Frequency-degree correlation

Dg(ω)dω

D²+ ω²(δ^inc(0, D))². (318) IfD≈ D^c= 1/2 then δ^inc(m, D)≈ 0. When D ≈ 0 at fixed m, δnoiseless^inc (m, D) satisfies

1 = πg(0) 2δ^inc_noiseless −m

2 Z ∞

−∞

g(ω)dω

1 + m²(δ^inc_noiseless)²ω². (319) Komarov et al. [313] studied a generic model in the presence of inertia, noise, and phase shift, i.e.

miθ¨i=− ˙θⁱ+ ωi+ λr sin (ψ− θⁱ− ϕ) + ξⁱ(t), (320) where the inertiami is distributed according to the density functionf (m), and ϕ is the phase shift. Rich phenomena emerge via considering various symmetry and asymmetry distributions off (m) and g(ω), e.g., the derivation of an exact solution of the self-consistent solution of the order parameter shows nontrivial phase transitions to synchrony due to correlations between natural frequencies and the moments of inertia [313].

Note that the recent review by Gupta et al. [314] provided a general mean-field analysis framework of the second-order Kuramoto model with noise, focus-ing on the equilibrium and out-of-equilibrium aspects of its dynamics from a statistical physics point of view.

7.1.4. Frequency-degree correlation

Let us now turn to effects of network topologies on dynamics. As illustrated in Sec. 5, the correlation between the dynamics and the structure can induce the emergence of dynamical abrupt phase transitions. In this case, the natural frequency distribution becomes asymmetric. Basnarkov and Urumov [233] in-vestigated the first-order Kuramoto model with natural frequencies distributed according to a unimodal asymmetric function and showed that a first-order phase transition occurs if the distribution has a sufficiently large flat section.

The MFA method [300, 301] (shown in the section 7.1.2) is provided for the second-order Kuramoto model (292) with symmetric frequency distribution, but the method for the model with asymmetric distribution is still open.

Ji et al. [244, 315] substantially extended the first-order Kuramoto model with frequency-degree correlation as discussed in Sec. 5 to the Kuramoto model with inertia. By consideringωi of each oscillator i proportional to its degree with zero mean, i.e. ωi= B(ki− hki) so thatP

iωi= 0, the original dynamics becomes

θ¨i =−α ˙θⁱ+ B(ki− hki) + λ

N

X

j=1

Aijsin (θj− θⁱ), (321) where B is a proportionality constant that weights the influence of the local structure on the natural frequencies. WhenN → ∞, and in uncorrelated net-works, after the transformation via replacing the coupling term by the imaginary

term of the continuum-limit version ofr (Eq. 175), Eq. (321) becomes

θ =¨ −α ˙θ + B(k − hki) + kλr sin(ψ − θ), (322) where the subscript_i is dropped in the continuum limit.

In the mean-field version (322), each oscillator appears to be uncoupled from the others but interacts through the mean-field properties(r, ψ) and the phase θ is pulled toward ψ by the coupling strength kλr. Natural frequencies are proportional to degrees, and since the degree distribution is not necessary sym-metric,ψ cannot be set as a constant, but rather oscillates periodically. Here, we assume that r is at the steady state, otherwise, complex phenomena could occur, e.g. secondary synchronization [300, 307]. To derive sufficient condi-tions for synchronization, for convenience, a new rotating reference is defined as φ = θ− ψ. Substituting this into Eq. (322) yields

φ =¨ −α ˙φ + B[k − hki − C(λr)] − kλr sin φ, (323) whereC(λr) ≡ ( ¨ψ + α ˙ψ)/B. In this case, each oscillator can be treated sepa-rately and behaves independently, i.e. either synchronizes to the mean-field or runs periodically with frequency given by Eq. (300), which further depends on the parameter combination that includes the dissipation coefficientα, the new natural frequencyB[k− hki − C(λr)] and the new coupling strength kλr.

Provided that nodes with degree within the range[k1, k2] are synchronized, i.e. ˙φ = 0 and ¨φ = 0, their phases are k-dependent with φ = arcsinB(k−hki−C(λr)) [k1, k2]. After substituting the density function into the definition of the order

parameter (see Sec. 2), the locked order parameterrlockfollows rlock= 1 and its real term becomes

rlock= 1 where kmin denotes the minimal degree and kmax the maximal degree. These drifting nodes rotate with the period ˆT and the frequency ˆω = ^2π_Tˆ in the sta-tionary state. As the density ρdrift(φ, t|k) is proportional to | ˙φ|⁻¹ [300, 301]

and H ρdrift(φ|k)dφ = RT^ˆ

0 ρdrift(φ|k) ˙φdt = 1, we get ρ^drift(φ|k) = ˆT⁻¹| ˙φ|⁻¹ =

ˆ ω

2π| ˙φ|⁻¹. Therefore, the drifting order parameterrdrift becomes rdrift= 1

= 20

Figure 37: Parameter space of one-node model for the increasing coupling strength (a) and for the decreasing coupling strength (b). The red (dark) region indicates the existence of the stable fixed point (the stable limit cycle). Adapted with permission from [315]. Copyrighted by the American Physical Society.

As nodes with negative (positive) natural frequency oscillate over (under) the locked group, one can assume that ˙φ < 0 for k ∈ [k^min, k1] and ˙φ > 0 for k∈ [k², kmax] without loss of generality. A perturbation approximation of the self-consistent equations enables us to get a series expression of the periodic solutionφ(t) using the Poincare-Lindstead method and approximate cos(φ(t)) using Bessel functions [301]. After performing some manipulations on Eq. (326) motivated by [300], one gets the final solution of the real part ofrdriftas follows

rdrift= − The self-consistent equation of r sums the contribution rlock (325) from oscil-lators locked to the mean-field and the contribution rdrift (327) from the rest, i.e. r = rlock+ rdrift. To solve this self-consistent equation, three parameters remain to be solved: constant C and the range of the degree of synchronized nodes [k1, k2]. Considering the complex order parameter summing Eqs. (324) and (326) and following a similar procedure to expressRT^ˆ

0 cos φ(t)dt [301] for the integralRT^ˆ

0 sin φ(t)dt in its imaginary term, we yield the self-consistent equation

0 = 1

In order to determine these quantities, for notational simplicity, we setβ ≡ α/√

kλr and I ≡ B(k − hki − C(λr))/(kλr). As depicted in panel (a) of Fig. 37, initially all nodes are in the region of the stable limit cycle with increasingλ until the onset of synchronizationλ^I_c. At λ^I_c, the homoclinic bifurcation occurs and

0

Figure 38: Analytical (in blue) and numerical (in red) results of the order parameterr with increasing and decreasing strength (a) andC(λr) with increasing coupling strength (b) for synchronization diagrams. Adapted with permission from [244]. Copyrighted by the American Physical Society.

nodes within the synchronization boundary start synchronizing to the mean-field. For small value ofβ, [k^I₁, k₂^I] must fulfil two conditions: |B(k−hki−C(λr))|

kλr ≤

1 and |B(k−hki−C(λr))|

kλr ≤ _π^√4α_kλr.

For decreasingλ, nodes start from the phase-locked synchronous state (panel (b) in Fig. 37). With decreasing in λ, nodes reach the asychronous state at λ^D_c, at which a saddle-node bifurcation occurs. Therefore, the phase-locked oscillators satisfy|B(k−hki−C(λr))|

kλr ≤ 1 and the synchronization boundary follows [k^D1, k2^D] ≡ h_hki+C(λr)

1+^λr_B ,^hki+C(λr)₁₋λr B

i. With the above synchronization boundary [k1, k2], the self-consistent equation of the order parameter can be obtained as a sum of Eqs. 325 and 327 as a function of C, k1 and k2. Additionally, the imaginary term of the complex order parameter (328) should then be used to obtain the dependenceC = C(λr). The comparison between analytical results of the order parameter and simulations are shown in Fig. 38 (a).

To further uncover the first-order phase transition [Fig. 38 (a)], the average frequencyhωikof nodes with the same degreek (here called cluster) is visualized as a function of the coupling strengthλ, and its calculation follows

hωik = X

[i|k_i=k]

h ˙θⁱi^t/(N P (k)), (329)

whereh ˙θⁱi^t =Rt+T

t ˙θi(τ )dt/T . Unlike explosive synchronization in [30] and as discussed in Sec. 5 (see Fig. 19), where all nodes synchronize abruptly at the same coupling, a new phenomenon was found. Oscillators join the synchronous component grouped into clusters of nodes with the same degree, where small degree nodes form the synchronous component simultaneously, whereas high de-gree nodes synchronize successively (see Fig. 38). This phenomenon was termed cluster explosive synchronization [244, 315].