Synchronization in multilayer networks: when good links go bad

Igor Belykh, Douglas Carter Jr., and Russell JeterDepartment of Mathematics and Statistics and Neuroscience Institute,

Georgia State University, P.O. Box 4110, Atlanta, Georgia, 30302-410, USA(Dated: August 8, 2018)

Many complex biological and technological systems can be represented by multilayer networkswhere the nodes are coupled via several independent networks. Despite its signi�cance from boththe theoretical and application perspectives, synchronization in multilayer networks and its depen-dence on the network topology remain poorly understood. In this paper, we develop a universalconnection graph-based method which removes a long-standing obstacle to studying synchronizationin dynamical multilayer networks. This method opens up the possibility of explicitly assessing criti-cal multilayer-induced interactions which can hamper network synchronization and reveals striking,counterintuitive e�ects caused by multilayer coupling. It demonstrates that a coupling which isfavorable to synchronization in single-layer networks can reverse its role and destabilize synchro-nization when used in a multilayer network. This property is controlled by the tra�c load on agiven edge when the replacement of a lightly loaded edge in one layer with a coupling from anotherlayer can promote synchronization, but a similar replacement of a highly loaded edge can breaksynchronization, forcing a �good� link go �bad.� This method can be transformative in the highlyactive research �eld of synchronization in multilayer engineering and social networks, especially inregard to hidden e�ects not seen in single-layer networks.

PACS numbers: 05.45.Xt, 87.19.La

I. INTRODUCTION

Complex networks are common models for many sys-tems in physics, biology, engineering, and the social sci-ences [1�3]. Signi�cant attention has been devoted toalgebraic, statistical, and graph theoretical propertiesof networks and their relationship to network dynamics(see a review [4] and references therein). The strongestform of network cooperative dynamics is synchronizationwhich has been shown to play an important role in thefunctioning of a wide spectrum of technological and bio-logical networks [5�14], including adaptive and evolvingnetworks [15�22].

Despite the vast literature to be found on networkdynamics and synchronization, the majority of researchactivities have been focused on oscillators connectedthrough single-layer network (one type of coupling) [23�41]. However, in many realistic biological and engineer-ing systems the units can be coupled via multiple, in-dependent systems and networks. Neurons are typicallyconnected through di�erent types of couplings such asexcitatory, inhibitory, and electrical synapses, each cor-responding to a di�erent circuitry whose interplay a�ectsnetwork function [42, 43]. Pedestrians on a lively bridgeare coupled via several layers of communication, includ-ing people-to-people interactions and a feedback from thebridge that can lead to complex pedestrian-bridge dy-namics [44�47]. In engineering systems, examples of inde-pendent networks include coupled grids of power stationsand communication servers where the failure of nodes inone network may lead to the failure of dependent nodesin another network [48]. Such interconnected networkscan be represented as multiplex or multilayer networks[49, 50] which include multiple systems and layers of

connectivity. Multilayer-induced correlations can havesigni�cant rami�cations for the dynamical processes onnetworks, including the e�ects on the speed of diseasetransmission in social networks [51] and the role of re-dundant interdependencies on the robustness of multi-plex networks to failure [52].Typically, in single-layer networks of continuous time

oscillators, synchronization becomes stable when the cou-pling strength between the oscillators exceeds a thresholdvalue [23, 30]. This threshold depends on the individualoscillator dynamics and the network topology. In thiscontext, a central question is to determine the criticalcoupling strength so that the stability of synchroniza-tion is guaranteed. The master stability function [23] orthe connection graph method [30, 31] are usually usedto answer this question in single-layer networks. Bothmethods reduce the dimensionality of the problem suchthat synchronization in a large, complex network can bepredicted from the dynamics of the individual node andthe network structure.Synchronization in multilayer networks has been stud-

ied in [53�56]; however, its critical properties and ex-plicit dependence on intralayer and interlayer networkstructures remain poorly understood. This is in particu-lar due to the inability of the existing eigenvalue meth-ods, including the master stability function [23] to givedetailed insight into the stability condition of synchro-nization as the eigenvalues, corresponding to connectiongraphs composing a multilayer network, must be calcu-lated via simultaneous diagonalization of two or moreconnectivity matrices. Simultaneous diagonalization oftwo or more matrices is impossible in general, unless thematrices commute [53, 54]. A nice approach based onsimultaneous block diagonalization of two connectivitymatrices was proposed in [54]. This most successful ap-

2

plication of the eigenvalue-based approach allows one toreduce the dimensionality of a large network to a smallernetwork whose synchronization condition can be used toevaluate the stability of synchronization in the large net-work. For some network topologies, this technique yieldsa substantial reduction of the dimensionality; however,this reduction is less signi�cant in general. The reducednetwork typically contains weighted positive and nega-tive connections, including self loops such that the roleof multilayer network topologies and the location of anedge remain di�cult to evaluate.

In this paper, we report our signi�cant progress to-wards removing this long-standing obstacle to study-ing synchronization in multilayer networks. We developa new general stability approach, called the MultilayerConnection Graph method, which does not depend onexplicit knowledge of the spectrum of the connectivitymatrices and can handle multilayer networks with arbi-trary network topologies, which are out of reach for theexisting approaches. An example of a multilayer networkin this study is a network of Lorenz systems where someof the oscillators are coupled through the x variable (�rstlayer), some through the y variable (second layer), andsome through both (interlayer connections). Our Multi-layer Connection Graph method originates from the con-nection graph method [30, 31] for single-layer networks;however, this extension is highly non-trivial and requiresovercoming a number of technically challenging issues.This includes the fact that the oscillators from two x andy layers in the networks of Lorenz systems are connectedthrough the intrinsic, nonlinear equations of the Lorenzsystem. As a result, multilayer networks can have dras-tically di�erent synchronization properties from those ofsingle-layer networks. In particular, our method showsthat an interlayer tra�c load on a link is the crucial quan-tity which can be used to foster or hamper synchroniza-tion in a nonlinear fashion. For example, it demonstratesthat replacing a link with a light interlayer tra�c load bya stronger pairwise converging coupling (a �good� link)via another layer may lower the synchronization thresh-old and improve synchronizability. At the same time,such a replacement of a highly loaded link can make thenetwork unsynchronizable, forcing the pairwise stabiliz-ing �good� link go �bad.�

The layout of this paper is as follows. First, in Sec.II, we present and discuss the network model. In Sec.III, we start with a motivating example of why the re-placement of some links in a multilayer network can im-prove or break network synchronization. And what isan underlying mechanism? In Sec. IV, we formulatethe Multilayer Connection Graph method for predictingsynchronization in multilayer networks. In Secs. V andVI, we show how to apply the general method to speci�cnetwork topologies. In Sec. VII, a brief discussion of theobtained results is given. Finally, the Appendix containsthe complete derivation of the general method. MAT-LAB code for algorithms used for calculating networktra�c loads are given in the Supplement.

II. NETWORK MODEL AND PROBLEM

STATEMENT

We start with a general network of n oscillators withtwo connectivity layers:

dxidt

= F(xi)+

n∑j=1

cijPxj+

n∑j=1

dijLxj , i = 1, ..., n, (1)

where xi = (x1i , ..., xsi ) is the s state vector containing the

coordinates of the i-th oscillator, F : Rs → Rs describesthe oscillators' individual dynamics, cij and dij are thecoupling strengths. C = (cij) and D = (dij) are n × nLaplacian connectivity matrices with zero-row sums andnonnegative o�-diagonal elements cij = cji and dij = dji,respectively [30]. These connectivity matrices C and Dde�ne two di�erent connection layers (also denoted byC and D, with m and l edges, respectively). The in-ner matrices P and L determine which variables couplethe oscillators within the C and D layers, respectively.Without loss of generality, we will be considering the os-cillators of dimension s = 3 and xi = (xi, yi, zi). There-fore, the C graph with the inner matrix P = diag(1, 0, 0)will correspond to the �rst-layer connections via x, whilethe D graph with the inner matrix L = diag(0, 1, 0 ) willindicate the second-layer connections via y. Overall, theoscillators of the network are connected through a com-bination of the two layers (see Fig. 1 for an example of acombined two-layer graph). The graphs are assumed tobe undirected [30]. Oscillators, comprising the network(1), can be periodic or chaotic. As chaotic oscillators aredi�cult to synchronize, they are usually used as test bedexamples for probing the e�ectiveness of a given stabilityapproach. The oscillators used in the numerical veri�ca-tion of our stability method are chaotic Lorenz [57] anddouble scroll oscillators [58].In this paper, we are interested in the stability of

complete synchronization de�ned by the synchronizationmanifold M = {x1 = x2 = ... = xn}. Our main objec-tive is to determine a threshold value for the couplingstrengths required for the global stability of synchroniza-tion. We seek to predict this threshold or the absencethereof in the general network (1) from synchronizationin the simplest two-node network and graph propertiesof the two-layer network structures.It is important to emphasize that only Type I oscilla-

tors [26] are capable of synchronizing globally and retain-ing synchronization for any coupling strengths exceedingthe synchronization threshold. Most known oscillators,including the Lorenz and double scroll oscillators belongto Type I systems. A much more narrow Type II classof oscillators contains x-coupled Rössler systems [23] inwhich synchronization becomes only locally stable andeventually looses its stability with an increase of cou-pling [24]. As a result, we limit our consideration toType I networks in this paper; however, a numerically-assisted extension of our method to Type II networks canbe performed with moderate e�ort and remains a subject

3

of future study.

(a)

(b)

(c)

(d)

FIG. 1. The puzzle: why do �good� links go �bad�? Syn-chronization in six-node networks of Lorenz systems (2). (a)Single-layer network, with all x edges (black). (b) The re-placement of x edge 5-6 with a presumably better converg-ing y coupling (blue) improves synchronization, as expected(see (d)). (c) A similar replacement of x edge 2-3 with a yedge (red) makes synchronization impossible by pushing thethreshold to in�nity (see (d)). (d). Systematic study of thecoupling threshold c∗ as a function of the y edge locationthat replaces an x edge in the original x-coupled network (a).The blue solid line indicates numerically calculated thresh-olds. The black solid line depicts the interlayer tra�c bint forthe respective y edge. Note a signi�cant increase of bint thatcauses the network to become unsynchronizable as predictedby the method. The predicted coupling thresholds (blue dot-ted line) are computed from (3) using the exponential �t inFig. 3 and scaling factors β = 0.18 and γ = 0.7180.

III. A MOTIVATING EXAMPLE AND A

PUZZLE

To illustrate the complexity of assessing the role ofmultilayer connections and their controversial role in fos-

tering or hindering synchronization, we begin with two-layer networks of chaotic Lorenz oscillators, depicted inFig. 1(a-c). For the Lorenz oscillators, the vector equa-tion (1) can be written in a more reader-friendly scalarform:

xi = σ(yi − xi) +n∑j=1

cijxj ,

yi = rxi − yi − xizi +n∑j=1

dijyj ,

zi = −bzi + xiyi, i = 1, ..., n,

(2)

where the connectivity matrix C = (cij) describes thetopology of x connections (black edges in Fig. 1(a-c)),and matrix D = (dij) describes the location of one yedge (blue or red edge). The parameters of the individ-ual Lorenz oscillator are standard: σ = 10, r = 28, andb = 8/3. The strengths of the x and y coupling are homo-geneous (cij = c, dij = d) and varied uniformly (c = d).We are interested in the question of how the replace-

ment of an x edge in the network of Fig. 1(a) with a yedge can a�ect synchronization. To address this ques-tion, we �rst need to understand synchronization prop-erties of two-node single-layer networks of x-coupled andy-coupled Lorenz systems. It is well-known that if thecoupling in a single-layer network (2) with either all x orall y connections exceeds a critical threshold, then syn-chronization becomes stable and persists for any c > c∗and d > d∗, respectively. A rigorous upper bound for thecritical coupling c∗, explicit in parameters of the Lorenzoscillator can be found in [30].Calculated numerically, these coupling thresholds are

for c∗ ≈ 3.81 for the two-node x-coupled network andd∗ ≈ 1.42 for the y-coupled network. As the synchro-nization threshold d∗ is signi�cantly lower, then one couldexpect that replacing an x edge with a presumably betterconverging y coupling improves synchronization. This istrue for the network of Fig. 1(b) when x edge 5-6 is re-placed with a y edge, yielding a minor reduction in thesynchronization threshold from c∗ ≈ 17.94 in the single-layer x-coupled network of Fig. 1(a) to c∗ ≈ 17.74 inthe multilayer network of Fig. 1(b). A surprise comesfrom the network of Fig. 1(c) where the replacement ofx edge with a y edge, that would be expected to improvesynchronization, on the contrary makes the network un-synchronizable (see the coupling threshold jumping toin�nity in Fig. 1(d)).What is the origin of this highly counterintuitive e�ect?

Why do edges in a multilayer network reverse their stabi-lizing roles depending on the edge location whereas theyare well behaved in single-layer networks? As the con-nectivity matrices for x and y coupling in the networksof Fig. 1(b) and Fig. 1(c) do not commute and, there-fore, the predictive power of the master stability functionbased methods [53�55] is severely impaired, this puzzlecalls for an explanation and utlimately motivates the de-velopment of an e�ective, general method for assessingthe stability of synchronization in multilayer networks.In the following, we will develop such a method that

4

identi�es the location of critical interlayer links whichcontrol stable synchronization and reveals its explicit de-pendence on an interlayer tra�c load on a given edge.

IV. MULTILAYER CONNECTION GRAPH

METHOD

In this section, we �rst present the main result of thispaper and formulate the method. We will then demon-strate how to apply the method to speci�c network con-�gurations.Theorem 1 [The method]. Synchronization in themultilayer network (1) is globally stable if for each edgek = 1, ...,m (l)

cij ≡ ck > 1n

{ax · bxk + ωx · bintk + αxk

}dij ≡ dk > 1

n

{ay · byk + ωy · bintk + αyk

},

(3)

where bxk =n∑

j>i; k∈Pij∈C|Pij | is the sum of the lengths

of all chosen paths Pij between any pair of oscillatorsi and j which belong to the x graph C and go througha given x edge. The path length |Pij | is the number ofedges comprising the path. Similarly, byk corresponds tothe sum of the lengths of all chosen paths which belongto the y graph D and go through a given y edge. bintk isthe sum of the lengths of all chosen paths which containx and y edges and belong to the combined, two-layer xygraph and go through a given edge k which may be an x ory edge. Constant ax (ay) is the double coupling strengthsu�cient for synchronization in the network of two x-coupled (y-coupled) oscillators, composing the network.The constants ωx and ωy represent a combination of thedouble coupling strengths for c12 and d12 su�cient forsynchronization in the two-oscillator network with both xand y connections. Finally, the constants αkx and αky arechosen large enough such that they can globally stabilizethe auxiliary stability systems written for the di�erencevariables that correspond to an edge k : Xk = Xij =xj − xi :

for αkx : Xk =

[1∫0

DF(βxj + (1− β)xidβ]Xk+

ωxbintk LXk − (ax + αxk)PXk,

(4)

for αky : Xk =

[1∫0

DF(βxj + (1− β)xidβ]Xk+

ωybintk PXk − (ay + αyk)LXk,

(5)

where the Jacobian DF can be calculated explicitly viathe parameters of the individual oscillator.Proof. The proof closely follows the notations and stepsin the derivation of the connection graph method [30] forsingle-layer networks up to a point where the stability ar-gument becomes drastically di�erent and yields the newterms ωxb

intk and ωyb

intk which play a pivotal role in syn-

chronization of multilayer networks. The complete proof

is given in the Appendix. �Remark 1. In terms of tra�c networks, the graph the-oretical quantities bxk, b

yk and bintk represent the total

lengths of the chosen roads that go through a given edge kwhich can be loosely analogized as a busy street. There-fore, we refer to them as �tra�c� loads. In this view, thequantity bintk is a tra�c load on edge k, caused by in-terlayer travelers. It is important to notice that positiveterms +ωxb

intk LXk in the equation (4) and +ωyb

intk PXk

in (5) play a destabilizing role such that a heavily loadededge with a high bintk can represent a potential problemfor making the systems (4) and (5) stable at all. This ob-servation has dramatic consequences for synchronizationin speci�c multilayer networks described in the followingsections and also is a key to solving the puzzle of Fig. 1.Remark 2. If both x and y connection graphs are con-nected such that all oscillators are coupled via bothgraphs, the stability of synchronization can be simplyassessed by applying the connection graph method [30]for each of the x and y connected graphs and combin-ing the two conditions as follows: cij + dij = ck + dk >1n {ax · b

xk + ay · byk} (cf. the condition (37) in the Ap-

pendix). As a result, one should not expect the e�ectsdue to the multilayer coupling discussed in the motivat-ing example.Remark 3. While the stability criterion (3) is completelyrigorous, the theoretically calculated bounds may givelarge overestimates on the threshold coupling strength.As a result, we suggest to use a semi-analytical approachwhich combines numerically calculated exact bounds ax,ay, ωx, ωy, α

kx, and α

ky with graph theoretical quantities

bx, by, and bint. In this way, this method becomes ane�ective, predictive tool and plays the role of the masterstability function for the general multilayer network (1)where a synchronization threshold or the absence thereofcan be deduced from the properties of the individual os-cillators comprising the network and the network topolo-gies of the connection layers. In this numerically-assistedapplication of our method, the conservative estimate ontra�c loads bx, by, and bint, that come from the Cauchy-Schwartz inequality (see the Appendix), can be replacedby βbx, βby, and γbint, where the scaling factors β and γare introduced to provide tighter numerical bounds.

Computing the stability conditions (3) and its con-stants is a complicated, multi-step process which involvescalculations of (i) two stability diagrams (for a pair ofωx, ωy and αxk, α

y) and (ii) graph theoretical quantitiesbx, by, and bint. In the following section, we will walk thereader through a step-by-step computation of the sta-bility conditions (3) for speci�c multilayer networks andillustrate their implications for the stability of synchro-nization.

5

FIG. 2. Stability of synchronization in a two-node networkof xy-coupled Lorenz systems as a function of the x coupling,ωx and the y coupling, ωy. Yellow depicts instability (non-zero synchronization error) and purple depicts stability (zerosynchronization error). The white dot indicates the pair (ωx,ωy) used in the predictions shown in Fig. 1(d) and Fig. 4(c).

V. APPLICATION OF THE METHOD:

SOLVING THE PUZZLE

Armed with the predictive method, we �rst return tothe puzzle to better understand synchronization prop-erties of the six-node networks of Lorenz oscillators (2)from Fig. 1. Below we give a step-by-step recipe of howour numerically-assisted method can be applied to thesenetworks.Step 1: Calculate ax, ay, ωx, and ωy.Consider the simplest two-node network (2) with both xand y coupling: c12 = c21 = c and d12 = d21 = d. Deter-mine threshold coupling strengths that guarantee stablesynchronization in (i) the two-node x-coupled networkwith c∗ = ax/2 and d = 0; (ii) the two-node y-couplednetwork with d∗ = ay/2 and c = 0; and (iii) the two-nodexy network with c∗ = ωx and d∗ = ωy.

The numerically calculated thresholds for the x-coupled and y-coupled two-node network (2) reportedin Sec. III are c∗ ≈ 7.62/2 and d∗ ≈ 2.84/2, respec-tively. This yields the double coupling strength constantsax = 7.62 and ax = 2.84 to be used in (3).

Note that di�erent combinations of c = ωx and d = ωyin the xy-coupled network yield stable synchronization(see Fig. 2). Without loss of generality, we choosec = ωx = 5 and d = ωy = 0.5 as a point on the sta-bility boundary in Fig. 2 and keep these values �xed forthe prediction of the synchronization threshold in largertwo-layer networks (2) with arbitrary topologies. Thischoice of the pair (ωx, ωy) is somewhat arbitrary; how-ever, it dictates the choice of constants in the stabilitydiagrams for αxk and αyk (Fig. 3). It is often a good ideato choose ωx and ωy such that both are non-zero and liesomewhere in the middle range of (ωx, ωy) to balance outthe stability conditions (4) and (5).

Step 2: Calculate the stability diagrams to determine αxkand αyk.The auxiliary stability systems (4) and (5) for the scalardi�erence variables of coupled Lorenz oscillators (2)Xk = Xij = xj−xi, Yk = Yij = yj−yi, Zk = Zij = zj−zican be rewritten in a more convenient form:

Xk = σ(Yk −Xk)− (ax + αxk)Xk,

Yk =(r − U (z)

k

)Xk − Yk − U (x)

k Zk + ωxbintk Yk

Zk = U(y)k Xk + U

(x)k Yk − bZk,

(6)

Xk = σ(Yk −Xk) + ωybintk Xk

Yk =(r − U (z)

k

)Xk − Yk − U (x)

k Zk − (ay + αyk)Yk,

Zk = U(y)k Xk + U

(x)k Yk − bZk,

(7)

where U(ξ)ij = (ξi + ξj)/2 for ξ = x, y, z are the corre-

sponding sum variables. As in (3), the auxiliary sta-bility system (6) corresponds to the di�erences betweenthe nodes connected by an x edge, and (7) correspondsto the di�erences between nodes coupled via a y edge.If the connection layers overlap and the same nodes areconnected through both x and y edges, then the auxiliarysystems (6) and (7) should be applied to the correspond-ing x and y edges independently. Their contributions willthen appear in the general stability conditions (3) for ckand dk for the same edge k.The auxiliary systems (6) and (7) are used to yield

global stability conditions for the analytical criterion (3),where bounds on the sum di�erences U (x), U (y), and U (z)

can be placed similar to the analysis of synchronizationin single-layer networks [30]. A rigorous analysis of globalstability of (6) and (7) will be performed in a more tech-nical publication.Here, we take a more practical route towards devel-

oping a numerically-assisted approach which simpli�esthe evaluation of the stability of (6) and (7). We lin-earize both systems (6) and (7) by making the di�erencesXk, Yk, and Zk in�nitesimal and replacing U (x) = x(t),U (y) = y(t), and U (z) = z(t), where (x(t), y(t), z(t)) is thesynchronous solution governed by the uncoupled Lorenzoscillator. The linearized stability systems (6) and (7)take the form:

Xk = σ(Yk −Xk)− (ax + αxk)Xk,

Yk = (r − z(t))Xk − Yk − x(t)Zk + ωxbintk Yk

Zk = y(t)Xk + x(t)Yk − bZk,(8)

Xk = σ(Yk −Xk) + ωybintk Xk

Yk = (r − z(t))Xk − Yk − x(t)Zk − (ay + αyk)Yk,

Zk = y(t)Xk + x(t)Yk − bZk.(9)

Notice that αxk [αyk] must be large enough to stabilize (8)[(9)] in the presence of ωxb

intk [ωyb

intk ]. While ax, ay, ωx,

and ωy are chosen and �xed in Step 1, the tra�c loadbintk on a given edge k (which will be determined in Step

6

(a)

(b)

FIG. 3. Stability of the auxiliary systems (6) and (7) for cou-pled Lorenz oscillators. Yellow depicts instability of the originwhile purple indicates its stability. The dependence of the sta-bilizing term, αx, on b

intk , ωx, and βx is estimated by the ex-

ponential function 0.4645 exp(2.408βxωxb

intk

)(dashed curve),

and is used to predict synchronization thresholds in networksof Fig. 1 and Fig. 4.

3) controls the choice of αxk and αyk. Thus, if the edgeis highly loaded, then the contribution of the positiveterm +ωxb

intk Yk in the Yk-equation of system (8) can-

not always be compensated by increasing −αxkXk in theXk-equation. Typically, this happens when the positiveterm exceeds the proper negative linear terms such as−Yk (technically, through a combination of terms in theRouth-Hurwitz criterion). Therefore, the auxiliary sys-tem (8) can become unstable, independently of how largethe stabilization coe�cient αxk is. The same argument re-lates to the destabilization of the auxiliary system (9) viathe positive term +ωyb

intk Xk.

The complex relationship between these terms in re-gard to stabilizing (8) and (9) is shown in Fig. 3. Noticethe coe�cient β on the βωxb

intk and [βωyb

intk ] axes. Be-

cause of the Cauchy-Schwarz inequality used in the proof,bintk provides an overestimate for the terms added to theauxiliary system and the scaling factor β is added to com-

pensate for this overestimate. The diagrams of Fig. 3con�rm the existence of threshold values for βωxb

intk and

[βωybintk ] such that even in�nitely large values of αx and

αy cannot compensate for the caused instability and sta-bilize systems (8) and (9).It is important to emphasize that the stability di-

agrams of Fig. 3, which later will be used for pre-dicting thresholds in large networks, need to be calcu-lated through simulation of two three-dimensional sys-tems (8) and (9) driven by the synchronous solution(x(t), y(t), z(t)). Here, the terms βωxb

intk and βωyb

intk are

treated as single, aggregated control parameters. There-fore, the threshold value for αxk [αyk] required to stabilizethe auxiliary system (8) [(9)] for a given edge k with bintkcan simply be read o� from Fig. 3. To better quantifythis dependence to be used in predicting synchronizationthresholds in networks of Lorenz oscillators, we approxi-mate the stability boundary in Fig. 3(a) by the exponen-tial function

αx = 0.4645 exp(2.408βωxb

intk

). (10)

The stability diagrams of Fig. 3 along with Fig. 2 ac-count for the role of the individual oscillators composingthe networks and the way these oscillators are coupled(through x and y coupling) in the stability of synchro-nization. These diagrams represent an analog of the mas-ter stability function in single-layer networks [23] andhelp in solving, once and for all, the question of stabil-ity for synchronization in two-layer networks involvingthe Lorenz oscillator through the criterion (3), where therole of multilayer network topologies is assessed via thecalculation of pure graph theoretical quantities as shownin the next step.Step 3: Calculate tra�c loads bxk, b

yk, and b

intk .

This calculation is similar to that of the connectiongraph method for single-layer networks [30], except thatthe tra�c load should be partitioned into three groups:intralayer tra�c loads bxk and byk within the x and ylayer, respectively, and interlayer tra�c load bintk be-tween the layers. To do so, we �rst choose a set of paths{|Pij | i, j = 1, ..., n, j > i}, one for each pair of ver-tices i, j, and determine their lengths |(Pij |, the numberof edges in each Pij . Then, for each edge k of the x (y)layer graph, we calculate the sum bxk (byk) of the lengthsof all Pij that are composed of only x (y) edges and passthrough k.We repeat the same procedure to calculate thesum bintk of the lengths of all Pij that contain both x andy edges and pass through k. These constants depend onthe choice of the paths Pij . Usually, one uses the short-est path from vertex i to vertex j. Sometimes, however,a di�erent choice of paths can lead to lower bounds [34].We use the six-node multilayer network of Fig. 1(b) as

an example for calculating bxk, byk, and b

intk . To compute

all of the paths that pass through a given edge, it isrecommended that the reader algorithmically �nds theshortest path between every pair of oscillators, and takenote of the paths that go through edge k and di�erentiatethe paths that entirely belong to only the x or y layers

7

and the ones that contain a combination of x and y edges.As a result, we can �nd each edge's tra�c loads as follows

bx12 = |P12|+ |P13|+ |P14|+ |P15|+ |P16| =1 + 2 + 3 + 3 + 4 = 13,

bx23 = |P13|+ |P14|+ |P15|+ |P16|+ |P23|+|P24|+ |P25|+ |P26| = 20,

bx34 = |P14|+ |P16|+ |P24|+ |P26|+ |P34| = 15bx35 = |P15|+ |P25|+ |P35| = 6, bx46 = |P16|+ |P46| = 5,by56 = |P56| = 1, bint12 = 0, bint23 = 0, bint34 = 0,bint35 = |P36| = 2, bint46 = |P45| = 2,bint56 = |P36|+ |P45| = 4.

(11)Note that the maximum interlayer tra�c load on thisnetwork is fairly low and due to our choice of paths isbint56 = 4 although it could have also been minimized tozero, provided that all paths to node 6 bypass edge 56.At the same time, the interlayer tra�c load in the net-

work of Fig. 1(c) is signi�cantly higher since there areno alternatives to go around the �bottle neck� edge 2-3when traveling from nodes 1 and 2 to nodes 4, 5, and 6.For the same choice of shortest paths, we get bint23 = 19.The remaining bxk, b

yk for the network of Fig. 1(c) can be

calculated similarly to (11).Step 4: Putting pieces together to solve the puzzle. Giventhe stability diagram of Fig. 3 with abrupt threshold de-pendences of αxk and αyk on increasing interlayer tra�cload bxk, the e�ect of synchrony breaking when a highlyloaded x edge is replaced with a better pairwise stabi-lizing y (see Sec. III ) is no longer a puzzle and directlyfollows from the application of our stability method. Ac-tually, in a historical retrospective, we �rst developedthe general method that revealed this and other highlycounterintuitive e�ects due to the multilayer structureand then constructed the network examples. To makethe presentation more appealing before it becomes tootechnical, we have decided to put forward the motivat-ing example. As our exhaustive study of various net-work con�gurations suggests, we hypothesize that six-node networks of Fig. 1 are minimum size networks ofLorenz oscillators that exhibit the synchrony breakingphenomenon.To test the predictive power of our approach with the

constants identi�ed in Steps 1-3, we perform a systematicstudy of how one edge replacement, in which we replaceonly one x edge in the single-layer, x-coupled network ofFig. 1(a) with a y edge, a�ects synchronization. The edgereplacement is performed in the order of the increasinginterlayer tra�c load on this edge, bintk . After comput-ing the coupling threshold required to synchronize thenew network, this edge reverts back to being an x edge.This results in multiple networks of �ve x edges and oney edge. The two multilayer networks of Fig. 1(b) andFig. 1(c) with the drastically di�erent synchronizationproperties are two instances of this replacement process.Fig. 1(d) presents the actual synchronization thresholdvalues (blue solid line), the interlayer tra�c loads bint

(black line) calculated similarly to (11), and the thresh-old values predicted by the numerically-assisted criterion

(3) with constants ax = 7.20, ay = 2.63, ωx = 5.00 andωy = 0.50 chosen above. The constants αxk and αyk areread o� from the diagrams of Fig. 3(a) and Fig. 3(b),respectively. As the stability system (8) is much moresensitive to the changes in bintk than (9) (cf. the onset ofinstability in Figs. 3(a-b)), the threshold values for cij inthe criterion (3) for the x layer largely dominate over dij .Thus, since the synchronization threshold for the entirenetwork (2) with uniform coupling c = d is de�ned bythe maximum of the thresholds cij or dij for each edgeof the multilayer graph, the maximum threshold valuespredicted by the method and depicted in Fig. 1(d) arethe ones corresponding to x edges with coupling c. Thesethreshold values are calculated using the numerically as-sisted modi�cation of (3)

c > maxk

{ck =

1

n

[γax · bxk + βωx · bintk + αxk(βωxb

intk )]}

,

(12)where αxk is de�ned by the stability diagram of Fig. 3(a)via the approximating function (10) for each edge k, andax, ωx, and b

intk are determined in Steps 1-3. Notice the

presence of an additional scaling factor γ which is chosento compensate for the conservative nature of the connec-tion graph stability method when the network is entirelycoupled through the x variable. γ scales down the termaxn to match the coupling needed to synchronize the six-node network of Fig. 1(a) with only x edges. The scal-ing factor β is then chosen for the network of Fig. 1(b)with the lowest bintk = 4 to match the actual synchro-nization threshold and then kept constant for predictingthe thresholds in the other six-node networks with onereplaced x edge. Figure 1(d) shows that the predictedthresholds are fairly close to the actual ones, and the cri-terion (12) correctly predicts an increase or decrease ofthe coupling threshold for each six-node network and ul-timately predicts synchrony break for the network withthe replaced x edge 2-3.As Fig. 1(d) indicates that when lightly loaded edges

(edges with fewer chosen paths passing through them)are replaced, the e�ect on the synchronization stabilityis fairly small. As discussed in the description of themotivating example, the replacement of x edge 5-6 witha y edge improves synchronization by slightly loweringthe synchronization threshold. According to our stabil-ity criterion (3), this happens due to a slight decreasein the tra�c load on the bottleneck edge bx23 (see (11)),compared to the original network of Fig. 1(a) with all xedges where one additional path P16 goes through edge2-3. As a result, it decreases the contribution of the dom-inating term axb

xk in (3). At the same time, the contribu-

tion of the other factors ωxbintk and αxk remain insignif-

icant, especially due to the fact that αxk still lies on a�at part of the approximating curve (10) before this ex-ponential curve takes o� at larger values of bintk . On theother hand, such a replacement of the bottleneck node2-3 in the network of Fig. 1(c) signi�cantly increases theintralayer tra�c load bintk , requiring in�nitely large αx23to stabilize the stability system (8) and causing synchro-

8

nization to break.

VI. LARGER NETWORKS

To demonstrate that similar synchrony breakdownphenomena occur in larger networks and can be e�ec-tively predicted by our method, we consider a 20-nodenetwork of Lorenz (and then double-scroll) oscillators de-scribed in Fig. 4(a). The network is initially coupled en-tirely through the x variable. To test our prediction thatreplacing edges with a high tra�c load can make the net-work unsynchronizable, we index the edges according totheir bxk. Edges similar to edge 10-12 have very few pathsthat pass through them, and subsequently have a low bxk(and in turn, bintk , shown as the black curve in Fig. 4(c)).We successively replace x edges (denoted by black edgesin Fig. 4(a)) with y edges (denoted by gray edges in Fig.4(b)), according to this ordering until the network is com-pletely connected through y edges. The values of bintkrange from 0 (for edge 10-12 with edge ranking index 1(see Fig. 4(c)), bypassed by all chosen interlayer paths)to 100 − 400 for highly loaded edges (for example, foredge 3-5 for which every path from node 3 of the x layergraph to any other node in the y-layer must pass throughit).

A. Networks of Lorenz oscillators

The coupling necessary to synchronize the x-coupledLorenz network (2) described in Fig. 4(a) is c ≈ 86.95.As outlying, low tra�c edges are replaced with y edges,there is almost no e�ect on the threshold for the couplingstrength required to synchronize the network, evidencedby the lack of change in the actual coupling thresholdfor the �rst eight edges replaced in Fig. 4(c). As suc-cessively more loaded edges are replaced in the network(indicated by the dramatic increase in bint), the networkbecomes more di�cult to synchronize, until edge 13 (edge3-5 which is depicted in red in Fig. 4(b)) is replaced. Af-ter which, synchronization is no longer feasible for thenetwork for any additional edge replacement, until edge25 (edge 14-15 in Fig. 4). Replacing edge 25, shown inFig. 4(b) corresponds to �nishing the successive edge re-placement process, and results in a graph identical to theone in Fig. 4(a), but in which all of the edges representy coupling (gray) instead of x coupling (black). This re-inforces our tra�c load predictions for the breakdown ofsynchrony in two ways: (i) after enough highly loadededges are replaced (even with a normally favorable cou-pling type), the network can no longer synchronize forany coupling strength, (ii) replacing only one edge thatis very highly loaded can make the network unsynchro-nizable, evidenced by the network having no synchroniz-ing coupling value even when all but one edge has beenreplaced (see edge index 24 in Fig. 4(c)).

(a)

(b)

(c)

FIG. 4. E�ect of successively replacing x coupling edges withy coupling edges on synchronization in a 20-node network ofLorenz oscillators (2). (a) Original x-coupled network beforereplacing edges according to their tra�c load. (b) Snapshotof the multilayer network before the edge 3-5, labeled as the13-th edge according to the tra�c load ranking, is replaced.The replacement of this critical edge (red) yields the break-down of synchronization in the network. Further successivereplacement of remaining x edges (gray) with higher tra�cload preserves the instability of synchronization, until all 25edges have been replaced with a y edge, yielding a single-layery-coupled network with bint = 0 that is able to synchronizeagain. (c) Actual (blue solid line) and predicted (blue dottedline) threshold for the coupling strength required to synchro-nize the network after the i-th x edge has been successivelyreplaced with a y edge. The black solid line depicts the inter-layer tra�c bint for the respective edge. The predicted cou-pling thresholds are computed from (3) using the exponential�t in Fig. 3 and scaling factors β = 0.0025 and γ = 0.5993.

As in the six-node example of Fig. 1(a), we have ob-tained a good �t in Fig. 4(c) which only focuses on plac-ing the stability conditions on x edges in (3), becausethe stability term αx required to stabilize the x stabilitysystem (8) must be signi�cantly higher than αy in they stability system (9) (compare Figs. 3(a-b)). We usethe same criterion (12) with the same constants ax, ωx,

9

and αxk to predict the synchronization threshold and onlyneed to identify the tra�c loads bintk and the scaling fac-tors γ and β for a better �t, once and for all variationsof the multilayer network topologies used in Fig. 4(c).In contrast to the six-node network example where traf-�c load bintk can be easily calculated by hand as in (11),computing bintk for the 20-node or larger networks is alaborious task which was performed by an algebraic al-gorithm, implemented as MATLAB code and given inthe Supplement. While the values of bintk heavily dependon the choice of paths from one node to another, ouralgorithm uses the natural choice of the shortest paths,computed via Dijkstra's algorithm [59]. Optimizing thechoices of not necessarily shortest paths that distributetra�c loads on edges more equally may yield even betterpredictions and �ts.

B. Networks of double scroll oscillators

To illustrate the generality of synchrony break phe-nomenon when �good� but highly-loaded links go �bad�,we apply our numerically-assisted method to networks(1), comprised by chaotic double-scroll oscillators [58]

xi = κ(yi − xi − h(x)) +n∑j=1

cijxj ,

yi = xi − yi + zi +n∑j=1

dijyj ,

zi = −λyi − µzi, i = 1, ..., n,

(13)

with

h(x) =

m1(x+ 1)−m0 x < −1m0x −1 ≤ x ≤ 1m1(x− 1) +m0 x > 1

and standard parameters κ = 10, m0 = −1.27, m1 =−0.68, λ = 15, and µ = 0.038.Similarly to networks of Lorenz oscillators (2), a pair of

double-scroll oscillators (13) can be synchronized througheither the x or y variable, and the minimum couplingstrength required for synchronization in a two-node y-coupled network, d∗ = 1.16 is much lower than thecoupling threshold in the two-node x-coupled network,c∗ = 5.94.In Fig. 5, we apply our method to predict the synchro-

nization thresholds in the 20-node network of Fig. 4 asin the same network of Lorenz oscillators. When succes-sively replacing x edges in the network, there is initiallya decrease in the coupling threshold for synchronization,when peripheral edges or edges in highly connected re-gions of the graph with low tra�c loads bintk are replacedwith more favorable y edges that provide better pairwiseconvergence to synchronization. Then, as with the net-work of Lorenz oscillators, when edge 13 (edge 3-5) isreplaced with a y edge, synchronization is no longer at-tainable. Synchronization then returns when the entire

FIG. 5. E�ect of successively replacing x edges with y edgeson the synchronization threshold in a 20 node network ofdouble-scroll oscillators. The network topology, edge replace-ment process, and notations are identical to those in Fig. 4(a-b). Notice the same location of critical links (edges 13 to 24)whose replacement leads to synchrony breaks as in the net-work of Lorenz oscillators (cf. Fig. 4(c)).

x-coupled network has been replaced with y edges. No-tably, the synchrony break occurs at the same edge asin the network of Lorenz oscillators, suggesting that crit-ical edges whose replacement hampers synchronizationare mainly controlled by the network multilayer topol-ogy rather than the individual properties of the intrin-sic oscillators, provided that the oscillators possess simi-lar synchronization properties as the Lorenz and doublescroll oscillators.

The solid curve in Fig. 5 displays the synchroniza-tion thresholds, calculated via the stability criterion(12) with ax = 5.94 × 2 = 11.88, ωx = 2.0, β =0.0041, γ = 0.282, and the approximating function αx =1.556 exp(3.711βωxb

intk ) with the same tra�c loads bintk

shown in Fig. 4. This approximating function is ob-tained from a stability diagram for coupled double scroll-oscillators which is computed similarly to Fig. 3 and dis-plays a similar threshold e�ect as in Fig. 3 [not shown].As in the Lorenz oscillator case, the auxiliary stabilitysystem (4) for αx is much more sensitive to increasingbintk than the stability system (5) for αy, therefore onecan only evaluate the stability condition (12) for the xcoupling c to identify a bottle-neck for the synchroniza-tion threshold in the entire network.

Going back to the puzzle example, we have also per-formed a similar analysis of the six-node network of Fig. 1where the Lorenz oscillators are replaced with the double-scroll oscillators [not shown]. Remarkably, this analy-sis indicates the same qualitative phenomena when thereplacement of the lightly loaded edge 5-6 slightly low-ers the synchronization threshold from c = 13.57 in theoriginal x-coupled single-layer network of Fig. 1(a) toc = 13.36, and predicts the breakdown of synchrony whenedge 2-3 is replaced as in the Lorenz network.

10

We have also simulated series of other 20-node net-works (2) and then networks (13) where all oscillatorswere connected via x layer graphs, whereas the y cou-pling only connected some of the oscillators. In contrastto the networks of Fig. 1 and Fig. 4 where the criticalhighly-loaded links separate the network into disjoint xand y graph components, these networks do not showthe e�ect of synchrony breaking as any pair of nodes iscoupled directly or indirectly via the x graph such thatthe coupling strengths c can be made strong enough tostabilize synchronization. However, the synchronizationthresholds in such networks depend on the location ofadded y edges in a nonlinear fashion. In support of thisclaim, we draw the reader's attention to the six-node ex-ample of Fig. 1(b) where the x graph connects all sixnodes and the replacement of edge 5-6 with an y edge low-ers the synchronization threshold. On the contrary, thereplacement of the x edge 3-5 with an edge y, which stillpreserves the connectedness of the x graph, increases thesynchronization threshold, as predicted by the method(see Fig. 1(c)). The 20-node networks with connectedx graphs yield similar e�ects. To avoid repetition, theseresults are not shown.

VII. CONCLUSIONS

While the study of synchronization in multilayer dy-namical networks has gained signi�cant momentum, thegeneral problem of assessing the stability of synchroniza-tion as a function of multilayer network topology re-mained practically untouched due to the absence of gen-eral predictive methods. The existing eigenvalue meth-ods, including the master stability function [23], whiche�ectively predict synchronization thresholds in single-layer networks cannot be applied to multilayer networksin general. This is due to the fact that the connectivitymatrices corresponding to two or more connection layersdo not commute in general, and therefore, the eigenvaluesof the connectivity matrices cannot be used. Therefore,synchronization in multilayer networks is usually studiedon a case by case basis either via (i) full-scale simulationsof all transversal Lyapunov exponents of the (n−1)×s-dimensional system of variational equations [55], wheren is the network size and s is the dimension of the intrin-sic node dynamics, or more e�ectively via (ii) simulta-neous block diagonalization of the connectivity matrices[54] which in some cases can reduce the problem of as-sessing synchronization in a large network to a smallernetwork which, however, contains positive and negativeconnections, including self loops such that the exact roleof multilayer network topology and the addition or ex-change of edges remains unclear.In this paper, we have made a breakthrough in un-

derstanding synchronization properties of multilayer net-works by developing a predictive method, called the Mul-tilayer Connection Graph method, which does not rely oncalculations of eigenvalues of the connectivity matrices,

and therefore can handle multilayer networks. Originatedfrom the connection graph method for synchronization insingle-layer networks [30], our method combines stabilitytheory with graph theoretical reasoning. Two key ingre-dients of the method are (i) the calculation of stabilitydiagrams for the auxiliary s-dimensional system whichindicate how the dynamics of the given oscillator com-prising the network can be stabilized via one variablecorresponding to one connection layer when an instabil-ity is introduced via the other variable from another con-nection layer and (ii) the calculation of tra�c loads viaa given edge on the multilayer connection graph. All to-gether, these quantities allow for predicting the synchro-nization threshold and identify critical links that controlsynchronization in the original, potentially large, multi-layer network.

Using the method, we have discovered striking, highlyunexpected phenomena not seen in single-layer networks.In particular, we have shown that replacing a link witha light interlayer tra�c load by a stronger pairwise con-verging coupling via another layer may improve synchro-nizability, as one would expect. At the same time, sucha replacement of a highly loaded link may essentiallyworsen synchronizability and make the network unsyn-chronizable, turning the pairwise stabilizing �good� linkinto a destabilizing connection (a �bad� link). The criti-cal links whose replacement can lead to synchrony breakare typically the ones that connect the layers such theoscillators from two layers become coupled through theintrinsic, nonlinear equations of the individual oscillatorthat correspond to a �relay� node passed by the only pathfrom one layer to the other. As a result, the intrinsic dy-namics of the individual node oscillator plays a pivotalrole in the stability of synchronization. In this paper, wehave limited our attention to Type I oscillators such asthe Lorenz and double scroll oscillators that yield globalsynchronization that remains stable in a single-layer net-work once the coupling exceeds a critical threshold. Re-markably, when used in a multilayer network, both oscil-lators have indicated similar synchronization properties,suggesting that the location of critical edges in the con-sidered network may remain unchanged for other TypeI oscillators. While our rigorous method for provingglobal synchronization is only applicable to Type I os-cillators, its semi-analytical version can be modi�ed tohandle Type II networks, including multilayer networksof Rössler systems [23]. This modi�cation is a subject offuture study.

To gain insight into the determining factors for theemergence of synchrony breaking, without potential con-founds associated with the interplay between multiplelayers and direction of links, we have considered two-layerundirected networks of identical oscillators. However,the extension of our results to multiple layers, directednetworks and non-identical oscillators is fairly straight-forward and will be reported elsewhere. In particular,the extension of our method to directed networks can beperformed by adapting the generalized connection graph

11

method [31, 60] for single-layer directed networks, wheredirected edges are symmetrized and assigned additionalweights according to the mean node unbalance.Our method can also be modi�ed to handle multi-

layer neuronal networks connected via electrical, excita-tory, and inhibitory synapses which exhibit a number ofcounterintuitive synergistic e�ects: when (i) the additionof pairwise repulsive inhibition to single-layer excitatorynetworks can promote synchronization [43] and (ii) com-bined electrical and inhibitory coupling can induce syn-chronization even though each coupling alone promotesan anti-phase rhythm [61]. Our method promises to allowan analytical treatment of these e�ects in large neuronalnetworks which has been impaired by the absence of rig-orous methods that can handle excitatory, inhibitory, andelectrical neuronal circuitries simultaneously.

VIII. ACKNOWLEDGMENTS

This work was supported by the National ScienceFoundation (USA) under Grant No. DMS-1616345and the U.S. Army Research O�ce under Grant No.W911NF-15-1-0267.

IX. APPENDIX: THE PROOF

In this appendix we derive the Multilayer ConnectionGraph method and prove Theorem 1. Our goal is toderive the conditions of global asymptotic stability ofthe synchronization manifold M in the system (1). Toachieve this goal and develop the stability method, we fol-low the steps of the proof of the connection graph method[30]. The concept is similar, up to a certain step where anew stability argument is used.In the network model (1) we introduce the di�erence

variable

Xij = xj − xi, i, j = 1, ..., n, (14)

whose convergence to zero will imply the transversal sta-bility of the synchronization manifold M.Subtracting the i-th equation from the j-th equation

in system (1), we obtain the equations for the transversalstability of M

Xij = F(xj)− F(xi) +n∑k=1

{cjkPXjk − cikPXik+

djkLXjk − dikLXik}, i, j = 1, ..., n.(15)

To obtain the explicit dependence of F(xj)−F(xi) onXij , we introduce the following vector notation

F(xj)− F(xi) =

1∫0

DF(βxj + (1− β)xi)dβ

Xij ,

where DF is the s × s Jacobian matrix of F. This no-tation is simply a compact form of the mean value theo-rem, f(B)− f(A) = f ′(C)(B−A), applied to the vectorfunctions F(xj) and F(xi), where the Jacobian DF isevaluated at some point C ∈ [xi,xj ].Therefore, the di�erence system (15) can be rewritten

in the form

Xij =

[1∫0

DF(βxj + (1− β)xi)dβ]Xij+

n∑k=1

{cjkPXjk − cikPXik + djkLXjk − dikLXik},

i, j = 1, ..., n.(16)

The �rst term with the brackets yields instability viathe divergence of trajectories within the individual, pos-sibly chaotic oscillators. The second (summation) term,which represents the contribution of the network connec-tions, may overcome the unstable term, provided thatthe coupling is strong enough.Notice that the stability of system (16) is redundant as

it contains all possible (n−1)n/2 non-zero di�erencesXij

along with n zero di�erences Xii = 0 which can be dis-regarded. At the same time, there are only n−1 linearlyindependent di�erences required to show the convergencebetween n variablesXij . However, this redundancy prop-erty and the consideration of all non-zero Xij are a keyingredient of our approach which allows for separatingthe di�erence variables later in the stability description,without diagonalizing the connectivity matrices.We strive to �nd conditions under which the trivial

�xed point {Xij = 0, i, j = 1, ..., n} of system (16) isglobally stable. This amounts to �nding conditions forglobal stability of synchronization in the network (1).We introduce the following terms AijXij , where A is

a 3× 3 matrix, such that

Aij =

axP = diag(ax, 0, 0) if oscillators i and jbelong to x layer C

ayL = diag(0, ay, 0) if oscillators i and jbelong to y layer D

K = diag(ωx, ωy, 0) if oscillators i and jbelong to di�erent layers C and D,

(17)where constants ax, ay, ωx, and ωy are to be determined.We add and subtract additional terms AijXij with ma-

trix Aij de�ned in (17) from the stability system (16) andobtain

Xij =

[1∫0

DF(βxj + (1− β)xi)dβ −Aij]Xij +AijXij

+n∑k=1

{cjkPXjk − cikPXik + djkLXjk − dikLXik}.

(18)The introduction of the terms AijXij allows for obtain-ing stability conditions of the trivial �xed point Xij = 0,i, j = 1, .., n in two steps. Note that the matrix −Acontributes to the stability of the �xed point and cancompensate for instabilities induced by eigenvalues with

12

nonnegative real parts of the Jacobian DF. This can beachieved by increasing parameters ax, ay, ωx, and ωy. Atthe same time, the instability originated from its posi-tively de�nite counterpart, matrix +A, can be dampedby the coupling terms through cij and dij .Step I. We make the �rst step by introducing the follow-ing auxiliary systems for i, j = 1, ..., n

Xij =

1∫0

DF(βxj + (1− β)xi) dβ −Aij

Xij . (19)

This system is identical to system (18) where the couplingterms are removed.Aij can take three di�erent values, depending on

whether oscillators i and j both belong to the x or ygraphs, or belong to di�erent graphs, for example, if ibelongs to the x graph, and j belongs to the y graph (see(17)). Therefore, we have three types of the auxiliarysystems

Xij =

[1∫0

DF(βxj + (1− β)xi) dβ − axP]Xij

if i and j both belong to x layer,

(20)

Xij =

[1∫0

DF(βxj + (1− β)xi) dβ − ayL]Xij

if i and j both belong to y layer,

(21)

Xij =

[1∫0

DF(βxj + (1− β)xi) dβ −K]Xij

if i and j each belong to di�erent layers.

(22)

Remarkably, the auxiliary system (20) coincides with thedi�erence system for the global stability of synchroniza-tion in a two-oscillator network (1) with only x coupling,where ax plays the role of the double coupling strengththat guarantees the stability (see [30] for a detailed dis-cussion on this relation).Similarly, the stability of auxiliary system (21) implies

global stability of synchronization in the two-oscillatornetwork (1) with only y coupling, where ay is the doublecoupling strength of the y connection. Lastly, the sta-bility of auxiliary system (22) guarantees globally stablesynchronization in the two-oscillator network with bothx and y coupling, where a combination of constants ωxand ωy, present in K, is a combination of the double cou-pling strengths of x and y connections that is su�cientto induce stable synchronization in the two-oscillator xand y coupled network.Therefore, our immediate goal is to �nd upper bounds

on the values of ax, ay, and ωx and ωy that make theorigin of the auxiliary systems (20)-(22) stable. Thisamounts to proving global synchronization in the threex-, y-, and (x, y)-coupled networks that are composed oftwo oscillators. As only Type I oscillators can be globally

synchronized, our approach based on the calculation ofax, ay, is thus limited to this class of oscillators.The proof of global stability in (20)-(22) and deriva-

tion of bounds ax, ay, and ωx and ωy involves the con-struction of a Lyapunov function along with the assump-tion of the eventual dissipativeness of the coupled sys-tem. Therefore, before advancing with the study of largernetworks (1), one has to prove that globally stable syn-chronization in the simplest x-, y-, and (x, y)-coupled,two-oscillator networks is achievable. The bound ax forx-coupled Lorenz oscillators was given in [30]. Upperbounds for ay, ωx, and ωy can be derived similarly.Having obtained the bounds ax, ay, and ωx and ωy, and

therefore proving the stability of the auxiliary systems(20)- (22), we can now take the second step in analyzingthe full stability system (18).

Step II. The bounds ax, ay, and ωx and ωy that sta-bilize the auxiliary systems (20)-(22) reduce the stabilityanalysis of system (18) to the following equations by ex-cluding the term in brackets

Xij = AijXij +n∑k=1

{cjkPXjk − cikPXik + djkLXjk−

dikLXik}, i, j = 1, ..., n.(23)

Note that the positive term AijXij , which contains theupper bounds ax, ay, and ωx and ω, is destabilizing andmust be compensated for by the coupling terms. To studythe stability of (23) we introduce a Lyapunov function ofthe form

V =1

4

n∑i=1

n∑j=1

XTij · I ·Xij , (24)

where I is an s× s identity matrix.Its time derivative with respect to system (23) becomes

V = 12

n∑i=1

n∑j=1

XTijAijXij−

12

n∑i=1

n∑j=1

n∑k=1

{cjkXTjkIPXjk − cikXT

ikIPXik}−

12

n∑i=1

n∑j=1

n∑k=1

{djkXTjkILXjk − dikXT

ikILXik}.

(25)We need to demonstrate the negative semi-de�niteness

of the quadratic form V . As (X2ii = 0, X2

ij = X2ji), the

�rst sum simpli�es to

S1 =

n−1∑i=1

n∑j>i

AijX2ij . (26)

This sum is always positive de�nite and its contributionmust be compensated for by the second and third sums

S2 = − 12

n∑i=1

n∑j=1

n∑k=1

{cjkXTjkIPXjk − cikXT

ikIPXik},

S3 = − 12

n∑i=1

n∑j=1

n∑k=1

{djkXTjkILXjk − dikXT

ikILXik}.

(27)

13

Due to the coupling symmetry, the two terms in bothS2 and S3 can be made identical by exchanging the in-dices i by j in the second terms such that

S2 = −n∑i=1

n∑j=1

n∑k=1

cjkXTjkIPXjk,

S3 =n∑i=1

n∑j=1

n∑k=1

djkXTjkILXjk.

(28)

Taking into account that Xjj = 0, we obtain

S2 = −n∑i=1

n−1∑k=1

n∑j>k

cjkXTjiIPXjk−

n∑i=1

n−1∑k=1

n∑j<k

cjkXTjiIPXjk,

S3 = −n∑i=1

n−1∑k=1

n∑j>k

djkXTjiIPXjk−

n∑i=1

n−1∑k=1

n∑j<k

djkXTjiIPXjk.

(29)

Again, exchanging j and k in the second terms of S2

and S3 and implying the symmetries of coupling cjk = ckjand djk = dkj , we obtain

S2 = −n∑i=1

n−1∑k=1

n∑j>k

cjk(XTji +XT

ik)IPXjk,

S3 = −n∑i=1

n−1∑k=1

n∑j>k

djk(XTji +XT

ik)ILXjk.

(30)

Since XTji +XT

ik =[xTi − xTj + xTk − xTi

]= XT

jk, we ob-tain

S2 = −n−1∑k=1

n∑j>k

ncjkXTjkIPXjk,

S3 = −n−1∑k=1

n∑j>k

ndjkXTjkILXjk.

(31)

Returning to the derivation of the Lyapunov function(25) and combining the sums S1, S2 and S3 yields the

condition which guarantees that V ≤ 0 :

S1+S2+S3 =

n−1∑i=1

n∑j>i

XTijI[Aij−ncijP−ndijL]Xij < 0.

(32)The most remarkable property of this condition is thatwe are able to eliminate the cross terms and formulatethe condition in terms of Xij . This is because we choseto consider the redundant system with all possible di�er-ences Xij , including linearly dependent ones.The condition (32) �nally transforms into

nn−1∑i=1

n∑j>i

[cijXTijIPXij + dijX

TijILXij ] >

>n−1∑i=1

n∑j>i

XTijIAijXij .

(33)

Notice that the left-hand side (LHS) of this inequalitycontains only the di�erences Xij between the oscillators

that belong to the edges on the connection graphs C andD: the �rst term on the LHS corresponds to the x layerand the second term is de�ned by the edges of the ylayer. At the same time, the variables on the right-handside (RHS) of (33) correspond to all possible di�erencesbetween pairs of oscillators that might or might not bede�ned by edges of the connection graphs. Hence, to getrid of the presence of the di�erences Xij and therefore�nd the conditions explicit in the parameters of the net-work model (1), we express the di�erences on the RHSvia the di�erences on the LHS such that we will be ableto cancel them.So far, we have closely followed the steps in the deriva-

tion of the connection graph method [30] for single-layernetworks. The inequality (33) is similar to that of theconnection graph method, except for the presence of thesecond term on the RHS and a modi�ed matrix Aij . Anew non-trivial observation, however, is that the totalnumber of oscillators, n, in the network (1), composed oftwo connection layers, appears as a factor in both sumson the RHS, corresponding to the x and y graphs, eventhough each graph itself may connect fewer oscillators.The stability argument which follows drastically di�ersfrom that of the connection graph method.Denote on the LHS (33), the di�erences Xij corre-

sponding to edges of the x graph by Xk, k = 1, ...,mand the di�erences Xij corresponding to edges of the y

graph by Yk, k = 1, ..., l. Recall that m and l are thenumber of edges on the x and y graphs, respectively. Inaddition, let Xk be a scalar from the vector Xk whichindicates the scalar di�erence between xi and xj , corre-sponding to an edge on the x graph. Similarly, let Yk bea scalar from the vector Yk, de�ned by the correspond-ing yi and yj . Using these notations, the di�erences Xij

on the RHS will now de�ne the scalars Xij = xj − xiand Yij = yj − yi, . If the inequality (33) is satis�ed interms of xi and yi (via the scalar di�erences Xk and Yk),then it will also be satis�ed for the remaining scalar zi.Recall that (xi, yi, zi) are the scalar coordinates of theindividual oscillator, composing the network (1).Using this notation, we can rewrite (33) as follows

nm∑k=1

ckX2k + n

l∑k=1

dkY2k > ax

n−1∑i=1

n∑j>i,(i,j)∈C

X2ij+

ayn−1∑i=1

n∑j>i,(i,j)∈D

Y 2ij +

n−1∑i=1

n∑j>i,(i,j)/∈C,D

[ωxX2ij + ωyY

2ij ],

(34)where ck = cikjk and dk = dikjk . Here, the RHS of (34)has three terms, obtained by splitting the di�erence vari-ables into three groups, according to the coe�cients ofAij (cf. (33) and (20)-(22)). The �rst sum on the RHS iscomposed of the di�erences that belong to the x graph C,the second sum corresponds to the y graph D, whereasthe third sum identi�es the di�erences between the oscil-lators which belong to di�erent graphs such that i ∈ Cand j ∈ D or vice versa.To recalculate the di�erence variables of the RHS via

the variables Xk and Yk, we should �rst choose a path

14

from oscillator i to oscillator j for any pair of oscillators(i, j).We denote this path by Pij . Its path length |Pij | isthe number of edges comprising the path. The importantproperty of the path Pij is that if, for example, it passesthrough oscillators with indices 1, 2, 3, and 4, then thecorresponding di�erenceX14 = x4−x1 = (x4−x3)+(x3−x2)+ (x2−x1) = X12+X23+X34, where the di�erencesX12, X23, and X34 correspond to the edges and the pathlength |P14| = 3.The choice of paths is not unique. We typically choose

a shortest path between any pair of i and j; however,a di�erent choice of paths can yield closer estimates, asdiscussed in [34] for single-layer networks.Once the choice of paths is made, we stick with it and

start recalculating the di�erence variables on the RHSof (34) via Xk and Yk. A potential problem is that wehave to deal not with the variables Xij , but with theirsquares X2

ij , coming from the calculations of the deriva-tive of the Lyapunov function (25). Therefore, we haveto apply the Cauchy-Schwartz inequality; applied to theabove example, it yields X2

14 = (X12 + X23 + X34)2 ≤

3(X212 +X2

23 +X234). Notice the appearance of the factor

3, indicating the number of edges comprising the path.Similarly, for any di�erence Xij and Yij we have

X2ij =

( ∑k∈Pij

Xk

)2

≤ |Pij |∑

k∈Pij

X2k ,

Y 2ij =

( ∑k∈Pij

Yk

)2

≤ |Pij |∑

k∈Pij

Y 2k ,

(35)

where once again |Pij | indicates the length of the chosenpath from oscillator i to oscillator j along the connectiongraph, combined of the x and y graphs. At this point, wedo not di�erentiate between paths containing only x ory edges, but we have to consider interlayer paths whennecessary.Applying this idea to each di�erence variable on the

RHS of (34), we obtain the following condition

nm∑k=1

ckX2k + n

l∑k=1

dkY2k >

m∑k=1

[axbxk + ωxb

intk ]X2

k+

l∑k=1

[aybyk + ωyb

intk ]Y 2

k +∑k∈D

[ωxbintk ]X2

k +∑k∈C

[ωybintk ]Y 2

k ,

(36)

where bxk =n∑

j>i; k∈Pij∈C|Pij | is the sum of the lengths

of all chosen paths which belong to the x graph Cand go through a given x edge k. Similarly, byk =

n∑j>i; k∈Pij∈D

|Pij | is the sum of the lengths of all chosen

paths which belong to the y graph D and go through a

given y edge k. Finally, bintk =n∑

j>i; k∈Pij∈(C,D)

|Pij | is the

sum of the lengths of all chosen paths which contain xand y edges and belong to the combined, two-layer xy-graph and go through a given edge k which may be an xor y edge.

Note that the two sums on the LHS of (36) correspondto the �rst sums on the RHS. In the simplest case whereboth C and D graphs are connected such that each graphcouples all n oscillators, bintk can always be set to 0, sincethere are always paths between any two nodes that en-tirely belong to either the x or y graph. As a result,the third and forth sums on the RHS disappear, and weimmediately obtain the stability conditions by droppingthe summation signs and the di�erence variables

ck + dk >1

n{ax · bxk + ay · byk} . (37)

In the case of disconnected graphs C and D where alloscillators are coupled through a combination of the twographs and bintk is non-zero, the third and forth sums arealways present on the RHS. This makes the argumentmuch more complicated but yields a number of surprisingimplications of the stability method to speci�c networksdiscussed in Sections V and VI.A major stability problem, associated with the third

and fourth terms, is rooted in the fact that, for exam-ple, the third sum

∑k∈D

[ωxbintk ]X2

k contains the di�er-

ence variables Xk that correspond to the edges of the

y graph. As a result, the �rst sum nm∑k=1

ckX2k on the

LHS which contains the variables Xk that correspond tox edges, cannot compensate the third sum on the RHSas they belong to di�erent graphs and therefore can-not be compared. At the same time, the second sum

nl∑

k=1

dkY2k on the LHS does belong to the y graph but

contains the variables Yk and not Xk needed to handlethe third sum. The same problem relates to the fourthsum

∑k∈C

[ωybintk ]Y 2

k which contains Yk variables, corre-

sponding to the �wrong� graph (x graph).How can we get around this problem as we simply do

not have means on the LHS to compensate for the trou-blesome sums on the RHS? A solution comes from eco-nomics: if you do not have means, borrow them! [Butact responsibly]. This remark is added to entertain thereader that might be tired of following the proof up tothis point.In fact, the only place to �borrow� these terms from is

the auxiliary stability systems (20) and (21) as they docontain the desired variables Xk and Yk, correspondingto the �right� graphs (the x and y graphs, respectively).Therefore, we need to go back and modify the auxiliarysystems (20) and (21) as follows

Xij =

[1∫0

DF(βxj + (1− β)xi) dβ − [ax + αxk]P +

+ωxbintk L

]Xij if i, j ∈ x edge k,

(38)

Xij =

[1∫0

DF(βxj + (1− β)xi) dβ − (ay + αyk)L+

ωybintk P

]Xij if i, j ∈ y edge k.

(39)

15

The addition of a positive term ωxbintk LXij to the auxil-

iary system (38) worsens its stability, therefore we haveto introduce an additional parameter αxk and make surethat it is su�ciently large to stabilize the new auxiliarysystem. A very important property is that, in (38), wehave to add the positive, destabilizing term ωxb

intk LXij

to the second equation for the (yj−yi) di�erence but tryto stabilize the system via increasing the additional pa-rameter αxk in the �rst equation for the (xj−xi) equation(note the di�erent inner matrices: P versus L in (38)).Depending on the individual oscillator, chosen as the in-dividual unit, this might not be possible, especially whentra�c load bintk on the edge k is high. This property isdiscussed in detail for the Lorenz and double scroll oscil-lator examples in Sections V and IV. A similar argumentcarries over to the auxiliary system (39) where we addthe destabilizing term ωyb

intk Xij to the (xj − xi) equa-

tion, but seek to stabilize the system via the additionalparameter αyk in the (yj − yi) equation.Notice that we only modify the auxiliary systems for x

and y edges. All the other auxiliary systems for Xij , thatdo not correspond to edges of either the x or y graphs,remain intact and de�ned via the original systems (20)-(22).

Thus, the modi�cations of (38) and (39) make the trou-blesome sums

∑k∈D

[ωxbintk ]X2

k and∑k∈C

[ωybintk ]Y 2

k in (36)

disappear at the expense of worsened stability conditionsof the corresponding auxiliary systems, which is re�ectedby the appearance of additional terms with αxk and αyk.

Therefore, (36) turns into

nm∑k=1

ckX2k + n

l∑k=1

dkY2k >

m∑k=1

[axbxk + ωxb

intk + αxk]X

2k+

+l∑

k=1

[aybyk + ωyb

intk + αxy ]Y

2k .

(40)Notice the new stabilizing constants αxk and αyk. Depend-ing on the individual oscillator dynamics and tra�c loadon edge k, these constants might have to be very large oreven in�nite.Comparing the terms containingXk and Yk on the LHS

and RHS of (40) and omitting the summation signs, weobtain the following conditions

nckX2k > [axb

xk + ωxb

intk + αxk]X

2k , k = 1, ...,m

ndkY2k > [ayb

yk + ωyb

intk + αxy ]Y

2k , k = 1, ..., l.

(41)

Finally, we omit the di�erence variables to obtain thebounds on coupling strengths, ck for x edges and dk for yedges, su�cient to make the derivative of the Lyapunovfunction (25) negative semi-de�nite, and therefore, en-sure global stability of synchronization in the network(1). It follows from (41) that these upper bounds are

ck >1n [axb

xk + ωxb

intk + αxk], k = 1, ...,m

dk >1n [ayb

yk + ωyb

intk + αxy ], k = 1, ..., l.

(42)

This completes the proof of Theorem 1. �

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