One node driving synchronisation

Abrupt changes of behaviour in complex networks can be triggered by a single node. This work describes the dynamical fundamentals of how the behaviour of one node affects the whole network formed by coupled phase-oscillators with heterogeneous coupling strengths. The synchronisation of phase-oscillators is independent of the distribution of the natural frequencies, weakly depends on the network size, but highly depends on only one key oscillator whose ratio between its natural frequency in a rotating frame and its coupling strength is maximum. This result is based on a novel method to calculate the critical coupling strength with which the phase-oscillators emerge into frequency synchronisation. In addition, we put forward an analytical method to approximately calculate the phase-angles for the synchronous oscillators.

i 0 i N , denote the vectors whose elements represent the oscillators' natural frequencies, instantaneous phases, and coupling weights, respectively. Define the frequency synchronisation (FS), i.e., the phase-locking state, of the phase-oscillators described by Eq. (1) as, Our goal is to find K C , as the oscillators emerge into FS for a large enough K with as K > K C . The mean field form of Eq.
where s(i) = ± 1, we have, from Eqs. (5) and (7), that, Define a function f as θ θ θ , and a set  as However, the conclusion in (9) is still effective for the general case where α i ≠ α j . Because the proof for this conclusion was independent of α , and the only restriction was ζ = 0 26 , which is fulfilled when α i ≠ α j . The conclusion in (9) Therefore, the m-th oscillator (the key oscillator) is the most "outside" one of all FS oscillators spreading on a unit circle, where the most inner oscillator possesses the smallest value of φ i among all oscillators. As K is decreased from a larger value that enables FS in the network to a smaller one, φ sin i (as well as and φ m ≠ 0. Therefore, K C can be obtained by the following optimisation (OPT) problem (12) can be numerically solved by selecting a small step for x, x step , then increasing x from x min to x max by x step , such that we get a series of values of Explosive synchronisation was studied in ref. 36 using a generalised Kuramoto model, which is a particular case of the model described in Eq.
In this case, we have ζ i = ± 1, and = > ζ Kr 0 x m from Eq. (7). Then OPT in (12) can be analytically solved, and the minimum of f(x) is 2, i.e., K C = 2 when = x 2 2 . This result remarkably coincides with the critical coupling strength proposed in ref. 36 for the backward process (namely, decrease K from a larger one to a smaller one) of the explosive behaviour. However, the critical coupling strength for the backward process is different from the one for the forward process (namely, increase K from a smaller one to a larger one) for the explosive synchronisation 36 . In this paper, we consider network configurations for which the critical coupling strength is the same for both the backward process and the forward process, i.e., no explosive synchronisation happens, then K C obtained by OPT in (12) is also the critical coupling strength for the onset of FS in the forward process.
We further find, numerically, that OPT in (12) obtains its solution at Kr 0 x m , an approximate K C can be analytically obtained by forcing Let us now numerically demonstrate the exactness of the OPT in (12) to calculate K C , and Eq. (13) to calculate K A as the approximation of K C , for different phase-oscillator networks.
indicates that all oscillators (not all oscillators) are in FS. The coupling weight α i > 0,  ∀ ∈ i N , is generated within 1,10 , without losing generality. Figure 1(a-c) show the results for three networks: Fig. 1(a), 10 oscillators with Ω following an exponential distribution; Fig. 1(b), 50 oscillators with Ω following a normal distribution; Fig. 1(c), 100 oscillators with Ω following a uniform distribution. We calculate K C by OPT in (12), and gradually decrease K from K = K C + 0.2 to K C − 0.2. The results show that if K > K C , δ = 0 with an acceptable error in numerical experiments for all cases, meaning that the oscillators are in FS. If K < K C , δ > 0 implying that the oscillators lose FS for all cases. We note that the oscillators lose FS abruptly at K = K C . This means that our method is effective to calculate K C for all cases. Figure 1(d-f) demonstrate the effectiveness of Eq. (13) to analytically calculate an approximate K C by forcing = x 1. Denote x opt as the value of x that provides K C by OPT in (12). We define the relative error between 1 and | | x opt as η ( ) = − | | | | x : is near K C , and | | x opt is close to 1 for all cases. This means Eq. (13) works well to approximately calculate K C for networks formed by arbitrary number of oscillators with any Ω distributions.
One node driving synchronisation. Below, we show that K C is independent of the Ω distribution, weakly depends on the network size N, and mainly depends only on the key ratio of the key oscillator. For networks with different frequency distributions, diverse network sizes and various key ratios, we verify the dependence of K C on the Ω distribution, the network size N and the key ratio ζ m . In order to present the results in a way such that they can be compared, we normalise ζ m for these networks by making a parametrisation of α m based on the value of ζ m for each network. The surprising result is that, when we normalise ζ m to be the same value for networks with different N and diverse Ω distributions, K C is roughly the same in these networks. Therefore, the key oscillator is the key factor for the behaviour of these networks. Next, we perform two sets of simulations to demonstrate this result. We use Ω ( ) where ζ ζ = | | s 0 is a constant, and γ is a varying parameter which is set to be equal to 1 in the first set of simulation and will vary in the second set of simulation. Note that, this parametrisation process will enlarge all ζ m except for ζ s , such that all of these oscillators maintain their status of key oscillators in their own networks. (vi), calculate and record K C for all the 594 networks. In the second set of simulation, we further parametrise α m as a function of γ for all the 594 key oscillators. We increase γ from its original value 1 to 20 by a small step, and simultaneously decrease each α m by a proper ratio, such that Eq. (14) still holds. For each value of γ, we calculate and record K C for all the 594 networks. , respectively. The surfaces representing K C are similar in all panels, which means that K C is independent of the Ω distribution. We note that K C depends on N when N is small, but K C is almost independent of N for most cases where ⩾ N 50. Thus, we say K C weakly depends on N. However, if we keep N unchanged, we observe that K C almost linearly increases with the growth of γ [i.e., the decrease of α ( ) ] for all cases. In other words, K C will increase if we decrease the coupling weight for only one key oscillator. The reason is that the key oscillator is the first one to lose FS when we decrease K, and a key oscillator with a smaller coupling weight is easier to lose FS, which in turn implies a larger K C . As a conclusion, the behaviour of the key oscillator determines the FS of all oscillators, and the key ratio ( ) for the first group and second group of oscillators, respectively. From Eqs. (6) and (7), we have . The non-negativity of μ 1 comes from the fact that ζ ζ ⩾ i j for any ′ ⩽ ⩽ i N 1 and any . For simplicity, we denote µ µ ≈ − ⩾ 0 1 2 for both cases. When the oscillators emerge into FS with a given K′ (K′ > K C ), our model treats the whole system as two frequency-synchronous oscillators coupled by a common coupling strength K′ , with natural frequencies μ 1 and μ 2 , respectively. We assume that the two-oscillator system also follows the model described by Eq. (3) with coupling weights α 1 = α 2 = 1 which results in ζ 1 = μ 1 , and ζ 2 = μ 2 . Thus, from Eq. (7), we have  N(N = 3 to 200) oscillators with Ω following exponential, normal, and uniform distributions, respectively. γ is the the parameter used to re-scale the key ratio. The surface represents the critical coupling, K C , for different N and γ.
where r′ is the order parameter of the two-oscillator system. From Eq. (7), we have where λ ′ ≈ r 1 1 and λ ′ ≈ r 2 2 indicate a locally stable branch and a locally unstable branch of the FS solution for the two-oscillator model, respectively (see Methods). We only consider the stable branch ( λ ′ ≈ r 1 1 ). Furthermore, we use the order parameter of the two-oscillator system to be an approximation of the order parameter [Eq. (5)] of the N-oscillator system, i.e., λ ≈ ′ ≈ r r 1 . Thus, the analytical approximation φ ( ′ ) for the master solution φ ( ) in Eq. (16) is, The corresponding approximate FS solution [Eq. (15)], is Figure 3(a) shows the numerical results of the order parameter for a network with 50 oscillators where Ω follows a normal distribution and α l ,  ∀ ∈ i N , is a random number within 1,10 . K C is indicated by the magenta dash-dot line. When ⩾ K K C , the approximate order parameter, λ 1 [Eq. (18)] is close to the numerical one, r [Eq. (5)]. This means λ 1 can effectively approximate r. Define an N × 1 vector, ε  Figure 3(b,c) show the results of the average absolute error ε and σ, respectively, at K = K C + 0.1 which ensures the emergence of FS. Networks are formed by N(N = 3 to 200) oscillators, with Ω following exponential, normal and uniform distributions. ε and σ are small for all cases, which means that the error between φ ′ i and φ i is small I ∀ ∈ i N in all cases. Moreover, the larger K is, the smaller the error between λ 1 and r is [ Fig. 3(a)], which will further imply a smaller error between φ ′ i and φ i ,  ∀ ∈ i N . This means our method is effective to solve the phase-angles for oscillators as they emerge into FS, for networks formed by an arbitrary number of oscillators with any Ω distribution.  . (b,c) show, for different networks, the change of the average absolute error ε ( ) between φ′ in Eq. (19) and φ in Eq. (16) and the standard deviation (σ) of ε  , as a function of K, respectively. Networks with N (from 3 to 200) oscillators with Ω following exponential (dash red line), normal (green solid line) and uniform (black line with "+ ") distributions, respectively.

Discussion
In this paper, we presented our studies on the synchronisation for a finite-size Kuramoto model with heterogeneous coupling strengths. We provided a novel method to accurately calculate [OPT in (12)] or analytically approximate [Eq. (13)] the critical coupling strength for the onset of synchronisation among oscillators. With this method, we find that the synchronisation of phase-oscillators is independent of the natural frequency distribution of the oscillators, weakly depends on the network size, but highly depends on only one node which has the maximum proportion of its natural frequency to its coupling strength. This helps us to understand the mechanism of "the one affects the whole" in complex networks.
In addition, we put forward a method to approximately calculate the phase-angles for the oscillators when they emerge into synchronisation. With our method, one can easily obtain the solution of phase-angles for frequency-synchronous oscillators, without numerically solving the differential equation.

Methods
Excluding the unstable solutions. The FS solution of Eq. (3), i.e., the solution of Eq. (7) is for the solution of the two-oscillator system in the paper.
However, the stability analysis of the FS solution for the general case where α i ≠ α j is difficult and is still an open problem. In our numerical experiments, the stable solution we obtained is only the one that Thus, we exclude the solutions that for ζ ⩾ 0 i , and that for ζ i < 0.
The stability analysis for the two-oscillator system. The two-oscillator system also follows the Kuramoto model with α 1 = α 2 = 1, namely, Let θ  eq be a FS solution of Eq. (22). Let θ θ θ = + ∆    eq , where θ ∆  is a small perturbation on θ  eq . Linearise Eq. (22) around θ  eq , we have, The two eigenvalues of J are e 1 = 0 and