We consider networks of coupled, nonidentical oscillators, mathematically described by
where N is the number of oscillators (nodes) in the network, xi is a d-dimensional vector of the dynamical variables of node i, Fi(xi) is the vector field of node i, ε is a coupling parameter, and G is a coupling matrix determined by the connection topology. The elements of G are Gij = -1, i ≠ j if oscillators i, j are coupled and Gij = 0 if they are not. The diagonal elements are given by Gii = -∑j≠i Gij in order to satisfy the condition ∑j=1N Gij = 0 for any i, where N is the network size. When all the oscillators are identical, a complete synchronized state defined by x1 = x2 = ... = xN = s is an exact solution of Eq. (1). The coupling matrix G, as determined by the network topology, can be diagonalized with a set of real eigenvalues {λi, i = 1, ..., N} and the corresponding set of eigenvectors e1, e2, ..., eN. Full connectivity of the network ensures that there is one zero eigenvalue and the eigenvalues can be sorted as 0 = λ1 < λ2 < ... < λN. The variational equations governing the time evolution of the set of infinitesimal vectors transverse to the synchronization manifold, δxi(t) ≡ xi(t) − s(t), are dδxi/dt = DF(s) · δxi − ε ∑j=1N Gij DH(s) · δxj, where DF(s) and DH(s) are the d × d Jacobian matrices of the corresponding vector functions evaluated at s(t). The transform δy = Q-1 · δx, where Q is a matrix whose columns are the set of eigenvectors of G, leads to the block-diagonally decoupled form of the variational equation: dδy/dt = [DF(s) − ελi DH(s)] · δyi. Let Ki = ελi(i = 2, ..., N) be a specific set of values of a normalized coupling parameter K. All blocks of the decoupled equation are structurally the same with only the factor of Ki being different, leading to the following generic form for all the decoupled blocks: dδy/dt = [DF(s) − KDH(s)] · δy.
The largest Lyapunov exponent of the above block-diagonal variational equation is the MSF Ψ(K). If Ψ(K) is negative, a small disturbance from the synchronization state will diminish exponentially so that the synchronous solution is stable, at least when the oscillators are initialized in its vicinity. The synchronous solution is unstable and cannot be realized physically if Ψ(K) is positive because small perturbations from the synchronous state will lead to trajectories that diverge from the state. For the coupled oscillator network Eq. (1), a necessary condition for synchronization is then that all normalized coupling parameters Kᵢ (i = 2, . . . , N) fall in an interval on the K axis where Ψ(K) is negative. The network is more synchronizable if the spread in the set of Kᵢ values, or equivalently, the spread in the eigenvalue spectrum λᵢ, is smaller. The MSF allows the synchronization interval to be determined, which depends only on the coupling function (H) but is independent of the network topology. In particular, the condition for complete synchronization is given by ελ₂ ≃ K₁, and the critical parameter for the onset of synchronization is K_c ≃ K₁/λ₂, where K₁ is the value of K at which Ψ(K) becomes negative from the positive side.
In Ref. [13], a stability analysis for synchronization of nearly identical oscillators was carried out, which was based
on the following extended master-stability function (eMSF):
The term in the square parentheses is the same as that in the block-diagonal form of the variational equation in the case of identical nodal dynamics. In the second term, D_μ is the Jacobian matrix with respect to the parameter vector μ and ψ = Σ_{j=1}^N u_j μ_j, where δμ_i = μ_i - μ̄ is the parameter mismatch with respect to mean of μ_i over all oscillators, and u_i is the ith eigenvector of G. The stability of Eq. (2) can then be determined as a function of the two parameters K and ψ, and we can decompose the problem into two separate parts: one that depends only on the nodal dynamics and the coupling function H, and another determined by the parameter mismatch among the oscillators.
Let Ψ(K, ψ) be the largest Lyapunov exponent of Eq. (2). In general, the value of K₁ depends on ψ. To be specific, we fix the number of oscillators and variation in the parameter mismatch, and focus on the scaling relation between K_c and the link density. In this case, δμᵢ obeys the same statistic as that for ψ in Eq. (2). For different link density, since the mismatch is bounded in the same range, the value of K₁ can be regarded as a constant, which is slightly larger than K₁|ψ=0. As the amount of mismatch is increased, i.e., with a larger standard deviation σ as in our case, K₁ also increases, which has been verified numerically. Since the oscillators are not identical, K₁ is not the critical coupling strength for complete synchronization, but the value for which the variation for the dynamical variables are bounded [13], so it is essentially the onset point of phase synchronization. Approximately, we can use K₁|ψ=0 to represent K₁ with parameter mismatches. In fact, we find numerically that the product K_cλ₂ for different networks is nearly constant and the values are comparable with those of identical oscillators. We thus have K_cλ₂ ≃ K₁.
For complex topologies we consider random and scale-free networks. For the former, the link density is determined by p, the probability that an arbitrary pair of oscillators in the network are coupled. For the latter, we consider those generated by the preferential-attachment mechanism [22]: starting from a small number m₀ of completely connected oscillators, a new oscillator is introduced into the network with m links according to the preferential-attachment rule. The parameter m thus determines the link density. For both types of networks, estimates of the value of λ₂ are available [23]. In particular, for random networks, we have λ₂ ~ Np - 2√Np(1-p), while for scale-free networks, we have λ₂ ~ Cm. The relationship K_cλ₂ ≃ K₁ gives, for random and scale-free networks, respectively, the following scaling law governing the onset of phase synchronization:
The scaling laws (3) are the main result of this paper. We note that, for large random networks that satisfy $Np \gg 1$, the scaling law becomes $K_c \sim p^{-1}$.
To provide numerical verification of Eq. (3), we consider networks of coupled Rössler chaotic oscillators. The vector field of the i-th oscillator (node i) is given by $\mathbf{F}_i(\mathbf{x}) = [-(\omega_i y + z), \omega_i x + 0.165y, 0.2 + z(x - 10)]$, where the parameter $\omega_i$ is