetrically around its mean value. For random inactivation, the active and inactive oscillators are mixed evenly. Thus the distributions of µ j in active and inactive subpopulations will approximately the same as in the whole system, i.e., we can approximately assume µ j = ρ for all j.
N j∈S I K j . Substitute these relations into Eqs. (8) and (9), we geṫ A = (a + iΩ − |A| 2 )A + σρK A (I − A), (12) where K A = 1 (1−ρ)N j∈S A K j and K I = 1 ρN j∈S I K j are the mean degrees of active and inactive subpopulations, respectively. Similarly, by applying a linear stability analysis, we can analytically obtain the critical point ρ c as: From this equation, we can immediately find that the dynamical robustness of the system are determined by the mean degrees of active and inactive subpopulations in the case of random inactivation. The point is, even though the mean degree of the whole network is fixed, there is still some degree of freedom for K A and K I as long as they satisfy the following constraint: On the parameter panel of circumstance, Eq. (15) degenerates as:  15,16 . Therefore, all these studies provides insights from different perspectives into the dynamical robustness of such networked system.

IV. CONCLUSION
In this work, we have studied the dynamical robustness in