Synchronization in multilayer networks: when good ...

Synchronization in multilayer networks: when good links go bad Igor Belykh, Douglas Carter Jr., and Russell Jeter

Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia, 30302-410, USA (Dated: May 31, 2018)

Many complex biological and technological systems can be represented by multilayer networks where the nodes are coupled via several independent networks. Despite its signicance from both the theoretical and application perspectives, synchronization in multilayer networks and its dependence on the network topology remain poorly understood.

In this paper, we develop a universal

connection graph-based method which removes a long-standing obstacle to studying synchronization in dynamical multilayer networks. This method opens up the possibility of explicitly assessing critical multilayer-induced interactions which can hamper network synchronization and reveals striking, counterintuitive eects caused by multilayer coupling.

It demonstrates that a coupling which is

favorable to synchronization in single-layer networks can reverse its role and destabilize synchronization when used in a multilayer network.

This property is controlled by the trac load on a

given edge when the replacement of a lightly loaded edge in one layer with a coupling from another layer can promote synchronization, but a similar replacement of a highly loaded edge can break synchronization, forcing a good link go bad. This method can be transformative in the highly active research eld of synchronization in multilayer engineering and social networks, especially in regard to hidden eects not seen in single-layer networks. PACS numbers: 05.45.Xt, 87.19.La

I.

INTRODUCTION

connectivity.

Multilayer-induced correlations can have

signicant ramications for the dynamical processes on Complex networks are common models for many systems in physics, biology, engineering, and the social sciences [13].

Signicant attention has been devoted to

algebraic, statistical, and graph theoretical properties of networks and their relationship to network dynamics (see a review [4] and references therein). The strongest form of network cooperative dynamics is synchronization which has been shown to play an important role in the functioning of a wide spectrum of technological and biological networks [514], including adaptive and evolving networks [1522].

networks, including the eects on the speed of disease transmission in social networks [51] and the role of redundant interdependencies on the robustness of multiplex networks to failure [52]. Typically, in single-layer networks of continuous time oscillators, synchronization becomes stable when the coupling strength between the oscillators exceeds a threshold value [23, 30]. This threshold depends on the individual oscillator dynamics and the network topology.

In this

context, a central question is to determine the critical coupling strength so that the stability of synchronization is guaranteed. The master stability function [23] or

Despite the vast literature to be found on network

the connection graph method [30, 31] are usually used

dynamics and synchronization, the majority of research

to answer this question in single-layer networks.

activities have been focused on oscillators connected

methods reduce the dimensionality of the problem such

Both

through single-layer network (one type of coupling) [23

that synchronization in a large, complex network can be

41]. However, in many realistic biological and engineer-

predicted from the dynamics of the individual node and

ing systems the units can be coupled via multiple, in-

the network structure.

dependent systems and networks. Neurons are typically

Synchronization in multilayer networks has been stud-

connected through dierent types of couplings such as

ied in [5356]; however, its critical properties and ex-

excitatory, inhibitory, and electrical synapses, each cor-

plicit dependence on intralayer and interlayer network

responding to a dierent circuitry whose interplay aects

structures remain poorly understood. This is in particu-

network function [42, 43]. Pedestrians on a lively bridge

lar due to the inability of the existing eigenvalue meth-

are coupled via several layers of communication, includ-

ods, including the master stability function [23] to give

ing people-to-people interactions and a feedback from the

detailed insight into the stability condition of synchro-

bridge that can lead to complex pedestrian-bridge dy-

nization as the eigenvalues, corresponding to connection

namics [4447]. In engineering systems, examples of inde-

graphs composing a multilayer network, must be calcu-

pendent networks include coupled grids of power stations

lated via simultaneous diagonalization of two or more

and communication servers where the failure of nodes in

connectivity matrices.

one network may lead to the failure of dependent nodes

two or more matrices is impossible in general, unless the

in another network [48].

matrices commute [53, 54].

Such interconnected networks

Simultaneous diagonalization of A nice approach based on

can be represented as multiplex or multilayer networks

simultaneous block diagonalization of two connectivity

[49, 50] which include multiple systems and layers of

matrices was proposed in [54]. This most successful ap-

2

plication of the eigenvalue-based approach allows one to

II.

NETWORK MODEL AND PROBLEM STATEMENT

reduce the dimensionality of a large network to a smaller network whose synchronization condition can be used to evaluate the stability of synchronization in the large network. For some network topologies, this technique yields a substantial reduction of the dimensionality; however, this reduction is less signicant in general. The reduced network typically contains weighted positive and negative connections, including self loops such that the role of multilayer network topologies and the location of an edge remain dicult to evaluate. In this paper, we report our signicant progress towards removing this long-standing obstacle to studying synchronization in multilayer networks. We develop a new general stability approach, called the Multilayer Connection Graph method, which does not depend on explicit knowledge of the spectrum of the connectivity matrices and can handle multilayer networks with arbitrary network topologies, which are out of reach for the existing approaches. An example of a multilayer network in this study is a network of Lorenz systems where some of the oscillators are coupled through the layer), some through the

y

x variable (rst

variable (second layer), and

some through both (interlayer connections). Our Multilayer Connection Graph method originates from the connection graph method [30, 31] for single-layer networks; however, this extension is highly non-trivial and requires overcoming a number of technically challenging issues. This includes the fact that the oscillators from two

y

x and

layers in the networks of Lorenz systems are connected

through the intrinsic, nonlinear equations of the Lorenz system. As a result, multilayer networks can have drastically dierent synchronization properties from those of single-layer networks.

In particular, our method shows

that an interlayer trac load on a link is the crucial quantity which can be used to foster or hamper synchronization in a nonlinear fashion. For example, it demonstrates that replacing a link with a light interlayer trac load by a stronger pairwise converging coupling (a good link) via another layer may lower the synchronization threshold and improve synchronizability.

At the same time,

such a replacement of a highly loaded link can make the network unsynchronizable, forcing the pairwise stabilizing good link go bad.

We start with a general network of

n

oscillators with

two connectivity layers:

n n X X dxi = F(xi ) + cij P xj + dij Lxj , i = 1, ..., n, dt j=1 j=1

(1)

xi = (x1i , ..., xsi ) is the s state vector containing the i-th oscillator, F : Rs → Rs describes the oscillators' individual dynamics, cij and dij are the coupling strengths. C = (cij ) and D = (dij ) are n × n where

coordinates of the

Laplacian connectivity matrices with zero-row sums and nonnegative o-diagonal elements

cij = cji

and

respectively [30]. These connectivity matrices

dij = dji , C and D

dene two dierent connection layers (also denoted by

C

and

D,

with

ner matrices

P

m

and

and

L

l

edges, respectively).

The in-

determine which variables couple

the oscillators within the

C

and

D

layers, respectively.

Without loss of generality, we will be considering the os-

xi = (xi , yi , zi ). ThereP = diag(1, 0, 0) will correspond to the rst-layer connections via x, while the D graph with the inner matrix L = diag(0, 1, 0 ) will indicate the second-layer connections via y. Overall, the

cillators of dimension fore, the

C

s=3

and

graph with the inner matrix

oscillators of the network are connected through a combination of the two layers (see Fig. 1 for an example of a combined two-layer graph). The graphs are assumed to be undirected [30]. Oscillators, comprising the network (1), can be periodic or chaotic. As chaotic oscillators are dicult to synchronize, they are usually used as test bed examples for probing the eectiveness of a given stability approach. The oscillators used in the numerical verication of our stability method are chaotic Lorenz [57] and double scroll oscillators [58]. In this paper, we are interested in the stability of complete synchronization dened by the synchronization manifold

M = {x1 = x2 = ... = xn }.

Our main objec-

tive is to determine a threshold value for the coupling strengths required for the global stability of synchronization.

We seek to predict this threshold or the absence

thereof in the general network (1) from synchronization in the simplest two-node network and graph properties

The layout of this paper is as follows.

First, in Sec.

of the two-layer network structures.

In Sec.

It is important to emphasize that only Type I oscilla-

III, we start with a motivating example of why the re-

tors [26] are capable of synchronizing globally and retain-

placement of some links in a multilayer network can im-

ing synchronization for any coupling strengths exceeding

prove or break network synchronization.

the synchronization threshold.

II, we present and discuss the network model.

And what is

Most known oscillators,

IV, we formulate

including the Lorenz and double scroll oscillators belong

the Multilayer Connection Graph method for predicting

to Type I systems. A much more narrow Type II class

synchronization in multilayer networks. In Secs. V and

of oscillators contains

VI, we show how to apply the general method to specic

which synchronization becomes only locally stable and

network topologies. In Sec. VII, a brief discussion of the

eventually looses its stability with an increase of cou-

obtained results is given. Finally, the Appendix contains

pling [24].

the complete derivation of the general method.

MAT-

Type I networks in this paper; however, a numerically-

LAB code for algorithms used for calculating network

assisted extension of our method to Type II networks can

trac loads are given in the Supplement.

be performed with moderate eort and remains a subject

an underlying mechanism?

In Sec.

x-coupled

Rössler systems [23] in

As a result, we limit our consideration to

3

of future study.

tering or hindering synchronization, we begin with twolayer networks of chaotic Lorenz oscillators, depicted in

(a)

Fig. 1(a-c). For the Lorenz oscillators, the vector equation (1) can be written in a more reader-friendly scalar form:

n P

x˙ i = σ(yi − xi ) +

cij xj ,

j=1

(b)

n P

y˙ i = rxi − yi − xi zi +

(2)

dij yj ,

j=1

z˙i = −bzi + xi yi , i = 1, ..., n, C = (cij ) describes the x connections (black edges in Fig. 1(a-c)), D = (dij ) describes the location of one y

where the connectivity matrix topology of

(c)

and matrix

edge (blue or red edge). The parameters of the individ-

σ = 10, r = 28, and b = 8/3. The strengths of the x and y coupling are homogeneous (cij = c, dij = d) and varied uniformly (c = d).

ual Lorenz oscillator are standard:

(d)

We are interested in the question of how the replacement of an

x

y

edge in the network of Fig. 1(a) with a

edge can aect synchronization.

To address this ques-

tion, we rst need to understand synchronization properties of two-node single-layer networks of

y -coupled

Lorenz systems.

x-coupled and

It is well-known that if the

coupling in a single-layer network (2) with either all all

y

x

or

connections exceeds a critical threshold, then syn-

c > c∗ d > d∗ , respectively. A rigorous upper bound for the ∗ critical coupling c , explicit in parameters of the Lorenz chronization becomes stable and persists for any

and

oscillator can be found in [30]. Calculated numerically, these coupling thresholds are

c∗ ≈ 3.81 d∗ ≈ 1.42 for

x-coupled network and y -coupled network. As the synchro∗ nization threshold d is signicantly lower, then one could expect that replacing an x edge with a presumably better converging y coupling improves synchronization. This is true for the network of Fig. 1(b) when x edge 5-6 is replaced with a y edge, yielding a minor reduction in the ∗ synchronization threshold from c ≈ 17.94 in the single∗ layer x-coupled network of Fig. 1(a) to c ≈ 17.74 in

for

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 placement of ing

y

x

edge

5-6

x

edges (black).

(b) The re-

with a presumably better converg-

coupling (blue) improves synchronization, as expected

(see (d)). (c) A similar replacement of

x

edge

2-3

with a

y

edge (red) makes synchronization impossible by pushing the threshold to innity (see (d)). (d). Systematic study of the ∗ coupling threshold c as a function of the y edge location that replaces an

x

edge in the original

x-coupled

network (a).

The blue solid line indicates numerically calculated threshint olds. The black solid line depicts the interlayer trac b for int the respective y edge. Note a signicant increase of b that causes the network to become unsynchronizable as predicted by the method. The predicted coupling thresholds (blue dotted line) are computed from (3) using the exponential t in Fig. 3 and scaling factors

β = 0.18

and

γ = 0.7180.

for the two-node the

the multilayer network of Fig. 1(b).

A surprise comes

from the network of Fig. 1(c) where the replacement of

x edge with a y

edge, that would be expected to improve

synchronization, on the contrary makes the network unsynchronizable (see the coupling threshold jumping to innity in Fig. 1(d)). What is the origin of this highly counterintuitive eect? Why do edges in a multilayer network reverse their stabilizing roles depending on the edge location whereas they are well behaved in single-layer networks? nectivity matrices for

x

and

y

As the con-

coupling in the networks

of Fig. 1(b) and Fig. 1(c) do not commute and, therefore, the predictive power of the master stability function III.

A MOTIVATING EXAMPLE AND A PUZZLE

based methods [5355] is severely impaired, this puzzle calls for an explanation and utlimately motivates the development of an eective, general method for assessing

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

the stability of synchronization in multilayer networks. In the following, we will develop such a method that

4

identies the location of critical interlayer links which

is given in the Appendix.

control stable synchronization and reveals its explicit de-

Remark 1.

pendence on an interlayer trac load on a given edge.

oretical quantities



In terms of trac networks, the graph the-

bxk , byk

and

bint k

represent the total

lengths of the chosen roads that go through a given edge

k

which can be loosely analogized as a busy street. ThereIV.

MULTILAYER CONNECTION GRAPH

fore, we refer to them as trac loads. In this view, the

METHOD

quantity

bint k

is a trac load on edge

k,

caused by in-

terlayer travelers. It is important to notice that positive In this section, we rst present the main result of this

terms

+ωx bint k LXk

in the equation (4) and

+ωy bint k P Xk

paper and formulate the method. We will then demon-

in (5) play a destabilizing role such that a heavily loaded

strate how to apply the method to specic network con-

edge with a high

gurations.

for making the systems (4) and (5) stable at all. This ob-

Theorem 1 [The method].

servation has dramatic consequences for synchronization

Synchronization in the multilayer network (1) is globally stable if for each edge k = 1, ..., m (l)

in specic multilayer networks described in the following

n P

|Pij |

j>i; k∈Pij ∈C

Remark 2. (3)

is the sum of the lengths

of all chosen paths Pij between any pair of oscillators i and j which belong to the x graph C and go through a given x edge. The path length |Pij | is the number of edges comprising the path. Similarly, byk corresponds to the sum of the lengths of all chosen paths which belong to the y graph D and go through a given y edge. bint is k the sum of the lengths of all chosen paths which contain x and y edges and belong to the combined, two-layer xy graph and go through a given edge k which may be an x or y edge. Constant ax (ay ) is the double coupling strength sucient for synchronization in the network of two xcoupled (y-coupled) oscillators, composing the network. The constants ωx and ωy represent a combination of the double coupling strengths for c12 and d12 sucient for synchronization in the two-oscillator network with both x and y connections. Finally, the constants αxk and αyk are chosen large enough such that they can globally stabilize the auxiliary stability systems written for the dierence variables that correspond to an edge k : Xk = Xij = xj − xi : k for αx

can represent a potential problem

sections and also is a key to solving the puzzle of Fig. 1.

 x cij ≡ ck > n1 ax · bxk + ωx · bint k + αk y dij ≡ dk > n1 ay · byk + ωy · bint k + αk ,

where bxk =

bint k

If both

x

and

y

connection graphs are con-

nected such that all oscillators are coupled via both graphs, the stability of synchronization can be simply assessed by applying the connection graph method [30] for each of the

x

and

y

connected graphs and combin-

cij + dij = ck + dk > y 1 x {a · b + a · b } (cf. the condition (37) in the Apx y k k n pendix). As a result, one should not expect the eects

ing the two conditions as follows:

due to the multilayer coupling discussed in the motivating example.

Remark 3.

While the stability criterion (3) is completely

rigorous, the theoretically calculated bounds may give large overestimates on the threshold coupling strength. As a result, we suggest to use a semi-analytical approach which combines numerically calculated exact bounds ax , ay , ωx , ωy , αxk , and αyk with graph theoretical quantities bx , by , and bint . In this way, this method becomes an eective, predictive tool and plays the role of the master stability function for the general multilayer network (1) where a synchronization threshold or the absence thereof can be deduced from the properties of the individual oscillators comprising the network and the network topologies of the connection layers. In this numerically-assisted application of our method, the conservative estimate on trac loads

bx , by , and bint , that come from the Cauchy-

Schwartz inequality (see the Appendix), can be replaced

˙k = : X

1 R

by

 DF(βxj + (1 − β)xi dβ Xk +

0

(4)

βbx , βby , and γbint , where the scaling factors β

and

γ

are introduced to provide tighter numerical bounds.

x ωx bint k LXk − (ax + αk )P Xk ,

for

˙k = αyk : X

1 R

 DF(βxj + (1 − β)xi dβ Xk +

0

(5)

y ωy bint k P Xk − (ay + αk )LXk ,

where the Jacobian DF can be calculated explicitly via the parameters of the individual oscillator. Proof. The proof closely follows the notations and steps

Computing the stability conditions (3) and its constants is a complicated, multi-step process which involves calculations of (i) two stability diagrams (for a pair of

in the derivation of the connection graph method [30] for

ω x , ωy and αkx , αy ) and (ii) graph theoretical quantities bx , by , and bint . In the following section, we will walk the

single-layer networks up to a point where the stability ar-

reader through a step-by-step computation of the sta-

gument becomes drastically dierent and yields the new

bility conditions (3) for specic multilayer networks and

terms

ωx bint k

and

ωy bint k

which play a pivotal role in syn-

chronization of multilayer networks. The complete proof

illustrate their implications for the stability of synchronization.

5

Step 2: Calculate the stability diagrams to determine αkx and αky . The auxiliary stability systems (4) and (5) for the scalar dierence variables of coupled Lorenz oscillators (2)

Xk = Xij = xj −xi , Yk = Yij = yj −yi , Zk = Zij = zj −zi can be rewritten in a more convenient form:

X˙ k =σ(Yk − Xk ) − (ax + αkx )Xk , (z) (x) Y˙ k = r − Uk Xk − Yk − Uk Zk + ωx bint k Yk (y) (x) Z˙ k = U Xk + U Yk − bZk ,

(6)

k

k

X˙ k =σ(Yk − Xk ) + ωy bint k Xk (z) (x) Y˙ k = r − U Xk − Yk − U Zk − (ay + αy )Yk , k

FIG. 2. of

ωx

Stability of synchronization in a two-node network

xy -coupled and the y

Lorenz systems as a function of the coupling,

ωy .

x

coupling,

Yellow depicts instability (non-

zero synchronization error) and purple depicts stability (zero

k

k

(y) (x) Z˙ k = Uk Xk + Uk Yk − bZk , (7) where

(ξ)

Uij

= (ξi + ξj )/2

for

ξ = x, y, z

are the corre-

synchronization error). The white dot indicates the pair (ωx ,

sponding sum variables.

ωy )

bility system (6) corresponds to the dierences between

used in the predictions shown in Fig. 1(d) and Fig. 4(c).

As in (3), the auxiliary sta-

the nodes connected by an

x

edge, and (7) corresponds

to the dierences between nodes coupled via a V.

APPLICATION OF THE METHOD:

connected through both

SOLVING THE PUZZLE

the puzzle to better understand synchronization properties of the six-node networks of Lorenz oscillators (2) from Fig. 1. Below we give a step-by-step recipe of how our numerically-assisted method can be applied to these networks.

Step 1: Calculate ax , ay , ωx , and ωy . coupling:

c12 = c21 = c

and

d12 = d21 = d.

x

Deter-

mine threshold coupling strengths that guarantee stable synchronization in (i) the two-node

x-coupled network two-node y -coupled

c∗ = ax /2 and d = 0; (ii) the ∗ network with d = ay /2 and c = 0; and (iii) the two-node xy network with c∗ = ωx and d∗ = ωy . The numerically calculated thresholds for the xcoupled and y -coupled two-node network (2) reported ∗ in Sec. III are c ≈ 7.62/2 and d∗ ≈ 2.84/2, respec-

with

tively. This yields the double coupling strength constants

ax = 7.62

and

ax = 2.84

in the

xy -coupled

(see Fig. 2).

c = ωx = 5

c = ωx

and

and

d = ωy = 0.5

dk

for the same edge

where bounds on the sum dierences

in single-layer networks [30]. A rigorous analysis of global stability of (6) and (7) will be performed in a more technical publication. Here, we take a more practical route towards developing a numerically-assisted approach which simplies the evaluation of the stability of (6) and (7).

to choose

ωx

and

and

ωy

αky

This

is somewhat arbitrary; how-

(Fig. 3). It is often a good idea

such that both are non-zero and lie

somewhere in the middle range of

We lin-

earize both systems (6) and (7) by making the dierences

Xk , Yk , and Zk innitesimal and replacing U (x) = x(t), U (y) = y(t), and U (z) = z(t), where (x(t), y(t), z(t)) is the synchronous solution governed by the uncoupled Lorenz oscillator.

The linearized stability systems (6) and (7)

take the form:

X˙ k = σ(Yk − Xk ) − (ax + αkx )Xk , Y˙ k = (r − z(t)) Xk − Yk − x(t)Zk + ωx bint k Yk Z˙ k = y(t)Xk + x(t)Yk − bZk ,

(8)

X˙ k = σ(Yk − Xk ) + ωy bint k Xk ˙ Yk = (r − z(t)) Xk − Yk − x(t)Zk − (ay + αky )Yk , Z˙ k = y(t)Xk + x(t)Yk − bZk .

(9)

ever, it dictates the choice of constants in the stability

αkx

U (x) , U (y) , and U (z)

as a point on the sta-

two-layer networks (2) with arbitrary topologies.

diagrams for

ck

k.

The auxiliary systems (6) and (7) are used to yield

we choose

the prediction of the synchronization threshold in larger

(ωx , ωy )

Their contributions will

global stability conditions for the analytical criterion (3),

d = ωy

bility boundary in Fig. 2 and keep these values xed for

choice of the pair

x and y edges independently.

network yield stable synchronization

Without loss of generality, and

ing

then appear in the general stability conditions (3) for

to be used in (3).

Note that dierent combinations of

x and y edges, then the auxiliary

can be placed similar to the analysis of synchronization

Consider the simplest two-node network (2) with both

y

edge.

systems (6) and (7) should be applied to the correspond-

Armed with the predictive method, we rst return to

and

y

If the connection layers overlap and the same nodes are

(ωx , ωy ) to balance out

the stability conditions (4) and (5).

Notice that

αkx [αky ]

must be large enough to stabilize (8)

[(9)] in the presence of and

ωy

bint k

on a given edge

int ωx bint k [ωy bk ].

While

ax , ay , ωx ,

are chosen and xed in Step 1, the trac load

k

(which will be determined in Step

6 (a)

pensate for this overestimate.

The diagrams of Fig. 3

βωx bint k and int [βωy bk ] such that even innitely large values of αx and αy cannot compensate for the caused instability and staconrm the existence of threshold values for

bilize systems (8) and (9). It is important to emphasize that the stability diagrams of Fig. 3,

which later will be used for pre-

dicting thresholds in large networks, need to be calculated through simulation of two three-dimensional systems (8) and (9) driven by the synchronous solution

(x(t), y(t), z(t)). Here, the terms βωx bint k

and

βωy bint k

are

treated as single, aggregated control parameters. Therefore, the threshold value for

αkx [αky ]

required to stabilize

the auxiliary system (8) [(9)] for a given edge

k

with

bint k

can simply be read o from Fig. 3. To better quantify this dependence to be used in predicting synchronization

(b)

thresholds in networks of Lorenz oscillators, we approximate the stability boundary in Fig. 3(a) by the exponential function


αx = 0.4645 exp 2.408βωx bint . k

(10)

The stability diagrams of Fig. 3 along with Fig. 2 account for the role of the individual oscillators composing the 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 master stability function in single-layer networks [23] and help in solving, once and for all, the question of stability for synchronization in two-layer networks involving the Lorenz oscillator through the criterion (3), where the role of multilayer network topologies is assessed via the FIG. 3. Stability of the auxiliary systems (6) and (7) for coupled Lorenz oscillators. Yellow depicts instability of the origin while purple indicates its stability. The dependence of the staint bilizing term, αx , on bk , ωx , and βx is estimated by the ex
int ponential function 0.4645 exp 2.408βx ωx bk (dashed curve), and is used to predict synchronization thresholds in networks of Fig. 1 and Fig. 4.

calculation of pure graph theoretical quantities as shown in the next step.

Step 3: Calculate trac loads bxk , byk , and bint k . This calculation is similar to that of the connection graph method for single-layer networks [30], except that the trac load should be partitioned into three groups: intralayer trac loads

bxk

and

byk

within the

layer, respectively, and interlayer trac load 3) controls the choice of

αkx

and

αky .

Thus, if the edge

is highly loaded, then the contribution of the positive term

+ωx bint k Yk

in the

Yk -equation

of system (8) can-

not always be compensated by increasing

Xk -equation.

−αkx Xk

in the

Typically, this happens when the positive

term exceeds the proper negative linear terms such as

−Yk

(technically, through a combination of terms in the

Routh-Hurwitz criterion). Therefore, the auxiliary system (8) can become unstable, independently of how large the stabilization coecient

αkx

is. The same argument re-

lates to the destabilization of the auxiliary system (9) via the positive term

+ωy bint k Xk .

x and y bint bek

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 vertices i, j, and determine their lengths |(Pij |, the number of edges in each Pij . Then, for each edge k of the x (y ) y x layer graph, we calculate the sum bk (bk ) of the lengths of all Pij that are composed of only x (y ) edges and pass through k. We repeat the same procedure to calculate the int sum bk of the lengths of all Pij that contain both x and y edges and pass through k. These constants depend on the choice of the paths Pij . Usually, one uses the shortest path from vertex i to vertex j. Sometimes, however, a dierent choice of paths can lead to lower bounds [34]. We use the six-node multilayer network of Fig. 1(b) as

The complex relationship between these terms in re-

an example for calculating

bxk , byk ,

and

bint k .

To compute

gard to stabilizing (8) and (9) is shown in Fig. 3. Notice

all of the paths that pass through a given edge, it is

βωx bint k

β

int

and [βωy bk

] axes. Be-

recommended that the reader algorithmically nds the

cause of the Cauchy-Schwarz inequality used in the proof,

shortest path between every pair of oscillators, and take

bint k

note of the paths that go through edge

the coecient

on the

provides an overestimate for the terms added to the

auxiliary system and the scaling factor

β is added to com-

k and dierentiate x or y layers

the paths that entirely belong to only the

7

and the ones that contain a combination of

x and y edges.

As a result, we can nd each edge's trac 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 | = 15 bx35 = |P15 | + |P25 | + |P35 | = 6, bx46 = |P16 | + |P46 | = 5, int int by56 = |P56 | = 1, bint 12 = 0, b23 = 0, b34 = 0, int int b35 = |P36 | = 2, b46 = |P45 | = 2, bint 56 = |P36 | + |P45 | = 4. (11) Note that the maximum interlayer trac load on this network is fairly low and due to our choice of paths is

bint 56 = 4

although it could have also been minimized to

zero, provided that all paths to node 6 bypass edge

56.

At the same time, the interlayer trac load in the net-

(3) with constants

ωy = 0.50

ax = 7.20, ay = 2.63, ωx = 5.00 and αkx and αky are

chosen above. The constants

read o from the diagrams of Fig. 3(a) and Fig. 3(b), respectively.

As the stability system (8) is much more

bint k

sensitive to the changes in

than (9) (cf. the onset of

cij in x layer largely dominate over dij .

instability in Figs. 3(a-b)), the threshold values for the criterion (3) for the

Thus, since the synchronization threshold for the entire network (2) with uniform coupling the maximum of the thresholds

cij

c = d is dened by dij for each edge

or

of the multilayer graph, the maximum threshold values predicted by the method and depicted in Fig. 1(d) are the ones corresponding to

x edges with coupling c. These

threshold values are calculated using the numerically assisted modication of (3)



  1 x int x int c > max ck = γax · bk + βωx · bk + αk (βωx bk ) , k n

work of Fig. 1(c) is signicantly higher since there are

(12)

αkx

2-3 when traveling from nodes 1 and 2 to nodes 4, 5, and 6. int For the same choice of shortest paths, we get b23 = 19. y x The remaining bk , bk for the network of Fig. 1(c) can be

where

calculated similarly to (11).

to compensate for the conservative nature of the connec-

no alternatives to go around the bottle neck edge

Step 4: Putting pieces together to solve the puzzle.

Given

the stability diagram of Fig. 3 with abrupt threshold dependences of

αkx

and

αky

on increasing interlayer trac

bxk , the eect of synchrony breaking when a highly loaded x edge is replaced with a better pairwise stabilizing y (see Sec. III ) is no longer a puzzle and directly load

is dened by the stability diagram of Fig. 3(a)

via the approximating function (10) for each edge

ax , ωx ,

bint k

and

k,

and

are determined in Steps 1-3. Notice the

presence of an additional scaling factor

γ

which is chosen

tion graph stability method when the network is entirely

x variable. γ scales down the term ax to match the coupling needed to synchronize the sixn node network of Fig. 1(a) with only x edges. The scal-

coupled through the

ing factor

β

is then chosen for the network of Fig. 1(b)

with the lowest

bint = 4 k

to match the actual synchro-

follows from the application of our stability method. Ac-

nization threshold and then kept constant for predicting

tually, in a historical retrospective, we rst developed

the thresholds in the other six-node networks with one

the general method that revealed this and other highly

replaced

counterintuitive eects due to the multilayer structure

thresholds are fairly close to the actual ones, and the cri-

and then constructed the network examples.

x

edge.

Figure 1(d) shows that the predicted

To make

terion (12) correctly predicts an increase or decrease of

the presentation more appealing before it becomes too

the coupling threshold for each six-node network and ul-

technical, we have decided to put forward the motivat-

timately predicts synchrony break for the network with

ing example.

the replaced

As our exhaustive study of various net-

x

edge

2-3.

work congurations suggests, we hypothesize that six-

As Fig. 1(d) indicates that when lightly loaded edges

node networks of Fig. 1 are minimum size networks of

(edges with fewer chosen paths passing through them)

Lorenz oscillators that exhibit the synchrony breaking

are replaced, the eect on the synchronization stability

phenomenon.

is fairly small.

To test the predictive power of our approach with the

As discussed in the description of the

motivating example, the replacement of

y

x

edge

5-6

with

constants identied in Steps 1-3, we perform a systematic

a

study of how one edge replacement, in which we replace

the synchronization threshold. According to our stabil-

only one

x

edge in the single-layer,

Fig. 1(a) with a

x-coupled

network of

y edge, aects synchronization.

The edge

replacement is performed in the order of the increasing interlayer trac load on this edge,

bint k .

After comput-

edge improves synchronization by slightly lowering

ity criterion (3), this happens due to a slight decrease in the trac load on the bottleneck edge

bx23

(see (11)),

compared to the original network of Fig. 1(a) with all edges where one additional path

P16

x

goes through edge

The two multilayer networks of Fig. 1(b) and

2-3. As a result, it decreases the contribution of the domx inating term ax bk in (3). At the same time, the contribuint x tion of the other factors ωx bk and αk remain insignifx icant, especially due to the fact that αk still lies on a

Fig. 1(c) with the drastically dierent synchronization

at part of the approximating curve (10) before this ex-

properties are two instances of this replacement process.

ponential curve takes o at larger values of

Fig. 1(d) presents the actual synchronization threshold

other hand, such a replacement of the bottleneck node

ing the coupling threshold required to synchronize the new network, this edge reverts back to being an This results in multiple networks of ve

y

edge.

x

x

edge.

edges and one

values (blue solid line), the interlayer trac loads

bint

2-3

bint k .

On the

in the network of Fig. 1(c) signicantly increases the

bint k ,

requiring innitely large

x α23

(black line) calculated similarly to (11), and the thresh-

intralayer trac load

old values predicted by the numerically-assisted criterion

to stabilize the stability system (8) and causing synchro-

8 (a)

nization to break.

VI.

LARGER NETWORKS

To demonstrate that similar synchrony breakdown phenomena occur in larger networks and can be eectively predicted by our method, we consider a

20-node

network of Lorenz (and then double-scroll) oscillators de-

(b)

scribed in Fig. 4(a). The network is initially coupled entirely through the

x variable.

To test our prediction that

replacing edges with a high trac load can make the network unsynchronizable, we index the edges according to their

bxk .

Edges similar to edge

10-12 have very few paths bxk

that pass through them, and subsequently have a low (and in turn,

bint k ,

shown as the black curve in Fig. 4(c)).

We successively replace in Fig. 4(a)) with

y

x

edges (denoted by black edges

edges (denoted by gray edges in Fig.

(c)

4(b)), according to this ordering until the network is com-

y 10-12

pletely connected through range from

0

(for edge

edges.

bint k index 1

The values of

with edge ranking

(see Fig. 4(c)), bypassed by all chosen interlayer paths)

100 − 400 for highly 3-5 for which every

to

edge

loaded edges (for example, for

3 of the x layer y -layer must pass through

path from node

graph to any other node in the it).

A.

Networks of Lorenz oscillators

x-coupled c ≈ 86.95. with y edges,

The coupling necessary to synchronize the Lorenz network (2) described in Fig. 4(a) is As outlying, low trac edges are replaced

there is almost no eect on the threshold for the coupling strength required to synchronize the network, evidenced

FIG. 4. Eect of successively replacing

y

x

coupling edges with

20-node network of x-coupled network before

coupling edges on synchronization in a

Lorenz oscillators (2). (a) Original

replacing edges according to their trac load. (b) Snapshot of the multilayer network before the edge

3-5,

labeled as the

by the lack of change in the actual coupling threshold

13-th

for the rst eight edges replaced in Fig.

As suc-

The replacement of this critical edge (red) yields the break-

cessively more loaded edges are replaced in the network

down of synchronization in the network. Further successive

(indicated by the dramatic increase in

4(c).

bint ),

the network

becomes more dicult to synchronize, until edge 13 (edge

3-5 which is depicted in red in Fig.

4(b)) is replaced. Af-

ter which, synchronization is no longer feasible for the network for any additional edge replacement, until edge 25 (edge 14-15 in Fig. 4). Replacing edge 25, shown in Fig. 4(b) corresponds to nishing the successive edge re-

edge according to the trac load ranking, is replaced.

replacement of remaining

x

edges (gray) with higher trac

load preserves the instability of synchronization, until all

25

edges have been replaced with a y edge, yielding a single-layer y -coupled network with bint = 0 that is able to synchronize again. (c) Actual (blue solid line) and predicted (blue dotted line) threshold for the coupling strength required to synchronize the network after the

i-th x

edge has been successively

placement process, and results in a graph identical to the

replaced with a y edge. The black solid line depicts the interint layer trac b for the respective edge. The predicted cou-

one in Fig. 4(a), but in which all of the edges represent

pling thresholds are computed from (3) using the exponential

y

t in Fig. 3 and scaling factors

coupling (gray) instead of

x

coupling (black). This re-

β = 0.0025

and

γ = 0.5993.

inforces our trac load predictions for the breakdown of synchrony in two ways: (i) after enough highly loaded edges are replaced (even with a normally favorable cou-

As in the six-node example of Fig. 1(a), we have ob-

pling type), the network can no longer synchronize for

tained a good t in Fig. 4(c) which only focuses on plac-

any

coupling strength, (ii) replacing only one edge that

ing the stability conditions on

αx

x

edges in (3), because

x stability αy in the

is very highly loaded can make the network unsynchro-

the stability term

nizable, evidenced by the network having no synchroniz-

system (8) must be signicantly higher than

ing coupling value even when all but one edge has been

y

stability system (9) (compare Figs. 3(a-b)).

We use

replaced (see edge index 24 in Fig. 4(c)).

the same criterion (12) with the same constants

ax , ωx ,

required to stabilize the

9

and

αkx

to predict the synchronization threshold and only

need to identify the trac loads tors

γ

and

β

bint k

and the scaling fac-

for a better t, once and for all variations

of the multilayer network topologies used in Fig. 4(c). In contrast to the six-node network example where trafc load

bint k

computing

can be easily calculated by hand as in (11),

bint k

for the

20-node

or larger networks is a

laborious task which was performed by an algebraic algorithm, implemented as MATLAB code and given in the Supplement. While the values of

bint k

heavily depend

on the choice of paths from one node to another, our algorithm uses the natural choice of the shortest paths, computed via Dijkstra's algorithm [59]. Optimizing the choices of not necessarily shortest paths that distribute trac loads on edges more equally may yield even better predictions and ts.

FIG. 5. Eect of successively replacing on the synchronization threshold in a

B.

x edges with y edges 20 node network of

double-scroll oscillators. The network topology, edge replace-

Networks of double scroll oscillators

ment process, and notations are identical to those in Fig. 4(ab). Notice the same location of critical links (edges

To illustrate the generality of synchrony break phenomenon when good but highly-loaded links go bad,

13

to

24)

whose replacement leads to synchrony breaks as in the network of Lorenz oscillators (cf. Fig. 4(c)).

we apply our numerically-assisted method to networks (1), comprised by chaotic double-scroll oscillators [58]

x-coupled n P

x˙ i = κ(yi − xi − h(x)) +

cij xj ,

y˙ i = xi − yi + zi +

y

edges. No-

in the network of Lorenz oscillators, suggesting that crit-

j=1 n P

network has been replaced with

tably, the synchrony break occurs at the same edge as (13)

dij yj ,

ical edges whose replacement hampers synchronization are mainly controlled by the network multilayer topol-

j=1

ogy rather than the individual properties of the intrin-

z˙i = −λyi − µzi , i = 1, ..., n,

sic oscillators, provided that the oscillators possess similar synchronization properties as the Lorenz and double

with

scroll oscillators.

  m1 (x + 1) − m0 h(x) = m0 x  m (x − 1) + m 1 0

x < −1 −1 ≤ x ≤ 1 x>1

The solid curve in Fig. 5 displays the synchronization thresholds,

κ = 10, m0 = −1.27, m1 = µ = 0.038.

and standard parameters

−0.68, λ = 15,

and

calculated via the stability criterion

ax = 5.94 × 2 = 11.88, ωx = 2.0, β = 0.0041, γ = 0.282, and the approximating function αx = int 1.556 exp(3.711βωx bint k ) with the same trac loads bk (12) with

shown in Fig. 4.

This approximating function is ob-

Similarly to networks of Lorenz oscillators (2), a pair of

tained from a stability diagram for coupled double scroll-

double-scroll oscillators (13) can be synchronized through

oscillators which is computed similarly to Fig. 3 and dis-

either the

x

or

y

variable, and the minimum coupling

plays a similar threshold eect as in Fig. 3 [not shown].

y-

As in the Lorenz oscillator case, the auxiliary stability

strength required for synchronization in a two-node coupled network,

d∗ = 1.16

is much lower than the

coupling threshold in the two-node

x-coupled

network,

c∗ = 5.94.

bint k

αx

is much more sensitive to increasing

than the stability system (5) for

αy ,

therefore one

can only evaluate the stability condition (12) for the

In Fig. 5, we apply our method to predict the synchronization thresholds in the

20-node

network of Fig. 4 as

in the same network of Lorenz oscillators. When successively replacing

system (4) for

x

coupling

c

x

to identify a bottle-neck for the synchroniza-

tion threshold in the entire network. Going back to the puzzle example, we have also per-

edges in the network, there is initially

formed a similar analysis of the six-node network of Fig. 1

a decrease in the coupling threshold for synchronization,

where the Lorenz oscillators are replaced with the double-

when peripheral edges or edges in highly connected re-

scroll oscillators [not shown].

gions of the graph with low trac loads with more favorable

y

bint k

are replaced

Remarkably, this analy-

sis indicates the same qualitative phenomena when the

edges that provide better pairwise

replacement of the lightly loaded edge

convergence to synchronization. Then, as with the net-

ers the synchronization threshold from

work of Lorenz oscillators, when edge replaced with a

y

13

(edge

3-5)

is

edge, synchronization is no longer at-

tainable. Synchronization then returns when the entire

5-6 slightly lowc = 13.57 in the

x-coupled single-layer network of Fig. 1(a) to c = 13.36, and predicts the breakdown of synchrony when edge 2-3 is replaced as in the Lorenz network. original

10

We have also simulated series of other works (2) and then networks (13) where

x

20-node

all

net-

and therefore can handle multilayer networks. Originated

oscillators

from the connection graph method for synchronization in

y

cou-

single-layer networks [30], our method combines stability

pling only connected some of the oscillators. In contrast

theory with graph theoretical reasoning. Two key ingre-

to the networks of Fig. 1 and Fig. 4 where the critical

dients of the method are (i) the calculation of stability

were connected via

layer graphs, whereas the

highly-loaded links separate the network into disjoint

y

x

diagrams for the auxiliary

s-dimensional

system which

graph components, these networks do not show

indicate how the dynamics of the given oscillator com-

the eect of synchrony breaking as any pair of nodes is

prising the network can be stabilized via one variable

and

x

graph such that

corresponding to one connection layer when an instabil-

can be made strong enough to

ity is introduced via the other variable from another con-

stabilize synchronization. However, the synchronization

nection layer and (ii) the calculation of trac loads via

thresholds in such networks depend on the location of

a given edge on the multilayer connection graph. All to-

coupled directly or indirectly via the the coupling strengths

y

c

edges in a nonlinear fashion. In support of this

gether, these quantities allow for predicting the synchro-

claim, we draw the reader's attention to the six-node ex-

nization threshold and identify critical links that control

added

ample of Fig. 1(b) where the

x

graph connects all six

synchronization in the original, potentially large, multi-

5-6 with an y edge low-

layer network.

nodes and the replacement of edge

ers the synchronization threshold. On the contrary, the replacement of the

x

edge

3-5

with an edge

preserves the connectedness of the

x graph,

y,

which still

increases the

synchronization threshold, as predicted by the method (see Fig.

x

1(c)). The

20-node

networks with connected

graphs yield similar eects. To avoid repetition, these

results are not shown.

Using the method, we have discovered striking, highly unexpected phenomena not seen in single-layer networks. In particular, we have shown that replacing a link with a light interlayer trac load by a stronger pairwise converging coupling via another layer may improve synchronizability, as one would expect. At the same time, such a replacement of a highly loaded link may essentially worsen synchronizability and make the network unsynchronizable, turning the pairwise stabilizing good link

VII.

CONCLUSIONS

into a destabilizing connection (a bad link). The critical links whose replacement can lead to synchrony break

While the study of synchronization in multilayer dy-

are typically the ones that connect the layers such the

namical networks has gained signicant momentum, the

oscillators from two layers become coupled through the

general problem of assessing the stability of synchroniza-

intrinsic, nonlinear equations of the individual oscillator

tion as a function of multilayer network topology re-

that correspond to a relay node passed by the only path

mained practically untouched due to the absence of gen-

from one layer to the other. As a result, the intrinsic dy-

eral predictive methods. The existing eigenvalue meth-

namics of the individual node oscillator plays a pivotal

ods, including the master stability function [23], which

role in the stability of synchronization. In this paper, we

eectively predict synchronization thresholds in single-

have limited our attention to Type I oscillators such as

layer networks cannot be applied to multilayer networks

the Lorenz and double scroll oscillators that yield global

in general. This is due to the fact that the connectivity

synchronization that remains stable in a single-layer net-

matrices corresponding to two or more connection layers

work once the coupling exceeds a critical threshold. Re-

do not commute in general, and therefore, the eigenvalues

markably, when used in a multilayer network, both oscil-

of the connectivity matrices cannot be used. Therefore,

lators have indicated similar synchronization properties,

synchronization in multilayer networks is usually studied

suggesting that the location of critical edges in the con-

on a case by case basis either via (i) full-scale simulations

sidered network may remain unchanged for other Type

of

all

transversal Lyapunov exponents of the

(n − 1) × s-

I oscillators.

While our rigorous method for proving

dimensional system of variational equations [55], where

global synchronization is only applicable to Type I os-

n is the network size and s is the dimension of the intrin-

cillators, its semi-analytical version can be modied to

sic node dynamics, or more eectively via (ii) simulta-

handle Type II networks, including multilayer networks

neous block diagonalization of the connectivity matrices

of Rössler systems [23]. This modication is a subject of

[54] which in some cases can reduce the problem of as-

future study.

sessing synchronization in a large network to a smaller

To gain insight into the determining factors for the

network which, however, contains positive and negative

emergence of synchrony breaking, without potential con-

connections, including self loops such that the exact role

founds associated with the interplay between multiple

of multilayer network topology and the addition or ex-

layers and direction of links, we have considered two-layer

change of edges remains unclear.

undirected networks of identical oscillators.

However,

In this paper, we have made a breakthrough in un-

the extension of our results to multiple layers, directed

derstanding synchronization properties of multilayer net-

networks and non-identical oscillators is fairly straight-

works by developing a predictive method, called the Mul-

forward and will be reported elsewhere.

tilayer Connection Graph method, which does not rely on

the extension of our method to directed networks can be

calculations of eigenvalues of the connectivity matrices,

performed by adapting the generalized connection graph

In particular,

11

DF

tation is simply a compact form of the mean value theo-

weights according to the mean node unbalance.

rem,

Our method can also be modied to handle multi-

Therefore, the dierence system (15) can be rewritten in the form

of pairwise repulsive inhibition to single-layer excitatory networks can promote synchronization [43] and (ii) com-

This no-

f (B) − f (A) = f 0 (C)(B − A), applied to the vector functions F(xj ) and F(xi ), where the Jacobian DF is evaluated at some point C ∈ [xi , xj ].

tory, and inhibitory synapses which exhibit a number of counterintuitive synergistic eects: when (i) the addition

Jacobian matrix of

F.

where

directed edges are symmetrized and assigned additional

layer neuronal networks connected via electrical, excita-

is the

s×s

method [31, 60] for single-layer directed networks, where

1 R

˙ ij = X

bined electrical and inhibitory coupling can induce syn-

 DF(βxj + (1 − β)xi )dβ Xij +

chronization even though each coupling alone promotes

0 n P

an anti-phase rhythm [61]. Our method promises to allow

k=1

{cjk P Xjk − cik P Xik + djk LXjk − dik LXik }, i, j = 1, ..., n.

an analytical treatment of these eects in large neuronal networks which has been impaired by the absence of rig-

(16)

orous methods that can handle excitatory, inhibitory, and

The rst term with the brackets yields instability via the divergence of trajectories within the individual, pos-

electrical neuronal circuitries simultaneously.

sibly chaotic oscillators. The second (summation) term, which represents the contribution of the network connecVIII.

tions, may overcome the unstable term, provided that

ACKNOWLEDGMENTS

the coupling is strong enough.

This work was supported by the National Science Foundation

(USA)

under

Grant

No.

DMS-1616345

and the U.S. Army Research Oce under Grant No. W911NF-15-1-0267.

Notice that the stability of system (16) is redundant as

(n−1)n/2 non-zero dierences Xij Xii = 0 which can be disAt the same time, there are only n − 1 linearly

it contains all possible along with regarded.

n

zero dierences

independent dierences required to show the convergence between

n variables Xij . However, this redundancy propXij are a key

erty and the consideration of all non-zero IX.

ingredient of our approach which allows for separating

APPENDIX: THE PROOF

the dierence variables later in the stability description, In this appendix we derive the Multilayer Connection Graph method and prove Theorem 1.

derive the conditions of global asymptotic stability of the synchronization manifold

M

without diagonalizing the connectivity matrices. We strive to nd conditions under which the trivial

Our goal is to

in the system (1).

To

achieve this goal and develop the stability method, we fol-

xed point

new stability argument is used. In the network model (1) we introduce the dierence variable

Xij = xj − xi , i, j = 1, ..., n,

(14)

We introduce the following terms a

3×3

Subtracting the

i-th

M.

equation from the

j -th

equation

in system (1), we obtain the equations for the transversal stability of

M

where

A

is

 ax P =       ay L = Aij =     =  K

diag(ax , 0, 0) if oscillators

i and j x layer C diag(0, ay , 0) if oscillators i and j belong to y layer D diag(ωx , ωy , 0) if oscillators i and j belong to dierent layers C and D , belong to

(17) where constants

ax , ay , ωx ,

and

ωy

are to be determined.

We add and subtract additional terms trix

Aij

Aij Xij

with ma-

dened in (17) from the stability system (16) and

obtain

˙ ij = F(xj ) − F(xi ) + X

n P

{cjk P Xjk − cik P Xik +

˙ ij = X

k=1 (15) To obtain the explicit dependence of

F(xj ) − F(xi )

we introduce the following vector notation

 1  Z F(xj ) − F(xi ) =  DF(βxj + (1 − β)xi )dβ  Xij , 0

1 R

 DF(βxj + (1 − β)xi )dβ − Aij Xij + Aij Xij

0

djk LXjk − dik LXik }, i, j = 1, ..., n.

Xij ,

Aij Xij ,

matrix, such that

whose convergence to zero will imply the transversal stability of the synchronization manifold

of system (16) is

This amounts to nding conditions for

global stability of synchronization in the network (1).

low the steps of the proof of the connection graph method [30]. The concept is similar, up to a certain step where a

{Xij = 0, i, j = 1, ..., n}

globally stable.

on

+

n P

{cjk P Xjk − cik P Xik + djk LXjk − dik LXik }.

k=1 (18) The introduction of the terms

Aij Xij

allows for obtain-

ing stability conditions of the trivial xed point

i, j = 1, .., n

in two steps.

Xij = 0, −A

Note that the matrix

contributes to the stability of the xed point and can compensate for instabilities induced by eigenvalues with

12

nonnegative real parts of the Jacobian

DF. This can be ax , ay , ωx , and ωy . At

synchronized, our approach based on the calculation of

achieved by increasing parameters

ax , ay ,

the same time, the instability originated from its posi-

The proof of global stability in (20)-(22) and deriva-

tively denite counterpart, matrix by the coupling terms through

cij

+A, can dij .

be damped

and

Step I. We make the rst step by introducing the following auxiliary systems for

i, j = 1, ..., n

is thus limited to this class of oscillators.

tion of bounds

ax , ay ,

and

ωx

and

ωy

involves the con-

struction of a Lyapunov function along with the assumption of the eventual dissipativeness of the coupled system. Therefore, before advancing with the study of larger networks (1), one has to prove that globally stable syn-

 1  Z ˙ ij =  DF(βxj + (1 − β)xi ) dβ − Aij  Xij . X

chronization in the simplest (19)

x-coupled

0

Aij

can take three dierent values,

i

and

j

depending on

both belong to the

x

y if i

or

graphs, or belong to dierent graphs, for example,

ay , ωx ,

ωy

and

Upper

can be derived similarly.

Having obtained the bounds

ax , ay , and ωx and ωy , and

(20)- (22), we can now take the second step in analyzing the full stability system (18).

Step II.

The bounds

ax , ay ,

and

ωx

and

ωy

that sta-

graph (see

bilize the auxiliary systems (20)-(22) reduce the stability

Therefore, we have three types of the auxiliary

analysis of system (18) to the following equations by ex-

belongs to the (17)).

(x, y)-coupled, ax for

therefore proving the stability of the auxiliary systems

terms are removed. whether oscillators

and

Lorenz oscillators was given in [30].

bounds for This system is identical to system (18) where the coupling

x-, y -,

two-oscillator networks is achievable. The bound

x graph, and j

belongs to the

y

cluding the term in brackets

systems

1 R

˙ ij = X

n ˙ ij = Aij Xij + P {cjk P Xjk − cik P Xik + djk LXjk − X

 DF(βxj + (1 − β)xi ) dβ − ax P Xij

0 if i and

j

both belong to

x

k=1

(20)

dik LXik }, i, j = 1, ..., n.

layer,

(23)

Aij Xij , which contains the ωx and ω, is destabilizing and

Note that the positive term

˙ ij = X

1 R

 DF(βxj + (1 − β)xi ) dβ − ay L Xij

0 if

i

and

j

both belong to

y

upper bounds (21)

layer,

ax , a y ,

and

must be compensated for by the coupling terms. To study the stability of (23) we introduce a Lyapunov function of the form

n

˙ ij = X

1 R

V =

 DF(βxj + (1 − β)xi ) dβ − K Xij

0 if i and

j

(22) where

each belong to dierent layers.

I

is an

s×s

n

1 XX T X · I · Xij , 4 i=1 j=1 ij

(24)

identity matrix.

Its time derivative with respect to system (23) becomes Remarkably, the auxiliary system (20) coincides with the dierence system for the global stability of synchronization in a two-oscillator network (1) with only where

ax

x coupling,

plays the role of the double coupling strength

that guarantees the stability (see [30] for a detailed discussion on this relation). Similarly, the stability of auxiliary system (21) implies

V˙ =

1 2 1 2 1 2

n P n P i=1 j=1 n P n P

XTij Aij Xij − n P

i=1 j=1 k=1 n P n P n P i=1 j=1 k=1

{cjk XTjk IP Xjk − cik XTik IP Xik }− {djk XTjk ILXjk − dik XTik ILXik }.

global stability of synchronization in the two-oscillator network (1) with only

y

coupling, where

coupling strength of the

y

ay

connection.

is the double

(25) We need to demonstrate the negative semi-deniteness

Lastly, the sta-

of the quadratic form

bility of auxiliary system (22) guarantees globally stable

rst sum simplies to

V˙ .

synchronization in the two-oscillator network with both

x

y coupling, where a combination of constants ωx and ωy , present in K , is a combination of the double coupling strengths of x and y connections that is sucient to induce stable synchronization in the two-oscillator x and y coupled network. and

S1 =

ax , ay ,

and

ωx

and

ωy

that make the

origin of the auxiliary systems (20)-(22) stable.

This

amounts to proving global synchronization in the three

x-, y -,

and

(x, y)-coupled

networks that are composed of

two oscillators. As only Type I oscillators can be globally

n−1 n XX

= 0, X2ij = X2ji ),

Aij X2ij .

the

(26)

i=1 j>i This sum is always positive denite and its contribution must be compensated for by the second and third sums

Therefore, our immediate goal is to nd upper bounds on the values of

2

As (Xii

S2 = − 21 S3 = − 21

n P n P n P

{cjk XTjk IP Xjk − cik XTik IP Xik },

i=1 j=1 k=1 n P n P n P

{djk XTjk ILXjk − dik XTik ILXik }.

i=1 j=1 k=1

(27)

13

Due to the coupling symmetry, the two terms in both

S2

and

dices

i

S3

can be made identical by exchanging the in-

j

by

in the second terms such that

S2 = − S3 =

n P

n P

S2 = −

side (RHS) of (33) correspond to all possible dierences (28)

djk XTjk ILXjk .

Xjj = 0,

n n−1 n P P P

i=1 k=1 j>k n n−1 n P P P i=1 k=1 jk n n−1 n P P P

Again, exchanging and and

j

and

dened by edges of the connection graphs. Hence, to get rid of the presence of the dierences

we obtain

Xij

and therefore

work model (1), we express the dierences on the RHS via the dierences on the LHS such that we will be able

cjk XTji IP Xjk −

to cancel them. So far, we have closely followed the steps in the derivation of the connection graph method [30] for single-layer (29)

S3 = −

between pairs of oscillators that might or might not be

nd the conditions explicit in the parameters of the net-

cjk XTji IP Xjk ,

i=1 k=1 jk n n−1 n P P P i=1 k=1 j>k

Since

cjk (XTji

+

(30)

djk (XTji + XTik )ILXjk .

  XTji + XTik = xTi − xTj + xTk − xTi = XTjk ,

we ob-

tain

S2 = − S3 = −

n−1 P

n P

k=1 j>k n−1 n P P k=1 j>k

ncjk XTjk IP Xjk , (31)

ndjk XTjk ILXjk .

Returning to the derivation of the Lyapunov function (25) and combining the sums

S1 , S2 and S3 V˙ ≤ 0 :

yields the

condition which guarantees that

S1 +S2 +S3 =

n−1 X

n X

Xij corre˜ k , k = 1, ..., m X and the dierences Xij corresponding to edges of the y ˜ k , k = 1, ..., l. Recall that m and l are the graph by Y number of edges on the x and y graphs, respectively. In ˜ k which addition, let Xk be a scalar from the vector X indicates the scalar dierence between xi and xj , corresponding to an edge on the x graph. Similarly, let Yk be ˜ k , dened by the corresponda scalar from the vector Y ing yi and yj . Using these notations, the dierences Xij on the RHS will now dene the scalars Xij = xj − xi and Yij = yj − yi , . If the inequality (33) is satised in terms of xi and yi (via the scalar dierences Xk and Yk ), then it will also be satised for the remaining scalar zi . Recall that (xi , yi , zi ) are the scalar coordinates of the Denote on the LHS (33), the dierences

XTik )IP Xjk ,

sponding to edges of the

Using this notation, we can rewrite (33) as follows

XTij I[Aij −ncij P

−ndij L]Xij < 0. (32)

The most remarkable property of this condition is that

m P

n

the condition in terms of

Xij .

ay

n P

i=1 j>i,(i,j)∈D

Yij2 +

n−1 P

n−1 P

n P

i=1 j>i,(i,j)∈C n P

i=1 j>i,(i,j)∈C,D /

2 Xij +

2 [ωx Xij + ωy Yij2 ], (34)

where

[cij XTij IP Xij + dij XTij ILXij ] i=1 j>i n−1 n P P > XTij IAij Xij . i=1 j>i

and

dk = dik jk .

Here, the RHS of (34)

(cf. (33) and (20)-(22)). The rst sum on the RHS is

x graph C , D, whereas

composed of the dierences that belong to the

> (33)

Notice that the left-hand side (LHS) of this inequality

Xij

ck = cik jk

ables into three groups, according to the coecients of

Aij

n P

contains only the dierences

k=1

dk Yk2 > ax

has three terms, obtained by splitting the dierence vari-

including linearly dependent ones.

The condition (32) nally transforms into

n

l P

This is because we chose

to consider the redundant system with all possible dier-

n−1 P

ck Xk2 + n

k=1 n−1 P

we are able to eliminate the cross terms and formulate

Xij ,

graph by

individual oscillator, composing the network (1).

i=1 j>i

ences

x

between the oscillators

the second sum corresponds to the

y

graph

the third sum identies the dierences between the oscillators which belong to dierent graphs such that and

j∈D

i∈C

or vice versa.

To recalculate the dierence variables of the RHS via the variables

Xk

and

Yk ,

we should rst choose a path

14

i to oscillator j for any pair of oscillators (i, j). We denote this path by Pij . Its path length |Pij | is

to the rst sums on the RHS. In the simplest case where

the number of edges comprising the path. The important

both

Note that the two sums on the LHS of (36) correspond

from oscillator

property of the path

Pij

is that if, for example, it passes

1, 2, 3, and 4, then the X14 = x4 −x1 = (x4 −x3 )+(x3 − x2 ) + (x2 − x1 ) = X12 + X23 + X34 , where the dierences X12 , X23 , and X34 correspond to the edges and the path length |P14 | = 3.

C

D graphs are connected such that each graph n oscillators, bint k can always be set to 0, since

and

couples all

through oscillators with indices

there are always paths between any two nodes that en-

corresponding dierence

tirely belong to either the

i

and

j;

discussed in [34] for single-layer networks. Once the choice of paths is made, we stick with it and start recalculating the dierence variables on the RHS of (34) via

Xk

Yk .

and

A potential problem is that we

Xij , but with their 2 squares Xij , coming from the calculations of the derivative of the Lyapunov function (25). Therefore, we have have to deal not with the variables

to apply the Cauchy-Schwartz inequality; applied to the

2 2 above example, it yields X14 = (X12 + X23 + X34 ) ≤ 2 2 2 3(X12 + X23 + X34 ). Notice the appearance of the factor 3, indicating the number of edges comprising the path.

Xij !2

Similarly, for any dierence

2 Xij =

Yij2

Yij

and

we have

P Xk ≤ |Pij | Xk2 , k∈Pij k∈Pij !2 P P Yk ≤ |Pij | Yk2 , P

=

k∈Pij

(35)

necessary. RHS of (34), we obtain the following condition

n

k=1 l P k=1

ck Xk2 + n

[ay byk

+

l P k=1

m P

dk Yk2 >

2 ωy bint k ]Yk

+

k∈D

+

P k∈C

2 [ωy bint k ]Yk ,

|Pij |

is the sum of the lengths

x

|Pij |

edge

k.

x

C byk =

graph

Similarly,

is the sum of the lengths of all chosen

j>i; k∈Pij ∈D paths which belong to the edge

k.

y

int Finally, bk

graph

=

D and n P

y

y

|Pij |

implications of the stability method to specic networks discussed in Sections V and VI. A major stability problem, associated with the third and fourth terms, is rooted in the fact that, for exam-

2 [ωx bint k ]Xk contains the dierk∈D ence variables Xk that correspond to the edges of the m P y graph. As a result, the rst sum n ck Xk2 on the k=1 LHS which contains the variables Xk that correspond to x edges, cannot compensate the third sum on the RHS ple, the third sum

P

as they belong to dierent graphs and therefore can-

l P

At the same time, the second sum

dk Yk2 on the LHS does belong to the

k=1 contains the variables

Yk

and not

Xk

y

graph but

needed to handle

the third sum. The same problem relates to the fourth

2 [ωy bint k ]Yk which contains Yk variables, correk∈C sponding to the wrong graph (x graph). sum

P

not have means on the LHS to compensate for the trou[But

this point. In fact, the only place to borrow these terms from is the auxiliary stability systems (20) and (21) as they do contain the desired variables to the right graphs (the

x xy an x

k

which may be

x

Xk

and

and

y

Yk ,

corresponding

graphs, respectively).

Therefore, we need to go back and modify the auxiliary systems (20) and (21) as follows

˙ ij = X

1 R

DF(βxj + (1 − β)xi ) dβ − [ax + αkx ]P +  +ωx bint k L Xij if i, j ∈ x edge k , 0

(38)

is the

edges and belong to the combined, two-layer

edge.

is non-zero, the third and forth sums are

much more complicated but yields a number of surprising

j>i; k∈Pij ∈(C,D)

graph and go through a given edge or

bint k

always present on the RHS. This makes the argument

go through a

sum of the lengths of all chosen paths which contain and

graphs and

act responsibly]. This remark is added to entertain the

j>i; k∈Pij ∈C

and go through a given

y

where all

oscillators are coupled through a combination of the two

reader that might be tired of following the proof up to

2 [ωx bint k ]Xk

of all chosen paths which belong to the

given

(37)

nomics: if you do not have means, borrow them!

2 [ax bxk + ωx bint k ]Xk +

k=1

P

n P

bxk =

n P

1 {ax · bxk + ay · byk } . n disconnected graphs C and D

blesome sums on the RHS? A solution comes from eco-

(36) where

As a result,

How can we get around this problem as we simply do

Applying this idea to each dierence variable on the

m P

In the case of

n

|Pij | indicates the length of the chosen path from oscillator i to oscillator j along the connection graph, combined of the x and y graphs. At this point, we do not dierentiate between paths containing only x or y edges, but we have to consider interlayer paths when where once again

graph.

ck + dk >

not be compared.

k∈Pij

y

the summation signs and the dierence variables

however,

a dierent choice of paths can yield closer estimates, as

or

immediately obtain the stability conditions by dropping

The choice of paths is not unique. We typically choose a shortest path between any pair of

x

the third and forth sums on the RHS disappear, and we

˙ ij = X

1 R

DF(βxj + (1 − β)xi ) dβ − (ay + αky )L+ 0  ωy bint k P Xij if i, j ∈ y edge k. (39)

15

The addition of a positive term

ωx bint k LXij

to the auxil-

Therefore, (36) turns into

iary system (38) worsens its stability, therefore we have to introduce an additional parameter

αkx

and make sure

that it is suciently large to stabilize the new auxiliary

n

m P k=1

ck Xk2 + n

l P k=1

system. A very important property is that, in (38), we have to add the positive, destabilizing term to the second equation for the

αkx

in the rst equation for the

P

(xj −xi ) equation L in (38)).

versus

Depending on the individual oscillator, chosen as the individual unit, this might not be possible, especially when

bint k

on the edge

k

is high. This property is

discussed in detail for the Lorenz and double scroll oscillator examples in Sections V and IV. A similar argument carries over to the auxiliary system (39) where we add the destabilizing term

ωy bint k Xij

(xj − xi )

to the

equa-

tion, but seek to stabilize the system via the additional parameter

αky

in the

(yj − yi )

m P

x 2 [ax bxk + ωx bint k + αk ]Xk +

k=1 l P

x 2 [ay byk + ωy bint k + αy ]Yk .

k=1

(yj − yi ) dierence but try

(note the dierent inner matrices:

trac load

+

ωx bint k LXij

to stabilize the system via increasing the additional parameter

dk Yk2 >

(40)

Notice the new stabilizing constants

αkx

and

αky . Depend-

ing on the individual oscillator dynamics and trac load on edge

k , these constants might have to be very large or

even innite.

Xk and Yk on the LHS

Comparing the terms containing

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

x 2 nck Xk2 > [ax bxk + ωx bint k + αk ]Xk , k = 1, ..., m x 2 ndk Yk2 > [ay byk + ωy bint + α ]Y y k , k = 1, ..., l. k

(41)

equation.

x Xij , that y graphs,

edges, sucient to make the derivative of the Lyapunov

remain intact and dened via the original systems (20)-

function (25) negative semi-denite, and therefore, en-

(22).

sure global stability of synchronization in the network

Notice that we only modify the auxiliary systems for and

y edges.

All the other auxiliary systems for

do not correspond to edges of either the

x

or

Thus, the modications of (38) and (39) make the troublesome sums

P

2 [ωx bint k ]Xk and

P

2 [ωy bint k ]Yk in (36)

k∈D k∈C disappear at the expense of worsened stability conditions of the corresponding auxiliary systems, which is reected by the appearance of additional terms with

αkx

and

αky .

[1] Steven H Strogatz,  Exploring complex networks, Nature 410, 268 (2001). [2] Réka Albert and Albert-László Barabási,  Statistical

Finally, we omit the dierence variables to obtain the bounds on coupling strengths,

ck

for

x edges and dk

x ck > n1 [ax bxk + ωx bint k + αk ], k = 1, ..., m y 1 int dk > n [ay bk + ωy bk + αyx ], k = 1, ..., l. This completes the proof of Theorem 1.

Synchronization: a Universal Concept in Nonlinear Sciences, Vol. 12 (Cambridge University Press, 2003). Kurths,

[11] Theoden I Neto and Steven J Schi,  Decreased neuronal

Journal of Neuroscience 22, 72977307 (2002).

[4] Stefano Boccaletti, Vito Latora, Yamir Moreno, Martin Chavez, and D-U Hwang,  Complex networks: Structure and dynamics, Physics Reports 424, 175308 (2006). [5] John Buck and Elisabeth Buck,  Synchronous reies, Scientic American 234, 7485 (1976).

synchronization

during

experimental

seizures,

Chaos Control Theory and Applications Series: Lecture Notes in Control and Information Sciences, Vol. 292 (Springer, 2003). [13] Leon Glass and Michael C Mackey, From Clocks to Chaos: the Rhythms of Life (Princeton University Press, [12] Guanrong Chen and Y Xinghuo,

1988).

[14] Adilson E. Motter, Seth A. Myers, Marian Anghel,

[6] James J Collins and Ian N Stewart,  Coupled nonlinear oscillators and the symmetries of animal gaits, Journal of Nonlinear Science 3, 349392 (1993). [7] Michael R Guevara, Alvin Shrier,

(42)



Physics 74, 47 (2002). plex networks, SIAM Review 45, 167256 (2003).

y

(1). It follows from (41) that these upper bounds are

mechanics of complex networks, Reviews of Modern [3] Mark EJ Newman,  The structure and function of com-

for

and

Takashi Nishikawa,  Spontaneous synchrony in powergrid networks, Nature Physics 9, 191 (2013). [15] Igor Belykh,

Vladimir Belykh,

and Martin Hasler,

and Leon Glass,

 Blinking model and synchronization in small-world net-

 Phase-locked rhythms in periodically stimulated heart

works with a time-varying coupling, Physica D: Nonlin-

cell aggregates, American Journal of Physiology - Heart and Circulatory Physiology 254, H1H10 (1988).

ear Phenomena 195, 188206 (2004). [16] Maurizio Porri, Daniel J Stilwell, Erik M Bollt,

and

[8] J Honerkamp,  The heart as a system of coupled non-

Joseph D Skufca,  Random talk: Random walk and syn-

linear oscillators, Journal of Mathematical Biology 18,

chronizability in a moving neighborhood network, Phys-

6988 (1983).

ica D: Nonlinear Phenomena 224, 102113 (2006).

[9] Vincent Torre,  A theory of synchronization of heart

[17] Maurizio Porri and Daniel J Stilwell,  Consensus seek-

pace-maker cells, Journal of Theoretical Biology 61, 55

ing over random weighted directed graphs, IEEE Trans-

71 (1976). [10] Arkady

Pikovsky,

actions on Automatic Control 52, 17671773 (2007). Michael

Rosenblum,

and

Jürgen

[18] Francesco Sorrentino and Edward Ott,  Adaptive syn-

16

chronization of dynamics on evolving complex networks, Physical Review Letters 100, 114101 (2008). [19] Paul So, Bernard C Cotton,

(2008). [36] Jie Sun, Erik M Bollt,

and Ernest Barreto,  Syn-

chronization in interacting populations of heterogeneous oscillators with time-varying coupling, Chaos:

and Takashi Nishikawa,  Master

stability functions for coupled nearly identical dynamical systems, Europhysics Letters 85, 60011 (2009).

An In-

[37] Takashi Nishikawa and Adilson E Motter,  Network syn-

terdisciplinary Journal of Nonlinear Science 18, 037114

chronization landscape reveals compensatory structures,

(2008).

quantization, and the positive eect of negative interac-

[20] Russell Jeter and Igor Belykh,  Synchronization in ono stochastic networks: windows of opportunity, IEEE Transactions on Circuits and Systems I: Regular Papers 62, 12601269 (2015).

tions, Proceedings of the National Academy of Sciences 107, 1034210347 (2010).

[38] Hiroya

Nakao,

Tatsuo

Yanagita,

and

Yoji

Kawa-

mura,  Phase-reduction approach to synchronization of

[21] Igor Belykh, Mario di Bernardo, Jürgen Kurths,

and

Maurizio Porri,  Evolving dynamical networks, Physica D: Nonlinear Phenomena 267, 16 (2014). [22] Olga Golovneva, Russell Jeter, Igor Belykh,

spatiotemporal rhythms in reaction-diusion systems, Physical Review X 4, 021032 (2014). [39] J Aguirre, R Sevilla-Escoboza, R Gutiérrez, D Papo, and

and Maur-

JM Buldú,  Synchronization of interconnected networks:

izio Porri,  Windows of opportunity for synchronization

the role of connector nodes, Physical Review Letters

in stochastically coupled maps, Physica D: Nonlinear Phenomena 340, 113 (2017).

112, 248701 (2014).

[40] Liyue Zhang, Adilson E Motter, and Takashi Nishikawa,

[23] Louis M Pecora and Thomas L Carroll,  Master stability functions for synchronized coupled systems, Physical Review Letters 80, 2109 (1998).

 Incoherence-mediated remote synchronization, Physical Review Letters 118, 174102 (2017). [41] Takashi Nishikawa, Jie Sun, and Adilson E Motter,  Sen-

[24] James F Heagy, Louis M Pecora, and Thomas L Carroll,  Short wavelength bifurcations and size instabilities in coupled oscillator systems, Physical Review Letters 74, 4185 (1995).

sitive dependence of optimal network dynamics on network structure, Physical Review X 7, 041044 (2017). [42] Nancy Kopell and Bard Ermentrout,  Chemical and electrical synapses perform complementary roles in the syn-

[25] Chai Wah Wu and Leon O Chua,  On a conjecture re-

chronization of interneuronal networks, Proceedings of

garding the synchronization in an array of linearly cou-

the National Academy of Sciences of the United States

pled dynamical systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 43, 161165 (1996).

[26] Stefano

Boccaletti,

DL Valladares,

of America 101, 1548215487 (2004). [43] Igor Belykh, Reimbay Reimbayev, and Kun Zhao,  Synergistic eect of repulsive inhibition in synchronization

Jürgen

Kurths,

Grigory

Osipov,

and CS Zhou,  The synchronization of

chaotic systems, Physics Reports 366, 1101 (2002).

of excitatory networks, Physical Review E 91, 062919 (2015). [44] Steven H Strogatz, Daniel M Abrams, Allan McRobie,

[27] Mauricio Barahona and Louis M Pecora,  Synchroniza-

Bruno Eckhardt, and Edward Ott,  Theoretical mechan-

tion in small-world systems, Physical Review Letters 89,

ics: Crowd synchrony on the Millennium Bridge, Nature

054101 (2002).

Synchronization in Coupled Chaotic Circuits & Systems, Vol. 41 (World Scientic, 2002).

[28] Chai Wah Wu,

[29] Xiao Fan Wang and Guanrong Chen,  Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49, 5462 (2002). [30] Vladimir N Belykh, Igor V Belykh,

and Martin Hasler,

 Connection graph stability method for synchronized coupled chaotic systems, Physica D: Nonlinear Phenomena 195, 159187 (2004).

438, 4344 (2005).

[45] Bruno

Eckhardt,

Edward

Daniel M Abrams,

Ott,

Steven

H

Strogatz,

and Allan McRobie,  Modeling

walker synchronization on the Millennium Bridge, Physical Review E 75, 021110 (2007). [46] Igor V Belykh, Russell Jeter,

and Vladimir N Be-

lykh,  Bistable gaits and wobbling induced by pedestrianbridge interactions, Chaos: An Interdisciplinary Journal of Nonlinear Science 26, 116314 (2016). [47] Igor Belykh, Russell Jeter,

and Vladimir Belykh,  Foot

force models of crowd dynamics on a wobbly bridge,

[31] Igor Belykh, Vladimir Belykh, and Martin Hasler,  Gen-

Science Advances 3, e1701512 (2017).

eralized connection graph method for synchronization in

[48] Sergey V Buldyrev, Roni Parshani, Gerald Paul, H Eu-

asymmetrical networks, Physica D: Nonlinear Phenom-

gene Stanley, and Shlomo Havlin,  Catastrophic cascade

ena 224, 4251 (2006).

of failures in interdependent networks, Nature 464, 1025

[32] Chai Wah Wu,  Synchronization and convergence of linear dynamics in random directed networks, IEEE Transactions on Automatic Control 51, 12071210 (2006). [33] Zhi Li and Guanrong Chen,  Global synchronization and asymptotic stability of complex dynamical networks, IEEE Transactions on Circuits and Systems II: Express Briefs 53, 2833 (2006). [34] Igor Belykh, Martin Hasler, Menno Lauret,

(2010). [49] Mikko Kivelä, Alex Arenas, Marc Barthelemy, James P Gleeson, Yamir Moreno,

and Mason A Porter,  Multi-

layer networks, Journal of Complex Networks 2, 203271 (2014). [50] Stefano Boccaletti, Ginestra Bianconi, Regino Criado, Charo I Del Genio, Jesús Gómez-Gardenes, Miguel Ro-

and Henk

Nijmeijer,  Synchronization and graph topology, International Journal of Bifurcation and Chaos 15, 34233433 (2005). [35] Igor Belykh and Andrey Shilnikov,  When weak inhibition synchronizes strongly desynchronizing networks of bursting neurons, Physical Review Letters 101, 078102

mance, Irene Sendina-Nadal, Zhen Wang,

and Massim-

iliano Zanin,  The structure and dynamics of multilayer networks, Physics Reports 544, 1122 (2014). [51] Manlio De Domenico, Clara Granell, Mason A Porter, and Alex Arenas,  The physics of spreading processes in multilayer networks, Nature Physics 12, 901 (2016). [52] Filippo Radicchi and Ginestra Bianconi,  Redundant in-

17

terdependencies boost the robustness of multiplex networks, Physical Review X 7, 011013 (2017).

038701 (2015). [57] Edward

[53] Francesco Sorrentino,  Synchronization of hypernetworks of coupled dynamical systems, New Journal of Physics 14, 033035 (2012).

N

Lorenz,

 Deterministic

nonperiodic

ow,

Journal of the Atmospheric Sciences 20, 130141 (1963).

Chua's Circuit: a Paradigm for Chaos (World Scientic Publishing, 1993).

[58] Rabinder N Madan,

[54] Daniel Irving and Francesco Sorrentino,  Synchroniza-

[59] Edsger W Dijkstra,  A note on two problems in con-

tion of dynamical hypernetworks: Dimensionality reduc-

nexion with graphs, Numerische mathematik 1, 269271

tion through simultaneous block-diagonalization of matrices, Physical Review E 86, 056102 (2012).

(1959). [60] Igor Belykh, Vladimir Belykh, and Martin Hasler,  Syn-

[55] Charo I del Genio, Jesús Gómez-Gardeñes, Ivan Bona-

chronization in asymmetrically coupled networks with

massa, and Stefano Boccaletti,  Synchronization in net-

node balance, Chaos:

works with multiple interaction layers, Science Advances

Nonlinear Science 16, 015102 (2006).

2, e1601679 (2016).

[56] Xiyun Zhang, Stefano Boccaletti, Shuguang Guan,

An Interdisciplinary Journal of

[61] Reimbay Reimbayev, Kevin Daley,

and Igor Belykh,

and

 When two wrongs make a right: synchronized neuronal

Zonghua Liu,  Explosive synchronization in adaptive

bursting from combined electrical and inhibitory cou-

and multilayer networks, Physical Review Letters 114,

pling, Phil. Trans. R. Soc. A 375, 20160282 (2017).