Strong coupling for Kuramoto model to reach synchronization We know the Kuramoto model: 
$$\dot{\theta}_i = \omega_i - \frac{K}{n}\sum_{j=1}^n \sin(\theta_i-\theta_j)$$  
where $\omega$ is the angular frequency and $K$ is the coupling. $i=1,\ldots, n $ represents the $n$ nodes of a network.   
I just want to ask a fundamental question here:
 Why is it more difficult for a weakly coupled network to exhibit coherent behavior (synchrony), whereas a strongly coupled network is amenable to synchronization? Does it come from the point of view of the Lyapunov equation or can we just see this from the Kuramoto dynamics?
 A: First of all, the coupling of the Kuramoto model can be roughly understood as a diffusive coupling for phases: If the phases of two coupled oscillators are different, they are attracted to each other (except if the phase difference is exactly $π$). In particular, for phase differences smaller than $\frac{π}{2}$, the attraction raises with the difference (both in absolute values).
This already suffices for an intuitive understanding of synchronisation phenomena: The higher the coupling strength, the stronger the mechanisms that aligns phases, and hence the easier it is for synchrony to emerge.
For a more detailed understanding, let’s first deduce the behaviour of two extreme cases:


*

*If the oscillators have varying eigenfrequencies ($ω_i$) and are not coupled at all ($K=0$), they cannot possibly exhibit synchronous behaviour (as they do not interact), even if you set their initial phases to be identical.

*If the oscillators have identical eigenfrequencies ($ω_1=ω_2=…=ω$), the completely synchronised state is a solution of the differential equation: The coupling terms become zero and thus every oscillator can happily oscillate with its own eigenfrequency ($\dot{θ}_i = ω$). Moreover this state is stable: A slight variation of the phases will be “levelled out” by the diffusive coupling. Note that this may not be the only solution, e.g., you could also have two groups of oscillators with phases being identical within each group, but shifted by $π$ between the groups.
Now, the interesting scenarios lie somewhere in between the two, so let’s approach them by slightly modifying the above scenarios:


*

*If the oscillators have varying eigenfrequencies and the coupling is small, the eigenfrequencies will still dominate the dynamics. To illustrate this, consider two oscillators with eigenfrequencies $ω_1$ and $ω_2$. To obtain some sort of synchrony, the coupling terms must on average cancel the effect of the difference $|ω_2-ω_1|$. If the coupling strength $K$ is too small, this is not possible. Hence weakly coupled oscillators cannot synchronise.

*If the oscillators have similar eigenfrequencies and have similar phases (i.e., they are in synchrony for an instant), the coupling term can suffice to correct small differences in the eigenfrequencies and it is designed to do exactly this (see the first paragraph). Thus for a sufficiently strong coupling, the synchronous state will be stable. Note that this cannot complete synchrony as there has to be some interaction, which can only result from a difference of the oscillators. Instead you have lag synchrony: The oscillators with lower eigenfrequencies will be (consistently) behind the others.
Finally going to the general case, synchrony happens iff the coupling term is large enough to dominate the difference of eigenfrequencies (and the oscillators are not anti-phase). It’s hard to say when exactly this happens, but it is clear that larger coupling is beneficial for this.
For a good starting point into the vast literature, see From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, in particular Section 3.
