Consider any network of n identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least μ(n1) other oscillators. There is a critical value of the connectivity, μc, such that whenever μ>μc, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when μ<μc, there are networks with other stable states. The precise value of the critical connectivity remains unknown, but it has been conjectured to be μc=0.75. In 2020, Lu and Steinerberger proved that μc0.7889, and Yoneda, Tatsukawa, and Teramae proved in 2021 that μc>0.6838. This paper proves that μc0.75 and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.

The Kuramoto model of coupled oscillators offers an ideal playground for exploring how the structure of a network affects the dynamics it can display. In that spirit, we consider the simplest version of the model, in which each oscillator is assumed to have the same intrinsic frequency. The oscillators are coupled with unit strength along the edges of an undirected network that is connected but otherwise has an arbitrary topology. Even in this minimalist setup, mysteries abound. Specifically, what level of connectivity ensures that the system will almost always settle into a state of perfect synchrony, with all the oscillators running in phase? Here, we prove that if every oscillator is connected to at least 75% of the others, the system globally synchronizes, regardless of the details of its wiring diagram. Our proof uses trigonometric identities and optimization arguments to derive inequalities (involving the first two moments of the oscillator phase distribution) that must hold for any stable phase-locked state. If it is possible to improve the bound μc0.75 further, then one must contend with a remarkable sequence of networks that lie on the so-called razor’s edge of stability. The networks in this sequence have connectivities that approach the upper bound of 0.75 from below, yet have twisted phase-locked states whose eigenvalues are either negative or zero; as such, linear analysis says nothing either way about their stability. Thus, although our theorem brings us closer to pinpointing the critical connectivity μc, a quantity that has recently attracted intense theoretical interest, it does not settle the question. There still remains a large gap between the best known upper and lower bounds on μc, and we are starting to suspect that this gap cannot be closed by linear stability analysis on its own. Indeed, if μc turns out to be less than 0.75, then nonlinear stability analysis will be needed to prove that this is the case.

Coupled oscillators often synchronize spontaneously. This phenomenon appears throughout nature and technology, from flashing fireflies and neural populations to arrays of Josephson junctions and nanoelectromechanical oscillators.1–7 

Among the many questions raised by synchronization, one of the most natural questions is how network topology can either promote or prevent global synchronization.8–24 We say that a network of oscillators globally synchronizes if it converges to a state for which all the oscillators are in phase, starting from all initial conditions except a set of measure zero. Otherwise, we say that the network supports a pattern. One expects that dense networks are more inclined to be globally synchronizing, and sparser ones might support waves or more exotic patterns. However, trusting intuition here can be dangerous. For example, a network in which any two oscillators are connected by exactly one path (i.e., a tree) can be quite sparse; yet, all trees of identical Kuramoto oscillators are known to be globally synchronizing; conversely, there are dense Kuramoto networks that nevertheless support a pattern.9,15,23,24

In this paper, we study the homogeneous Kuramoto model introduced by Taylor,11 in which each oscillator has the same frequency ω. By going into a rotating frame at this frequency, we can set ω=0 without loss of generality. Then, phase-locked states in the original frame correspond to equilibrium states in the rotating frame. Therefore, to explore the question that concerns us, it suffices to study the following simplified system of identical Kuramoto oscillators:

dθjdt=k=1nAjksin(θkθj),1jn.
(1)

Here, θj(t) is the phase of oscillator j (in the rotating frame). The network’s topology is encoded in the adjacency matrix A, with entries Ajk=Akj=1 if oscillator j is coupled to oscillator k and Ajk=Akj=0 otherwise. It is customary to assume that the network has no self-loops so that oscillator i is not connected to itself; i.e., Aii=0. Thus, all interactions are assumed to be symmetric, equally attractive, and of unit strength. The adjacency matrix A is symmetric; therefore, (1) is a gradient system.8,9 Thus, all the attractors of (1) are equilibrium points, which means that we do not need to concern ourselves with the possibility of more complicated invariant sets such as limit cycles, tori, or strange attractors.

Taylor11 proved that the system (1) globally synchronizes if each oscillator is coupled to at least 93.95% of the others. This result started a flurry of research into finding the critical connectivity to guarantee global synchrony. To make this notion precise, we define the connectivityμ of a network of size n as the minimum degree of the nodes in the network, divided by n1, the total number of other nodes. From here, we define the critical connectivityμc as the smallest value of μ such that any network of n oscillators is globally synchronizing. For any network G, if μμc, the network is guaranteed to be globally synchronizing. Otherwise, there exists a network with μ<μc that supports a pattern. In these terms, Taylor’s result is μc0.9395.

The remarkable thing about the critical connectivity is that the wiring of the network can be arbitrary. Recently, Ling et al.21 strengthened Taylor’s result to show that μc0.7929, and Lu and Steinerberger22 further refined the argument to show μc0.7889. On the other hand, some very large dense networks with μ>0.6838 support a pattern.24 Thus, until the present paper, the best bounds were 0.6838<μc0.7889. In this paper, we improve the upper bound to μc0.75.

Before turning to our proof that μc0.75, we comment that any attempt to show μc<0.75 with linear analysis (in the style of our argument) is most likely futile. In particular, the argument below cannot be refined to get a better upper bound on μc. We know this because a sequence of networks exists with connectivity tending to 75% that lie on the razor’s edge of stability.23 Linear analysis alone cannot determine whether these networks are globally synchronizing or support a pattern; the difficulty is that the associated Jacobian of certain equilibrium states—the so-called twisted states—has multiple zero eigenvalues.23 Unfortunately, long-time dynamical simulations reveal that these networks are most likely globally synchronizing, but just barely; they seem to avoid supporting a pattern by a whisker. Any potential argument that shows that μc<0.75 must contend with this sequence of graphs.

It is standard to assume that the nodes of the network in (1) do not have self-loops. However, the absence of self-loops does cause a few peculiarities, which can be avoided if one assumes that each oscillator is connected to itself. The dynamical system in (1) is oblivious to self-loops as when j=k, we have sin(θkθj)=0. Thus, it is only convention to take Aii=0, and the dynamics does not change when one adds self-loops; i.e., Aii=1. However, self-loops do change how one measures the connectivity of a network as there are now n possible neighbors. When we add self-loops to every node of a network, the connectivity jumps up from μ to

μ~=μ(n1)+1n,
(2)

while the evolution of θ1,,θn over time is left unaltered.

There is a process known as twinning that is closely related and shows that the value of the critical connectivity is not affected by the presence or absence of self-loops. Given any graph G (without self-loops) and the complete graph Kτ on τ nodes for any integer τ, Canale and Monzón15 showed that the lexicographic product G[Kτ] has the same synchronizing behavior. That is, G is globally synchronizing if and only if G[Kτ] is. Also, G supports a pattern if and only if G[Kτ] does. The graph G[Kτ] is formed by replacing each node in G by a clique of size τ, and the nodes in different cliques are connected just like the parent nodes they replaced. The twinned graph G=G[Kτ] does not have self-loops and is denser than G. We find that μ=(τ1+τμ(n1))/(nτ1)>μ for any τ>1, where μ is the connectivity of G. Thus, any graph G with connectivity μ can be used as an initial seed to construct a sequence of denser graphs with a limiting edge density of ((n1)μ+1)/n, where each graph in the sequence has the same synchronizing behavior as G itself. The limiting edge density of this sequence is μ~: the same connectivity one can achieve by adding self-loops to all the nodes of G.

For this reason, the expressions we derive below simplify somewhat when we work with the parameter μ~ in (2) instead of μ. Our argument below shows that any network with μ~>0.75 is guaranteed to be globally synchronizing (see Theorem 5). Using (2), this means that any network of size n with connectivity μ>(0.75n1)/(n1) is globally synchronizing. Since μ(n1) must be an integer (as it is the minimum degree of a node), we know that the following connectivity is sufficient for global synchrony:

μ>1n13n41.
(3)

In Fig. 1, we plot the lower bound in (3) together with the connectivity of the densest known regular networks that support a pattern for 5n50. The bound for μ in (3) tends to 0.75, from below, and hence implies that μc0.75.

For the rest of this paper, we consider a network of n identical oscillators with self-loops that has connectivity μ~. If we say that θ is a stable equilibrium, then we mean that θ is a stable state with respect to the dynamical system in (1) for that network.

An important quantity in the study of Kuramoto oscillators is the so-called complex order parameter, ρ1. Its magnitude 0|ρ1|1 measures the overall level of synchrony of the oscillators, and its argument measures their average phase.3,25 In geometrical terms, the complex order parameter corresponds to the centroid of the oscillators’ positions on the unit circle, regarded as a subset of the complex plane.

In the analysis below, we find it useful to look at higher-order moments ρm of the oscillator distribution as well.14,26–28 These higher-order moments are sometimes called Daido order parameters. Daido26 introduced them in his analysis of a generalization of the Kuramoto model, in which the usual coupling function sin(θkθj) was replaced by a general periodic function f(θkθj) containing all possible Fourier harmonics. In Daido’s work, and in much of the subsequent work where the higher-order moments ρm came into play, the oscillators were assumed to be coupled all-to-all, corresponding to a complete graph. In what follows, we see that ρm can also be helpful when analyzing networks of arbitrary topology.

For an equilibrium θ=(θ1,,θn), we define the moments

ρm=1njeimθj,m=1,2,.

(From now on, we use the notation j to mean j=1n.) Without loss of generality, we may assume that the complex order parameter ρ1 is real-valued and non-negative. To see this, write ρ1=|ρ1|eiψ for some ψ. Then, θ^=(θ1ψ,,θnψ) is also an equilibrium of (1) with the same stability properties as θ since (1) is invariant under a global shift of all phases by ψ. Therefore, for the rest of this paper, we assume that ψ=0 for any equilibrium state being considered, with 0ρ11. When ρ1=1, the oscillators are all in phase, corresponding to perfect synchrony, whereas when ρ10, the pattern of phases is incoherent.

The higher-order moments ρ2,ρ3, reveal additional information about an equilibrium state. When |ρ2|=1, the oscillators form at most two groups: those in sync, with cos(θjθ1)=1, and those in anti-sync so that cos(θjθ1)=1. Similarly, |ρm|=1 reveals that the oscillators form at most m groups. The only equilibrium for which |ρm|=1 for all m1 is the all-in-phase state.

We are particularly interested in the size of ρ1 and |ρ2| for stable equilibrium states. For an equilibrium to be stable, its first two moments must satisfy an inequality that we now derive. The resulting inequality (and more convenient versions of it that we obtain later) plays a crucial role in our proof.

We begin by deriving a lower bound on ρ1. First, note that |ρm|2 satisfies

|ρm|2=1n2(keimkjeimj)=1n2j,kcos(m(θkθj)).

Since cos2(xy)=12(cos(2(xy))+1), we have

1n2j,kcos2(m(θkθj))=12n2j,k(cos(2m(θkθj))+1)=12(1+|ρ2m|2).
(4)

Ling et al.21 proved (see p. 1893 of their paper) that when θ is a stable equilibrium, then

j,kAjkcos(θkθj)+j,kAjkcos2(θkθj)0;

hence, by (4), we have

j,k(1Ajk)(cos(θkθj)cos2(θkθj))j,k(cos(θkθj)cos2(θkθj))=n22(2ρ12|ρ2|21).
(5)

In other words, for an equilibrium point to be stable, the following lower bound on ρ12 must hold:

ρ121+|ρ2|22+1n2j,k(1Ajk)(cos(θkθj)cos2(θkθj)).
(6)

However, this lower bound on ρ12 is not convenient for our purposes because it also involves the network’s adjacency matrix and the stable equilibrium θ. We adapt this lower bound to a more convenient one in Sec. V. To get there, we need to find a way to replace the dependence on the adjacency matrix and θ with more aggregated quantities, such as ρ1,ρ2, and μ~. That is the goal of Secs. IV and V.

Intuitively, for dense networks, one expects that it is impossible to have a large order parameter ρ1 associated with a stable equilibrium unless it is the all-in-phase state. The idea is that if ρ1 is large, then many of the θj’s must be close to 0, and since the network is also dense, many of these nearly in-phase oscillators must be connected to each other, which should make it easy for the whole network to fall into sync. One of the key steps to make this intuition precise is the following inequality:

Lemma 1
If θ is a stable equilibrium, then for any 1jn, we have
n(1μ~)2ρ12sin2(θj)k(1Ajk)|cos(θkθj)|0.
(7)
(This inequality holds when there are self-loops as Ajj=1. For networks without self-loops, the summation k should be replaced with k=1,kjn.)
Proof.
Select any j such that 1jn. From the fact that θ is an equilibrium, we have kAjksin(θkθj)=0 and hence
k(1Ajk)sin(θkθj)=nρ1sin(θj).
Since 1Ajk0, (1Ajk)sin(θkθj)=1Ajk1Ajksin(θkθj) and k(1Ajk)(n1)(1μ)=n(1μ~), the Cauchy–Schwarz inequality shows that
(k(1Ajk)sin(θkθj))2n(1μ~)k(1Ajk)sin2(θkθj).
(8)
By replacing sin2(θkθj) by 1cos2(θkθj) and using that k(1Ajk)n(1μ~), we find that
n2ρ12sin2(θj)n2(1μ~)2n(1μ~)k(1Ajk)cos2(θkθj).
Moreover, the Cauchy–Schwarz inequality shows that
(k(1Ajk)|cos(θkθj)|)2n(1μ~)k(1Ajk)cos2(θkθj).
Hence, we find that
n2ρ12sin2(θj)n2(1μ~)2(k(1Ajk)|cos(θkθj)|)2,
(9)
and the first inequality in (7) follows by rearranging (9) and taking square roots on both sides. The second inequality in (7) is trivial as each term in k(1Ajk)|cos(θkθj)| is non-negative.

Note that Lemma 1 implies that for all 1jn,

ρ1|sin(θj)|1μ~,
(10)

since by (7) we have (1μ~)2ρ12sin2(θj)0. (An equivalent inequality also appears on p. 1895 in the paper of Ling et al.21) This allows us to conclude that if θ is a stable equilibrium associated with a large ρ1, then θ is the all-in-phase state.

Corollary 2

Suppose that θ is a stable equilibrium such that ρ1>2(1μ~). Then, θ must be the all-in-phase state.

Proof.

By (10), we see that |sin(θj)|(1μ~)/ρ1 for all j. Since ρ1>2(1μ~), we also have that |sin(θj)|<1/2 for all j. Therefore, θ must be the all-in-phase state, from Proposition 5 of the paper by Ling et al.21 

Corollary 2 makes our intuition precise and shows us that the only way ρ1 can be large for a stable equilibrium is if it is the all-in-phase state.

We now set out to show that if μ~>3/4, then ρ1>2(1μ~) for all stable equilibria, which by Corollary 2 guarantees that the dense network is globally synchronizing.

Since |1cos(θ)|2, we find that |cos(x)cos2(x)|=|cos(x)(1cos(x))|2|cos(x)|, and hence, by Lemma 1,

k(1Ajk)(cos(θkθj)cos2(θkθj))2n(1μ~)2ρ12sin2(θj)

for all 1jn. Putting this together with (6) leads to the following lower bound on ρ1:

ρ121+|ρ2|222nj(1μ~)2ρ12sin2(θj).
(11)

This is a more convenient lower bound on ρ12 than (6) because (11) does not depend on the topology of the network through the adjacency matrix. However, we need to go further and remove the dependency on the θj’s.

If we use the fact that ρ12sin2(θj)0, then we find

ρ122(μ~34)+12|ρ2|2.
(12)

The inequality in (12) also implies the one found by Ling et al.21 (see [5.4] in their paper) by taking the trivial lower bound of |ρ2|20. [To see this, replace μ~ by (μ(n1)+1)/n and ρ1 by 1njeiθj.]

To improve the bound obtained by Ling et al.,21 we now seek a non-trivial lower bound on |ρ2|. In particular, any non-trivial bound on |ρ2| can be substituted into (11) to obtain a better lower bound on ρ1.

Lemma 3
Let θ be a stable equilibrium with ρ1>0. Then, for all 0x0min{1,(1μ~)2/ρ12}, the following inequality holds:
|ρ2|a+b(1+|ρ2|22ρ12),
(13)
with
a=1+2x04(1μ~)2ρ12,b=(1μ~)2ρ12x0ρ12.
(14)
Proof.
Since cos(2θj)=12sin2(θj), we have
n|ρ2||Re(nρ2)|=|jcos(2θj)|j(12sin2(θj)).
Looking at (11) for inspiration to derive a final lower bound on |ρ2| that does not involve θj, we seek a bound of the form
12sin2(θj)a+4b(1μ~)2ρ12sin2(θj).
(15)
Since we would like to have (15) hold for any possible θj, if we write x=sin2(θj) for brevity, then we need
12xa+4b(1μ~)2ρ12x
(16)
to hold for all 0xmin{1,(1μ~)2/ρ12}. [Note that we can restrict x by (10) and the expression for x.] A simple calculation shows that for the choice of a and b in (14), the graphs of
f(x)=12x{and}g(x)=a+4b(1μ~)2ρ12x
intersect tangentially at x=x0; i.e., f(x0)=g(x0) and f(x0)=g(x0). Moreover, the concavity of the function x((1μ~)2ρ12x)1/2 guarantees that the inequality in (16) holds for all required values of x. We conclude that
|ρ2|a+4bj(1μ~)2ρ12sin2(θj).
The statement of the lemma follows from (11).

A simple concrete lower bound on |ρ2| from Lemma 3 is obtained by using that fact that |ρ2|20; i.e.,

|ρ2|1+2x04(1μ~)2ρ12+(1μ~)2ρ12x0ρ12(12ρ12),

which holds for all 0x0min{1,(1μ~)2/ρ12}. This lower bound on |ρ2| can now be squared and substituted back into the right-hand side of (13) for an improved lower bound on |ρ2|. This process can be repeated to derive a sequence of successively improved lower bounds on |ρ2|. We avoid this because the expressions in these refined lower bounds quickly become cumbersome, and we are fortunate to find that such refinements are not necessary for our purposes.

It is essential in Lemma 3 that we assume that ρ1>0 as when ρ1=0, one can also have |ρ2|=0. What is surprising to us is that even a very small ρ1>0 can lead to a useful lower bound on |ρ2|. In fact, simplifying and then optimizing (over x0) the inequality in Lemma 3, we can show that |ρ2|1/2 for networks with μ~>3/4.

Lemma 4

Suppose that μ~>3/4 and θ is a stable equilibrium. Then, |ρ2|1/2.

Proof.

First, note that if 12ρ12<0, then we have ρ1>2/2>22(1μ~) as μ~>3/4. Therefore, by Corollary 2, θ must be the all-in-phase state and ρ2=1. Thus, for the remainder of this proof, we assume that 12ρ120.

When 12ρ120, we show that the inequalities (12) and (13) with a and b given in (14) cannot both be satisfied for all 0<x0min{1,(1μ~)2/ρ12} unless |ρ2|1/2. Since |ρ2|20, we have ρ122(μ~3/4)>0 from (12), and |ρ2|a+b(12ρ12) from (13). A simple calculation shows that the value of x0 that optimizes the lower bound |ρ2|a+b(12ρ12), where a and b are in (14), is given by
x0=(1μ~)2ρ12(12ρ12)216ρ12.
Clearly, x0(1μ~)2/ρ12, and we find that x00 because ρ122(μ~3/4). Moreover, x0<1 since 16(1μ~)2(12ρ12)2<16ρ1 for 3/4<μ~1. For this valid choice of x0, we find that
|ρ2|12(1μ~)2ρ12+(12ρ12)28ρ12=12x0.
The statement of the lemma follows by noting that 12x01/2 if and only if ρ14(16(1μ~)21)/4. This last inequality holds as (16(1μ~)21)/4 is negative when μ~>3/4.

Finally, we are ready to prove our main result.

Theorem 5

If μ~>3/4, then the only stable equilibrium is the all-in-phase state.

Proof.

By Lemma 4, we know that |ρ2|1/2. By (12), we find that ρ1212|ρ2|2 for μ~>3/4. Thus, ρ121/8. To conclude the proof, we just need to ensure that this implies that ρ1>2(1μ~) (see Corollary 2). One can easily see that 1/8>2(1μ~)2 when μ~>3/4.

Theorem 5 says that any network with self-loops and connectivity μ~>0.75 must be globally synchronizing. As μ~=(μ(n1)+1)/n, we know that any network of size n without self-loops, and μ>(0.75n1)/(n1) is also guaranteed to be globally synchronizing (see Sec. II). Finally, since (0.75n1)/(n1)0.75 as n from below, any network without self-loops and connectivity μ0.75 cannot support a pattern. We conclude that μc0.75.

This section discusses what our argument can say about the stable equilibria of dense networks whose connectivity is just below three-quarters. We will roughly sketch an argument here without attempting to make it precise.

At first glance, it might seem that we cannot say anything because when μ~0.75, the inequalities in (12) and (13) can both be satisfied with ρ1=0 and |ρ2|=0. However, we can describe the possible stable equilibria when μ~ is just below 0.75. For example, when μ~0.7495, one can see that if both the inequalities in (12) and (13) are satisfied, then either (i) ρ1>0.7065 or (ii) ρ1<0.03166 and |ρ2|<0.04474. [These bounds are computed by searching over (ρ1,ρ2)[0,1]2, optimizing for x0 in Lemma 3, and checking to see when both (12) and (13) are satisfied.]

In case (i), we see that ρ1>2(1μ~)0.35; therefore, by Corollary 2, this situation corresponds to the all-in-phase state. Case (ii) is more interesting as it represents the possibility of a stable pattern. From (5) and our bounds on ρ1 and |ρ2|, we find that

j,k(1Ajk)(cos(θkθj)cos2(θkθj))0.49900n21.9921(1μ~)n2.
(17)

Since cos(θkθj)cos2(θkθj)2 and j,k(1Ajk)=(1μ~)n2, the average contribution in (17) from two oscillators j and k that are not connected is at most 1.9921. This means that in the vast majority of cases where one has Ajk=0, we also have cos(θjθk)1; therefore, oscillators j and k must be nearly anti-synchronized with a phase difference of roughly π between them.

Therefore, there are at least two clusters of oscillators in the network of size 0.249n that are nearly in anti-sync. Inside the clusters, the phases differ by at most 0.146 rad (about 8.4°). The bound for the size of the two clusters comes from 1.9921(1μ~)/2, and the phase spread comes from doubling the smallest positive root of cos(πϕ)cos2(πϕ)=1.9921. The bounds on ρ1 and |ρ2| in (12) and (13) further imply that there are two more clusters of size 0.249n with phases that are shifted by approximately π/2 compared to the other pair of clusters. This is because the first identified pair of clusters makes a significant contribution to |ρ2|, but |ρ2| is tiny. Therefore, the only way for |ρ2| to remain small is if there is another pair of clusters that approximately cancel out the contribution of the first pair. Thus, for networks with μ~>0.7495, the vast majority of the oscillators fall into four clusters with phases that are at ϕ, ϕ+π/2, ϕ+π, and ϕ+3π/2 for some ϕ. In addition, there are at most n/250 rogue oscillators in the network that do not fit into those four clusters.

The upshot of this argument is that any network with connectivity 0.7495<μ~0.75 can either be globally synchronizing or can support a particular pattern of the type we have just described. All our attempts to construct a network in this regime with such a pattern have not been successful so far. We believe that such networks do not support linearly stable patterns of this type (or any other type), but we are currently unable to rule them out rigorously. If no such networks or patterns exist, or if the patterns do exist but are only weakly (nonlinearly) stable, then this would be surprising. Indeed, it would suggest something remarkable: that the gap between the lower and upper bound 0.6838<μc0.75 might not be bridgeable by linear stability analysis alone because of a minefield of patterns on their own razor’s edge of stability.

We thank Lee DeVille for pointing out the connection between self-loops and twinning in the summer of 2020, which we used here to simplify the presentation of our argument. We also thank Federico Fuentes for comments on a draft. This research was supported by the Simons Foundation grant (No. 713557) to M.K., NSF grants (Nos. DMS-1513179 and CCF-1522054) to S.H.S., and NSF grants (Nos. DMS-1818757, DMS-1952757, and DMS-2045646) to A.T.

The data that support the findings of this study are almost all available within the article. Any data that are not obtainable are available from the corresponding author upon reasonable request.

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