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: August 8, 2018) Many complex biological and technological systems can be represented by multilayer networks where the nodes are coupled via several independent networks. Despite its signi cance from both the 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 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 criti- cal 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 synchro- nization when used in a multilayer network. This property is controlled by the tra c 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 signi cant rami cations for the dynamical processes on networks, including the eects on the speed of disease Complex networks are common models for many sys- transmission in social networks [51] and the role of re- tems in physics, biology, engineering, and the social sci- dundant interdependencies on the robustness of multi- ences [13]. Signi cant attention has been devoted to plex networks to failure [52]. algebraic, statistical, and graph theoretical properties Typically, in single-layer networks of continuous time of networks and their relationship to network dynamics oscillators, synchronization becomes stable when the cou- (see a review [4] and references therein). The strongest pling strength between the oscillators exceeds a threshold form of network cooperative dynamics is synchronization value [23, 30]. This threshold depends on the individual which has been shown to play an important role in the oscillator dynamics and the network topology. In this functioning of a wide spectrum of technological and bio- context, a central question is to determine the critical logical networks [514], including adaptive and evolving coupling strength so that the stability of synchroniza- networks [1522]. tion 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. Both activities have been focused on oscillators connected methods reduce the dimensionality of the problem such 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. Simultaneous diagonalization of one network may lead to the failure of dependent nodes two or more matrices is impossible in general, unless the in another network [48]. Such interconnected networks matrices commute [53, 54]. 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 reduce the dimensionality of a large network to a smaller STATEMENT network whose synchronization condition can be used to evaluate the stability of synchronization in the large net- We start with a general network of n oscillators with work. For some network topologies, this technique yields two connectivity layers: a substantial reduction of the dimensionality; however, this reduction is less signi cant in general. The reduced n n dxi X X network typically contains weighted positive and nega- = F(xi)+ cijP xj + dijLxj, i = 1, ..., n, (1) dt tive connections, including self loops such that the role j=1 j=1 of multilayer network topologies and the location of an where 1 s is the state vector containing the edge remain di cult to evaluate. xi = (xi , ..., xi ) s coordinates of the i-th oscillator, F : s → s describes In this paper, we report our signi cant progress to- R R the oscillators' individual dynamics, c and d are the wards removing this long-standing obstacle to study- ij ij coupling strengths. C = (cij) and D = (dij) are n × n ing synchronization in multilayer networks. We develop Laplacian connectivity matrices with zero-row sums and a new general stability approach, called the Multilayer nonnegative o-diagonal elements c = c and d = d , Connection Graph method, which does not depend on ij ji ij ji respectively [30]. These connectivity matrices C and D explicit knowledge of the spectrum of the connectivity de ne two dierent connection layers (also denoted by matrices and can handle multilayer networks with arbi- C and D, with m and l edges, respectively). The in- trary network topologies, which are out of reach for the ner matrices P and L determine which variables couple existing approaches. An example of a multilayer network the oscillators within the C and D layers, respectively. in this study is a network of Lorenz systems where some Without loss of generality, we will be considering the os- of the oscillators are coupled through the variable ( rst x cillators of dimension s = 3 and x = (x , y , z ). There- layer), some through the variable (second layer), and i i i i y fore, the C graph with the inner matrix P = diag(1, 0, 0) some through both (interlayer connections). Our Multi- will correspond to the rst-layer connections via x, while layer Connection Graph method originates from the con- the D graph with the inner matrix L = diag(0, 1, 0 ) will nection graph method [30, 31] for single-layer networks; indicate the second-layer connections via y. Overall, the however, this extension is highly non-trivial and requires oscillators of the network are connected through a com- overcoming a number of technically challenging issues. bination of the two layers (see Fig. 1 for an example of a This includes the fact that the oscillators from two x and combined two-layer graph). The graphs are assumed to y layers in the networks of Lorenz systems are connected be undirected [30]. Oscillators, comprising the network through the intrinsic, nonlinear equations of the Lorenz (1), can be periodic or chaotic. As chaotic oscillators are system. As a result, multilayer networks can have dras- di cult to synchronize, they are usually used as test bed tically dierent synchronization properties from those of examples for probing the eectiveness of a given stability single-layer networks. In particular, our method shows approach. The oscillators used in the numerical veri ca- that an interlayer tra c load on a link is the crucial quan- tion of our stability method are chaotic Lorenz [57] and tity which can be used to foster or hamper synchroniza- double scroll oscillators [58]. tion in a nonlinear fashion. For example, it demonstrates In this paper, we are interested in the stability of that replacing a link with a light interlayer tra c load by complete synchronization de ned by the synchronization a stronger pairwise converging coupling (a good link) manifold M = {x1 = x2 = ... = xn}. Our main objec- via another layer may lower the synchronization thresh- tive is to determine a threshold value for the coupling old and improve synchronizability. At the same time, strengths required for the global stability of synchroniza- such a replacement of a highly loaded link can make the tion. We seek to predict this threshold or the absence network unsynchronizable, forcing the pairwise stabiliz- thereof in the general network (1) from synchronization ing good link go bad. 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. II, we present and discuss the network model. 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. And what is the synchronization threshold. Most known oscillators, an underlying mechanism? In Sec. 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 x-coupled Rössler systems [23] in VI, we show how to apply the general method to speci c 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]. As a result, we limit our consideration to 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 tra c loads are given in the Supplement. be performed with moderate eort and remains a subject 3 of future study. tering or hindering synchronization, we begin with two- layer networks of chaotic Lorenz oscillators, depicted in (a) Fig. 1(a-c). For the Lorenz oscillators, the vector equa- tion (1) can be written in a more reader-friendly scalar form:
n P x˙ i = σ(yi − xi) + cijxj, j=1 (b) n P (2) y˙i = rxi − yi − xizi + dijyj, j=1 z˙i = −bzi + xiyi, i = 1, ..., n,
where the connectivity matrix C = (cij) describes the (c) topology of x connections (black edges in Fig. 1(a-c)), and matrix D = (dij) describes the location of one y edge (blue or red edge). The parameters of the individ- ual Lorenz oscillator are standard: σ = 10, r = 28, and b = 8/3. The strengths of the x and y coupling are homo- geneous ( ) and varied uniformly ( ). (d) cij = c, dij = d 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 y edge can aect synchronization. To address this ques- tion, we rst need to understand synchronization prop- erties of two-node single-layer networks of x-coupled and y-coupled Lorenz systems. It is well-known that if the coupling in a single-layer network (2) with either all x or all 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 the critical coupling c∗, explicit in parameters of the Lorenz oscillator can be found in [30]. Calculated numerically, these coupling thresholds are for c∗ ≈ 3.81 for the two-node x-coupled network and d∗ ≈ 1.42 for the y-coupled network. As the synchro- nization threshold d∗ is signi cantly lower, then one could FIG. 1. The puzzle: why do good links go bad? Syn- expect that replacing an x edge with a presumably better chronization in six-node networks of Lorenz systems (2). (a) converging y coupling improves synchronization. This is Single-layer network, with all x edges (black). (b) The re- true for the network of Fig. 1(b) when edge - is re- placement of edge - with a presumably better converg- x 5 6 x 5 6 placed with a edge, yielding a minor reduction in the ing y coupling (blue) improves synchronization, as expected y synchronization threshold from ∗ in the single- (see (d)). (c) A similar replacement of x edge 2-3 with a y c ≈ 17.94 edge (red) makes synchronization impossible by pushing the layer x-coupled network of Fig. 1(a) to c∗ ≈ 17.74 in threshold to in nity (see (d)). (d). Systematic study of the the multilayer network of Fig. 1(b). A surprise comes coupling threshold c∗ as a function of the y edge location from the network of Fig. 1(c) where the replacement of that replaces an x edge in the original x-coupled network (a). x edge with a y edge, that would be expected to improve The blue solid line indicates numerically calculated thresh- synchronization, on the contrary makes the network un- olds. The black solid line depicts the interlayer tra c bint for synchronizable (see the coupling threshold jumping to the respective y edge. Note a signi cant increase of bint that in nity in Fig. 1(d)). causes the network to become unsynchronizable as predicted What is the origin of this highly counterintuitive eect? by the method. The predicted coupling thresholds (blue dot- Why do edges in a multilayer network reverse their stabi- ted line) are computed from (3) using the exponential t in lizing roles depending on the edge location whereas they Fig. 3 and scaling factors β = 0.18 and γ = 0.7180. are well behaved in single-layer networks? As the con- nectivity matrices for x and y coupling in the networks of Fig. 1(b) and Fig. 1(c) do not commute and, there- fore, the predictive power of the master stability function III. A MOTIVATING EXAMPLE AND A based methods [5355] is severely impaired, this puzzle PUZZLE calls for an explanation and utlimately motivates the de- velopment of an eective, general method for assessing To illustrate the complexity of assessing the role of the stability of synchronization in multilayer networks. multilayer connections and their controversial role in fos- In the following, we will develop such a method that 4 identi es the location of critical interlayer links which is given in the Appendix. control stable synchronization and reveals its explicit de- Remark 1. In terms of tra c networks, the graph the- pendence on an interlayer tra c load on a given edge. oretical quantities x y and int represent the total bk, bk bk lengths of the chosen roads that go through a given edge k which can be loosely analogized as a busy street. There- IV. MULTILAYER CONNECTION GRAPH fore, we refer to them as tra c loads. In this view, the METHOD quantity int is a tra c load on edge caused by in- bk k, terlayer travelers. It is important to notice that positive In this section, we rst present the main result of this terms int in the equation (4) and int +ωxbk LXk +ωybk 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 speci c network con- edge with a high int can represent a potential problem bk gurations. for making the systems (4) and (5) stable at all. This ob- Theorem 1 [The method]. Synchronization in the servation has dramatic consequences for synchronization multilayer network (1) is globally stable if for each edge in speci c multilayer networks described in the following k = 1, ..., m (l) sections and also is a key to solving the puzzle of Fig. 1. Remark 2. If both x and y connection graphs are con- 1 x int x cij ≡ ck > n ax · bk + ωx · bk + αk (3) nected such that all oscillators are coupled via both 1 y int y dij ≡ dk > n ay · bk + ωy · bk + αk , graphs, the stability of synchronization can be simply assessed by applying the connection graph method [30] n where x P is the sum of the lengths for each of the x and y connected graphs and combin- bk = |Pij| j>i; k∈Pij ∈C ing the two conditions as follows: cij + dij = ck + dk > of all chosen paths P between any pair of oscillators 1 x y (cf. the condition (37) in the Ap- ij n {ax · bk + ay · bk} i and j which belong to the x graph C and go through pendix). As a result, one should not expect the eects a given x edge. The path length |Pij| is the number of due to the multilayer coupling discussed in the motivat- edges comprising the path. Similarly, y corresponds to bk ing example. the sum of the lengths of all chosen paths which belong Remark 3. While the stability criterion (3) is completely to the graph and go through a given edge. int is y D y bk rigorous, the theoretically calculated bounds may give the sum of the lengths of all chosen paths which contain large overestimates on the threshold coupling strength. x and y edges and belong to the combined, two-layer xy As a result, we suggest to use a semi-analytical approach graph and go through a given edge which may be an or k x which combines numerically calculated exact bounds ax, edge. Constant ( ) is the double coupling strength k and k with graph theoretical quantities y ax ay ay, ωx, ωy, αx, αy su cient for synchronization in the network of two x- bx, by, and bint. In this way, this method becomes an coupled (y-coupled) oscillators, composing the network. eective, predictive tool and plays the role of the master The constants ωx and ωy represent a combination of the stability function for the general multilayer network (1) double coupling strengths for c12 and d12 su cient for where a synchronization threshold or the absence thereof synchronization in the two-oscillator network with both x can be deduced from the properties of the individual os- and connections. Finally, the constants k and k are y αx αy cillators comprising the network and the network topolo- chosen large enough such that they can globally stabilize gies of the connection layers. In this numerically-assisted the auxiliary stability systems written for the dierence application of our method, the conservative estimate on variables that correspond to an edge k : Xk = Xij = tra c loads bx, by, and bint, that come from the Cauchy- xj − xi : Schwartz inequality (see the Appendix), can be replaced by βbx, βby, and γbint, where the scaling factors β and γ 1 for k ˙ R are introduced to provide tighter numerical bounds. αx : Xk = DF(βxj + (1 − β)xidβ Xk+ (4) 0 int x ωxbk LXk − (ax + αk)P Xk,
1 for k ˙ R αy : Xk = DF(βxj + (1 − β)xidβ Xk+ (5) 0 ω bintP X − (a + αy)LX , y k k y k k Computing the stability conditions (3) and its con- where the Jacobian DF can be calculated explicitly via stants is a complicated, multi-step process which involves the parameters of the individual oscillator. calculations of (i) two stability diagrams (for a pair of Proof. The proof closely follows the notations and steps x and x y) and (ii) graph theoretical quantities ω , ωy αk, α in the derivation of the connection graph method [30] for 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 speci c multilayer networks and terms int and int which play a pivotal role in syn- illustrate their implications for the stability of synchro- ωxbk ωybk chronization of multilayer networks. The complete proof nization. 5
Step 2: Calculate the stability diagrams to determine x αk and y αk. 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 X˙ k = σ(Yk − Xk) − (ax + α )Xk, k ˙ (z) (x) int (6) Yk = r − Uk Xk − Yk − Uk Zk + ωxbk Yk ˙ (y) (x) Zk = Uk Xk + Uk Yk − bZk,
˙ int Xk = σ(Yk − Xk) + ωybk Xk ˙ (z) (x) y Yk = r − Uk Xk − Yk − Uk Zk − (ay + αk)Yk, FIG. 2. Stability of synchronization in a two-node network ˙ (y) (x) Zk = Uk Xk + Uk Yk − bZk, of xy-coupled Lorenz systems as a function of the x coupling, (7) and the coupling, . Yellow depicts instability (non- ωx y ωy where (ξ) for are the corre- zero synchronization error) and purple depicts stability (zero Uij = (ξi + ξj)/2 ξ = x, y, z sponding sum variables. As in (3), the auxiliary sta- synchronization error). The white dot indicates the pair (ωx, bility system (6) corresponds to the dierences between ωy) used in the predictions shown in Fig. 1(d) and Fig. 4(c). the nodes connected by an x edge, and (7) corresponds to the dierences between nodes coupled via a y edge. V. APPLICATION OF THE METHOD: If the connection layers overlap and the same nodes are SOLVING THE PUZZLE connected through both x and y edges, then the auxiliary systems (6) and (7) should be applied to the correspond- ing x and y edges independently. Their contributions will Armed with the predictive method, we rst return to then appear in the general stability conditions (3) for c the puzzle to better understand synchronization prop- k and dk for the same edge k. erties of the six-node networks of Lorenz oscillators (2) The auxiliary systems (6) and (7) are used to yield from Fig. 1. Below we give a step-by-step recipe of how global stability conditions for the analytical criterion (3), our numerically-assisted method can be applied to these where bounds on the sum dierences U (x),U (y), and U (z) networks. can be placed similar to the analysis of synchronization Step 1: Calculate ax, ay, ωx, and ωy. in single-layer networks [30]. A rigorous analysis of global Consider the simplest two-node network (2) with both x stability of (6) and (7) will be performed in a more tech- and y coupling: c12 = c21 = c and d12 = d21 = d. Deter- nical publication. mine threshold coupling strengths that guarantee stable Here, we take a more practical route towards devel- synchronization in (i) the two-node x-coupled network oping a numerically-assisted approach which simpli es ∗ with c = ax/2 and d = 0; (ii) the two-node y-coupled the evaluation of the stability of (6) and (7). We lin- ∗ network with d = ay/2 and c = 0; and (iii) the two-node earize both systems (6) and (7) by making the dierences ∗ ∗ xy network with c = ωx and d = ωy. (x) Xk,Yk, and Zk in nitesimal and replacing U = x(t), The numerically calculated thresholds for the x- U (y) = y(t), and U (z) = z(t), where (x(t), y(t), z(t)) is the coupled and y-coupled two-node network (2) reported synchronous solution governed by the uncoupled Lorenz in Sec. III are c∗ ≈ 7.62/2 and d∗ ≈ 2.84/2, respec- oscillator. The linearized stability systems (6) and (7) tively. This yields the double coupling strength constants take the form: ax = 7.62 and ax = 2.84 to be used in (3). X˙ = σ(Y − X ) − (a + αx)X , Note that dierent combinations of c = ωx and d = ωy k k k x k k ˙ int (8) in the xy-coupled network yield stable synchronization Yk = (r − z(t)) Xk − Yk − x(t)Zk + ωxbk Yk (see Fig. 2). Without loss of generality, we choose Z˙k = y(t)Xk + x(t)Yk − bZk, c = ωx = 5 and d = ωy = 0.5 as a point on the sta- bility boundary in Fig. 2 and keep these values xed for ˙ int the prediction of the synchronization threshold in larger Xk = σ(Yk − Xk) + ωybk Xk two-layer networks (2) with arbitrary topologies. This ˙ y (9) Yk = (r − z(t)) Xk − Yk − x(t)Zk − (ay + αk)Yk, choice of the pair is somewhat arbitrary; how- (ωx, ωy) Z˙k = y(t)Xk + x(t)Yk − bZk. ever, it dictates the choice of constants in the stability diagrams for x and y (Fig. 3). It is often a good idea Notice that x [ y] must be large enough to stabilize (8) αk αk αk αk to choose and such that both are non-zero and lie [(9)] in the presence of int [ int]. While ωx ωy ωxbk ωybk ax, ay, ωx, somewhere in the middle range of (ωx, ωy) to balance out and ωy are chosen and xed in Step 1, the tra c load the stability conditions (4) and (5). int on a given edge (which will be determined in Step bk k 6
(a) pensate for this overestimate. The diagrams of Fig. 3 con rm the existence of threshold values for int and βωxbk [ int] such that even in nitely large values of and βωybk αx 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 Here, the terms int and int are (x(t), y(t), z(t)). βωxbk βωybk treated as single, aggregated control parameters. There- fore, the threshold value for x [ y] required to stabilize αk αk the auxiliary system (8) [(9)] for a given edge with int k bk can simply be read o from Fig. 3. To better quantify (b) this dependence to be used in predicting synchronization thresholds in networks of Lorenz oscillators, we approxi- mate the stability boundary in Fig. 3(a) by the exponen- tial function
int (10) αx = 0.4645 exp 2.408βωxbk . The stability diagrams of Fig. 3 along with Fig. 2 ac- count 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 mas- ter stability function in single-layer networks [23] and help in solving, once and for all, the question of stabil- ity 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 cou- calculation of pure graph theoretical quantities as shown pled Lorenz oscillators. Yellow depicts instability of the origin in the next step. while purple indicates its stability. The dependence of the sta- Step 3: Calculate tra c loads x y and int bilizing term, , on int , and is estimated by the ex- bk, bk, bk . αx bk , ωx βx This calculation is similar to that of the connection ponential function int(dashed curve), 0.4645 exp 2.408βxωxbk and is used to predict synchronization thresholds in networks graph method for single-layer networks [30], except that of Fig. 1 and Fig. 4. the tra c load should be partitioned into three groups: intralayer tra c loads x and y within the and bk bk x y layer, respectively, and interlayer tra c load int be- bk tween the layers. To do so, we rst choose a set of paths 3) controls the choice of x and y. Thus, if the edge αk αk {|P | i, j = 1, ..., n, j > i}, one for each pair of ver- is highly loaded, then the contribution of the positive ij tices i, j, and determine their lengths |(Pij|, the number term +ω bintY in the Y -equation of system (8) can- x k k k of edges in each Pij. Then, for each edge k of the x (y) not always be compensated by increasing x in the y −αkXk layer graph, we calculate the sum bx (b ) of the lengths -equation. Typically, this happens when the positive k k Xk of all P that are composed of only x (y) edges and pass term exceeds the proper negative linear terms such as ij through k. We repeat the same procedure to calculate the (technically, through a combination of terms in the −Yk sum bint of the lengths of all P that contain both x and Routh-Hurwitz criterion). Therefore, the auxiliary sys- k ij y edges and pass through k. These constants depend on tem (8) can become unstable, independently of how large the choice of the paths P . Usually, one uses the short- the stabilization coe cient x is. The same argument re- ij αk est path from vertex i to vertex j. Sometimes, however, lates to the destabilization of the auxiliary system (9) via a dierent choice of paths can lead to lower bounds [34]. the positive term int +ωybk Xk. We use the six-node multilayer network of Fig. 1(b) as The complex relationship between these terms in re- an example for calculating x y and int To compute bk, bk, bk . gard to stabilizing (8) and (9) is shown in Fig. 3. Notice all of the paths that pass through a given edge, it is the coe cient on the int and [ int] axes. Be- recommended that the reader algorithmically nds the β βωxbk βωybk cause of the Cauchy-Schwarz inequality used in the proof, shortest path between every pair of oscillators, and take int provides an overestimate for the terms added to the note of the paths that go through edge and dierentiate bk k auxiliary system and the scaling factor β is added to com- the paths that entirely belong to only the x or y layers 7
and the ones that contain a combination of x and y edges. (3) with constants ax = 7.20, ay = 2.63, ωx = 5.00 and As a result, we can nd each edge's tra c loads as follows chosen above. The constants x and y are ωy = 0.50 αk αk x read o from the diagrams of Fig. 3(a) and Fig. 3(b), b12 = |P12| + |P13| + |P14| + |P15| + |P16| = respectively. As the stability system (8) is much more 1 + 2 + 3 + 3 + 4 = 13, sensitive to the changes in int than (9) (cf. the onset of x bk b23 = |P13| + |P14| + |P15| + |P16| + |P23|+ instability in Figs. 3(a-b)), the threshold values for cij in |P24| + |P25| + |P26| = 20, x the criterion (3) for the x layer largely dominate over dij. b34 = |P14| + |P16| + |P24| + |P26| + |P34| = 15 x x Thus, since the synchronization threshold for the entire b35 = |P15| + |P25| + |P35| = 6, b46 = |P16| + |P46| = 5, y int int int network (2) with uniform coupling c = d is de ned by b56 = |P56| = 1, b12 = 0, b23 = 0, b34 = 0, the maximum of the thresholds or for each edge int int cij dij b35 = |P36| = 2, b46 = |P45| = 2, of the multilayer graph, the maximum threshold values int b56 = |P36| + |P45| = 4. predicted by the method and depicted in Fig. 1(d) are (11) the ones corresponding to edges with coupling These Note that the maximum interlayer tra c load on this x c. threshold values are calculated using the numerically as- network is fairly low and due to our choice of paths is sisted modi cation of (3) bint = 4 although it could have also been minimized to 56 zero, provided that all paths to node 6 bypass edge 56. 1 x int x int c > max ck = γax · bk + βωx · bk + αk(βωxbk ) , At the same time, the interlayer tra c load in the net- k n work of Fig. 1(c) is signi cantly higher since there are (12) no alternatives to go around the bottle neck edge - where x is de ned by the stability diagram of Fig. 3(a) 2 3 αk when traveling from nodes 1 and 2 to nodes 4, 5, and 6. via the approximating function (10) for each edge k, and For the same choice of shortest paths, we get int and int are determined in Steps 1-3. Notice the b23 = 19. ax, ωx, bk The remaining x y for the network of Fig. 1(c) can be presence of an additional scaling factor which is chosen bk, bk γ calculated similarly to (11). to compensate for the conservative nature of the connec- Step 4: Putting pieces together to solve the puzzle. Given tion graph stability method when the network is entirely the stability diagram of Fig. 3 with abrupt threshold de- coupled through the x variable. γ scales down the term pendences of x and y on increasing interlayer tra c ax to match the coupling needed to synchronize the six- αk αk n load x the eect of synchrony breaking when a highly node network of Fig. 1(a) with only edges. The scal- bk, x loaded x edge is replaced with a better pairwise stabi- ing factor β is then chosen for the network of Fig. 1(b) lizing (see Sec. III ) is no longer a puzzle and directly with the lowest int to match the actual synchro- y bk = 4 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 x edge. Figure 1(d) shows that the predicted counterintuitive eects due to the multilayer structure thresholds are fairly close to the actual ones, and the cri- and then constructed the network examples. 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. As our exhaustive study of various net- the replaced x edge 2-3. work con gurations 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. As discussed in the description of the To test the predictive power of our approach with the motivating example, the replacement of x edge 5-6 with constants identi ed in Steps 1-3, we perform a systematic a y edge improves synchronization by slightly lowering 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, x-coupled network of ity criterion (3), this happens due to a slight decrease Fig. 1(a) with a edge, aects synchronization. The edge in the tra c load on the bottleneck edge x (see (11)), y b23 replacement is performed in the order of the increasing compared to the original network of Fig. 1(a) with all x interlayer tra c load on this edge, int After comput- edges where one additional path goes through edge bk . P16 ing the coupling threshold required to synchronize the 2-3. As a result, it decreases the contribution of the dom- new network, this edge reverts back to being an edge. inating term x in (3). At the same time, the contribu- x axbk This results in multiple networks of ve edges and one tion of the other factors int and x remain insignif- x ωxbk αk edge. The two multilayer networks of Fig. 1(b) and icant, especially due to the fact that x still lies on a y αk 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 int On the bk . Fig. 1(d) presents the actual synchronization threshold other hand, such a replacement of the bottleneck node values (blue solid line), the interlayer tra c loads bint 2-3 in the network of Fig. 1(c) signi cantly increases the (black line) calculated similarly to (11), and the thresh- intralayer tra c load int requiring in nitely large x bk , α23 old values predicted by the numerically-assisted criterion to stabilize the stability system (8) and causing synchro- 8 nization to break. (a)
VI. LARGER NETWORKS
To demonstrate that similar synchrony breakdown phenomena occur in larger networks and can be eec- tively 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 en- tirely through the x variable. To test our prediction that replacing edges with a high tra c load can make the net- work unsynchronizable, we index the edges according to their x. Edges similar to edge - have very few paths bk 10 12 that pass through them, and subsequently have a low x bk (and in turn, int, shown as the black curve in Fig. 4(c)). bk We successively replace x edges (denoted by black edges in Fig. 4(a)) with edges (denoted by gray edges in Fig. y (c) 4(b)), according to this ordering until the network is com- pletely connected through edges. The values of int y bk range 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, for edge 3-5 for which every path from node 3 of the x layer graph to any other node in the y-layer must pass through it).
A. Networks of Lorenz oscillators
The coupling necessary to synchronize the x-coupled FIG. 4. Eect of successively replacing coupling edges with Lorenz network (2) described in Fig. 4(a) is c ≈ 86.95. x y coupling edges on synchronization in a 20-node network of As outlying, low tra c edges are replaced with y edges, Lorenz oscillators (2). (a) Original x-coupled network before there is almost no eect on the threshold for the coupling replacing edges according to their tra c load. (b) Snapshot strength required to synchronize the network, evidenced of the multilayer network before the edge 3-5, labeled as the by the lack of change in the actual coupling threshold 13-th edge according to the tra c load ranking, is replaced. for the rst eight edges replaced in Fig. 4(c). 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 bint), the network replacement of remaining x edges (gray) with higher tra c becomes more di cult to synchronize, until edge 13 (edge load preserves the instability of synchronization, until all 25 3-5 which is depicted in red in Fig. 4(b)) is replaced. Af- edges have been replaced with a y edge, yielding a single-layer ter which, synchronization is no longer feasible for the y-coupled network with bint = 0 that is able to synchronize network for any additional edge replacement, until edge again. (c) Actual (blue solid line) and predicted (blue dotted line) threshold for the coupling strength required to synchro- 25 (edge 14-15 in Fig. 4). Replacing edge 25, shown in nize the network after the -th edge has been successively Fig. 4(b) corresponds to nishing the successive edge re- i x replaced with a y edge. The black solid line depicts the inter- placement process, and results in a graph identical to the layer tra c bint 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 coupling (gray) instead of x coupling (black). This re- t in Fig. 3 and scaling factors β = 0.0025 and γ = 0.5993. inforces our tra c 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 x edges in (3), because is very highly loaded can make the network unsynchro- the stability term αx required to stabilize the x stability nizable, evidenced by the network having no synchroniz- system (8) must be signi cantly higher than αy in the 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, 9 and x to predict the synchronization threshold and only αk need to identify the tra c loads int and the scaling fac- bk tors γ and β 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 traf- c load int can be easily calculated by hand as in (11), bk computing int for the -node or larger networks is a bk 20 laborious task which was performed by an algebraic al- gorithm, implemented as MATLAB code and given in the Supplement. While the values of int heavily depend bk 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 tra c loads on edges more equally may yield even better predictions and ts. FIG. 5. Eect of successively replacing x edges with y edges on the synchronization threshold in a 20 node network of double-scroll oscillators. The network topology, edge replace- B. Networks of double scroll oscillators ment process, and notations are identical to those in Fig. 4(a- b). Notice the same location of critical links (edges 13 to 24) To illustrate the generality of synchrony break phe- whose replacement leads to synchrony breaks as in the net- nomenon when good but highly-loaded links go bad, work 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 network has been replaced with y edges. No- n P tably, the synchrony break occurs at the same edge as x˙ i = κ(yi − xi − h(x)) + cijxj, in the network of Lorenz oscillators, suggesting that crit- j=1 n ical edges whose replacement hampers synchronization P (13) y˙i = xi − yi + zi + dijyj, are mainly controlled by the network multilayer topol- j=1 ogy rather than the individual properties of the intrin- z˙ = −λy − µz , i = 1, ..., n, i i i sic oscillators, provided that the oscillators possess simi- with lar synchronization properties as the Lorenz and double scroll oscillators. The solid curve in Fig. 5 displays the synchroniza- m1(x + 1) − m0 x < −1 h(x) = m0x −1 ≤ x ≤ 1 tion thresholds, calculated via the stability criterion m1(x − 1) + m0 x > 1 (12) with ax = 5.94 × 2 = 11.88, ωx = 2.0, β = 0.0041, γ = 0.282, and the approximating function αx = and standard parameters int with the same tra c loads int κ = 10, m0 = −1.27, m1 = 1.556 exp(3.711βωxbk ) bk −0.68, λ = 15, and µ = 0.038. 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]. strength required for synchronization in a two-node y- As in the Lorenz oscillator case, the auxiliary stability ∗ coupled network, d = 1.16 is much lower than the system (4) for αx is much more sensitive to increasing coupling threshold in the two-node -coupled network, int than the stability system (5) for therefore one x bk αy, c∗ = 5.94. can only evaluate the stability condition (12) for the x In Fig. 5, we apply our method to predict the synchro- coupling c to identify a bottle-neck for the synchroniza- nization thresholds in the 20-node network of Fig. 4 as tion threshold in the entire network. in the same network of Lorenz oscillators. When succes- Going back to the puzzle example, we have also per- sively replacing x 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]. Remarkably, this analy- gions of the graph with low tra c loads int are replaced sis indicates the same qualitative phenomena when the bk with more favorable y edges that provide better pairwise replacement of the lightly loaded edge 5-6 slightly low- convergence to synchronization. Then, as with the net- ers the synchronization threshold from c = 13.57 in the work of Lorenz oscillators, when edge 13 (edge 3-5) is original x-coupled single-layer network of Fig. 1(a) to replaced with a y edge, synchronization is no longer at- c = 13.36, and predicts the breakdown of synchrony when tainable. Synchronization then returns when the entire edge 2-3 is replaced as in the Lorenz network. 10
We have also simulated series of other 20-node net- and therefore can handle multilayer networks. Originated works (2) and then networks (13) where all oscillators from the connection graph method for synchronization in were connected via x layer graphs, whereas the 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 highly-loaded links separate the network into disjoint x diagrams for the auxiliary s-dimensional system which and y 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 coupled directly or indirectly via the x graph such that corresponding to one connection layer when an instabil- the coupling strengths c 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 tra c loads via thresholds in such networks depend on the location of a given edge on the multilayer connection graph. All to- added y 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 ample of Fig. 1(b) where the x graph connects all six synchronization in the original, potentially large, multi- nodes and the replacement of edge 5-6 with an y edge low- layer network. ers the synchronization threshold. On the contrary, the Using the method, we have discovered striking, highly replacement of the x edge 3-5 with an edge y, which still unexpected phenomena not seen in single-layer networks. preserves the connectedness of the x graph, increases the In particular, we have shown that replacing a link with synchronization threshold, as predicted by the method a light interlayer tra c load by a stronger pairwise con- (see Fig. 1(c)). The 20-node networks with connected verging coupling via another layer may improve synchro- x graphs yield similar eects. To avoid repetition, these nizability, as one would expect. At the same time, such results are not shown. a replacement of a highly loaded link may essentially worsen synchronizability and make the network unsyn- chronizable, turning the pairwise stabilizing good link VII. CONCLUSIONS into a destabilizing connection (a bad link). The criti- cal 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 signi cant 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 modi ed 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 modi cation 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. In particular, 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 11 method [31, 60] for single-layer directed networks, where where DF is the s × s Jacobian matrix of F. This no- directed edges are symmetrized and assigned additional tation is simply a compact form of the mean value theo- weights according to the mean node unbalance. rem, f(B) − f(A) = f 0(C)(B − A), applied to the vector Our method can also be modi ed to handle multi- functions F(xj) and F(xi), where the Jacobian DF is layer neuronal networks connected via electrical, excita- evaluated at some point C ∈ [xi, xj]. tory, and inhibitory synapses which exhibit a number of Therefore, the dierence system (15) can be rewritten counterintuitive synergistic eects: when (i) the addition in the form of pairwise repulsive inhibition to single-layer excitatory 1 networks can promote synchronization [43] and (ii) com- R X˙ ij = DF(βxj + (1 − β)xi)dβ Xij+ bined electrical and inhibitory coupling can induce syn- 0 n chronization even though each coupling alone promotes P {cjkP Xjk − cikP Xik + djkLXjk − dikLXik}, an anti-phase rhythm [61]. Our method promises to allow k=1 an analytical treatment of these eects in large neuronal i, j = 1, ..., n. 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 electrical neuronal circuitries simultaneously. the 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 that VIII. ACKNOWLEDGMENTS the coupling is strong enough. Notice that the stability of system (16) is redundant as This work was supported by the National Science it contains all possible (n−1)n/2 non-zero dierences X Foundation (USA) under Grant No. DMS-1616345 ij along with n zero dierences X = 0 which can be dis- and the U.S. Army Research O ce under Grant No. ii regarded. At the same time, there are only n − 1 linearly W911NF-15-1-0267. independent dierences required to show the convergence
between n variables Xij. However, this redundancy prop- erty and the consideration of all non-zero Xij are a key IX. APPENDIX: THE PROOF ingredient of our approach which allows for separating the dierence variables later in the stability description, In this appendix we derive the Multilayer Connection without diagonalizing the connectivity matrices. Graph method and prove Theorem 1. Our goal is to We strive to nd conditions under which the trivial derive the conditions of global asymptotic stability of xed point {Xij = 0, i, j = 1, ..., n} of system (16) is the synchronization manifold M in the system (1). To globally stable. This amounts to nding conditions for achieve this goal and develop the stability method, we fol- global stability of synchronization in the network (1). low the steps of the proof of the connection graph method We introduce the following terms AijXij, where A is [30]. The concept is similar, up to a certain step where a a 3 × 3 matrix, such that new stability argument is used. axP = diag(ax, 0, 0) if oscillators i and j In the network model (1) we introduce the dierence belong to layer variable x C a L = diag(0, a , 0) if oscillators i and j A = y y ij belong to y layer D (14) Xij = xj − xi, i, j = 1, ..., n, diag if oscillators and K = (ωx, ωy, 0) i j belong to dierent layers and , whose convergence to zero will imply the transversal sta- C D (17) bility of the synchronization manifold M. where constants ax, ay, ωx, and ωy are to be determined. Subtracting the i-th equation from the j-th equation We add and subtract additional terms with ma- in system (1), we obtain the equations for the transversal AijXij trix Aij de ned in (17) from the stability system (16) and stability of M obtain
n 1 ˙ P R Xij = F(xj) − F(xi) + {cjkP Xjk − cikP Xik+ X˙ ij = DF(βxj + (1 − β)xi)dβ − Aij Xij + AijXij k=1 0 n djkLXjk − dikLXik}, i, j = 1, ..., n. P (15) + {cjkP Xjk − cikP Xik + djkLXjk − dikLXik}. k=1 To obtain the explicit dependence of F(xj) − F(xi) on (18) we introduce the following vector notation Xij, The introduction of the terms AijXij allows for obtain- ing stability conditions of the trivial xed point Xij = 0, 1 Z i, j = 1, .., n in two steps. Note that the matrix −A F(xj) − F(xi) = DF(βxj + (1 − β)xi)dβ Xij, contributes to the stability of the xed point and can compensate for instabilities induced by eigenvalues with 0 12 nonnegative real parts of the Jacobian DF. This can be synchronized, our approach based on the calculation of achieved by increasing parameters ax, ay, ωx, and ωy. At ax, ay, is thus limited to this class of oscillators. the same time, the instability originated from its posi- The proof of global stability in (20)-(22) and deriva- tively de nite counterpart, matrix +A, can be damped tion of bounds ax, ay, and ωx and ωy involves the con- by the coupling terms through cij and dij. struction of a Lyapunov function along with the assump- Step I. We make the rst step by introducing the follow- tion of the eventual dissipativeness of the coupled sys- ing auxiliary systems for i, j = 1, ..., n tem. Therefore, before advancing with the study of larger networks (1), one has to prove that globally stable syn- 1 Z chronization in the simplest x-, y-, and (x, y)-coupled, ˙ two-oscillator networks is achievable. The bound for Xij = DF(βxj + (1 − β)xi) dβ − Aij Xij. (19) ax -coupled Lorenz oscillators was given in [30]. Upper 0 x bounds for ay, ωx, and ωy can be derived similarly. This system is identical to system (18) where the coupling Having obtained the bounds ax, ay, and ωx and ωy, and terms are removed. therefore proving the stability of the auxiliary systems (20)- (22), we can now take the second step in analyzing Aij can take three dierent values, depending on whether oscillators i and j both belong to the x or y the full stability system (18). graphs, or belong to dierent graphs, for example, if i Step II. The bounds ax, ay, and ωx and ωy that sta- belongs to the x graph, and j belongs to the y graph (see bilize the auxiliary systems (20)-(22) reduce the stability (17)). Therefore, we have three types of the auxiliary analysis of system (18) to the following equations by ex- systems cluding the term in brackets
n 1 ˙ P ˙ R Xij = AijXij + {cjkP Xjk − cikP Xik + djkLXjk− Xij = DF(βxj + (1 − β)xi) dβ − axP Xij (20) k=1 0 if and both belong to layer, dikLXik}, i, j = 1, ..., n. i j x (23)
Note that the positive term AijXij, which contains the upper bounds and and is destabilizing and 1 ax, ay, ωx ω, ˙ R must be compensated for by the coupling terms. To study Xij = DF(βxj + (1 − β)xi) dβ − ayL Xij (21) 0 the stability of (23) we introduce a Lyapunov function of if i and j both belong to y layer, the form n n 1 X X T (24) 1 V = Xij · I · Xij, ˙ R 4 Xij = DF(βxj + (1 − β)xi) dβ − K Xij (22) i=1 j=1 0 if i and j each belong to dierent layers. where I is an s × s identity matrix. Its time derivative with respect to system (23) becomes Remarkably, the auxiliary system (20) coincides with the n n dierence system for the global stability of synchroniza- ˙ 1 P P T V = 2 XijAijXij− tion in a two-oscillator network (1) with only x coupling, i=1 j=1 n n n where ax plays the role of the double coupling strength 1 P P P T T 2 {cjkXjkIP Xjk − cikXikIP Xik}− that guarantees the stability (see [30] for a detailed dis- i=1 j=1 k=1 n n n cussion on this relation). 1 P P P T T 2 {djkXjkILXjk − dikXikILXik}. Similarly, the stability of auxiliary system (21) implies i=1 j=1 k=1 global stability of synchronization in the two-oscillator (25) network (1) with only y coupling, where ay is the double We need to demonstrate the negative semi-de niteness coupling strength of the connection. Lastly, the sta- of the quadratic form ˙ . As ( 2 2 2 ), the y V Xii = 0, Xij = Xji bility of auxiliary system (22) guarantees globally stable rst sum simpli es to synchronization in the two-oscillator network with both and coupling, where a combination of constants n−1 n x y ωx X X 2 (26) S1 = AijXij. and ωy, present in K, is a combination of the double cou- pling strengths of x and y connections that is su cient i=1 j>i to induce stable synchronization in the two-oscillator x This sum is always positive de nite and its contribution and y coupled network. must be compensated for by the second and third sums Therefore, our immediate goal is to nd upper bounds n n n 1 on the values of ax, ay, and ωx and ωy that make the P P P T T S2 = − 2 {cjkXjkIP Xjk − cikXikIP Xik}, origin of the auxiliary systems (20)-(22) stable. This i=1 j=1 k=1 n n n amounts to proving global synchronization in the three 1 P P P T T S3 = − 2 {djkXjkILXjk − dikXikILXik}. x-, y-, and (x, y)-coupled networks that are composed of i=1 j=1 k=1 two oscillators. As only Type I oscillators can be globally (27) 13
Due to the coupling symmetry, the two terms in both that belong to the edges on the connection graphs C and S2 and S3 can be made identical by exchanging the in- D: the rst term on the LHS corresponds to the x layer dices i by j in the second terms such that and the second term is de ned by the edges of the y layer. At the same time, the variables on the right-hand n n n P P P T side (RHS) of (33) correspond to all possible dierences S2 = − cjkXjkIP Xjk, i=1 j=1 k=1 between pairs of oscillators that might or might not be n n n (28) P P P T de ned by edges of the connection graphs. Hence, to get S3 = djkXjkILXjk. i=1 j=1 k=1 rid of the presence of the dierences Xij and therefore nd the conditions explicit in the parameters of the net- Taking into account that we obtain Xjj = 0, work model (1), we express the dierences on the RHS n n−1 n via the dierences on the LHS such that we will be able P P P T S2 = − cjkXjiIP Xjk− to cancel them. i=1 k=1 j>k So far, we have closely followed the steps in the deriva- n n−1 n P P P T tion of the connection graph method [30] for single-layer cjkXjiIP Xjk, i=1 k=1 j
The addition of a positive term int to the auxil- Therefore, (36) turns into ωxbk LXij iary system (38) worsens its stability, therefore we have to introduce an additional parameter x and make sure m l m αk P 2 P 2 P x int x 2 that it is su ciently large to stabilize the new auxiliary n ckXk + n dkYk > [axbk + ωxbk + αk]Xk + k=1 k=1 k=1 system. A very important property is that, in (38), we l y int + P [a b + ω bint + αx]Y 2. have to add the positive, destabilizing term ωxb LXij y k y k y k k k=1 to the second equation for the (yj − yi) dierence but try (40) to stabilize the system via increasing the additional pa- Notice the new stabilizing constants αx and αy. Depend- rameter x in the rst equation for the equation k k αk (xj −xi) ing on the individual oscillator dynamics and tra c load (note the dierent inner matrices: versus in (38)). P L on edge k, these constants might have to be very large or Depending on the individual oscillator, chosen as the in- even in nite. dividual unit, this might not be possible, especially when Comparing the terms containing X and Y on the LHS tra c load int on the edge is high. This property is k k bk k and RHS of (40) and omitting the summation signs, we discussed in detail for the Lorenz and double scroll oscil- obtain the following conditions lator examples in Sections V and IV. A similar argument carries over to the auxiliary system (39) where we add 2 x int x 2 nckXk > [axbk + ωxbk + αk]Xk , k = 1, ..., m the destabilizing term int to the equa- 2 y int x 2 (41) ωybk Xij (xj − xi) nd Y > [a b + ω b + α ]Y , k = 1, ..., l. tion, but seek to stabilize the system via the additional k k y k y k y k parameter y in the equation. αk (yj − yi) Notice that we only modify the auxiliary systems for x Finally, we omit the dierence variables to obtain the and edges. All the other auxiliary systems for , that y Xij bounds on coupling strengths, ck for x edges and dk for y do not correspond to edges of either the x or y graphs, edges, su cient to make the derivative of the Lyapunov remain intact and de ned via the original systems (20)- function (25) negative semi-de nite, and therefore, en- (22). sure global stability of synchronization in the network Thus, the modi cations of (38) and (39) make the trou- (1). It follows from (41) that these upper bounds are P int 2 P int 2 blesome sums [ωxb ]X and [ωyb ]Y in (36) k k k k c > 1 [a bx + ω bint + αx], k = 1, ..., m k∈D k∈C k n x k x k k (42) disappear at the expense of worsened stability conditions 1 y int x dk > [ayb + ωyb + α ], k = 1, ..., l. of the corresponding auxiliary systems, which is re ected n k k y by the appearance of additional terms with x and y This completes the proof of Theorem 1. αk αk.
[1] Steven H Strogatz, Exploring complex networks, Na- Kurths, Synchronization: a Universal Concept in Nonlin- ture 410, 268 (2001). ear Sciences, Vol. 12 (Cambridge University Press, 2003). [2] Réka Albert and Albert-László Barabási, Statistical [11] Theoden I Neto and Steven J Schi, Decreased neu- mechanics of complex networks, Reviews of Modern ronal synchronization during experimental seizures, Physics 74, 47 (2002). Journal of Neuroscience 22, 72977307 (2002). [3] Mark EJ Newman, The structure and function of com- [12] Guanrong Chen and Y Xinghuo, Chaos Control Theory plex networks, SIAM Review 45, 167256 (2003). and Applications Series: Lecture Notes in Control and [4] Stefano Boccaletti, Vito Latora, Yamir Moreno, Martin Information Sciences, Vol. 292 (Springer, 2003). Chavez, and D-U Hwang, Complex networks: Structure [13] Leon Glass and Michael C Mackey, From Clocks to and dynamics, Physics Reports 424, 175308 (2006). Chaos: the Rhythms of Life (Princeton University Press, [5] John Buck and Elisabeth Buck, Synchronous re ies, 1988). Scienti c American 234, 7485 (1976). [14] Adilson E. Motter, Seth A. Myers, Marian Anghel, and [6] James J Collins and Ian N Stewart, Coupled nonlinear Takashi Nishikawa, Spontaneous synchrony in power- oscillators and the symmetries of animal gaits, Journal grid networks, Nature Physics 9, 191 (2013). of Nonlinear Science 3, 349392 (1993). [15] Igor Belykh, Vladimir Belykh, and Martin Hasler, [7] Michael R Guevara, Alvin Shrier, 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 ear Phenomena 195, 188206 (2004). and Circulatory Physiology 254, H1H10 (1988). [16] Maurizio Por ri, 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 Por ri 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). actions on Automatic Control 52, 17671773 (2007). [10] Arkady Pikovsky, Michael Rosenblum, and Jürgen [18] Francesco Sorrentino and Edward Ott, Adaptive syn- 16
chronization of dynamics on evolving complex networks, (2008). Physical Review Letters 100, 114101 (2008). [36] Jie Sun, Erik M Bollt, and Takashi Nishikawa, Master [19] Paul So, Bernard C Cotton, and Ernest Barreto, Syn- stability functions for coupled nearly identical dynamical chronization in interacting populations of heterogeneous systems, Europhysics Letters 85, 60011 (2009). oscillators with time-varying coupling, Chaos: 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 on- tions, Proceedings of the National Academy of Sciences o stochastic networks: windows of opportunity, IEEE 107, 1034210347 (2010). Transactions on Circuits and Systems I: Regular Papers [38] Hiroya Nakao, Tatsuo Yanagita, and Yoji Kawa- 62, 12601269 (2015). mura, Phase-reduction approach to synchronization of [21] Igor Belykh, Mario di Bernardo, Jürgen Kurths, and spatiotemporal rhythms in reaction-diusion systems, Maurizio Por ri, Evolving dynamical networks, Phys- Physical Review X 4, 021032 (2014). ica D: Nonlinear Phenomena 267, 16 (2014). [39] J Aguirre, R Sevilla-Escoboza, R Gutiérrez, D Papo, and [22] Olga Golovneva, Russell Jeter, Igor Belykh, and Maur- JM Buldú, Synchronization of interconnected networks: izio Por ri, Windows of opportunity for synchronization the role of connector nodes, Physical Review Letters in stochastically coupled maps, Physica D: Nonlinear 112, 248701 (2014). Phenomena 340, 113 (2017). [40] Liyue Zhang, Adilson E Motter, and Takashi Nishikawa, [23] Louis M Pecora and Thomas L Carroll, Master stabil- Incoherence-mediated remote synchronization, Physi- ity functions for synchronized coupled systems, Physical cal Review Letters 118, 174102 (2017). Review Letters 80, 2109 (1998). [41] Takashi Nishikawa, Jie Sun, and Adilson E Motter, Sen- [24] James F Heagy, Louis M Pecora, and Thomas L Carroll, sitive dependence of optimal network dynamics on net- Short wavelength bifurcations and size instabilities in work structure, Physical Review X 7, 041044 (2017). coupled oscillator systems, Physical Review Letters 74, [42] Nancy Kopell and Bard Ermentrout, Chemical and elec- 4185 (1995). trical 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 of America 101, 1548215487 (2004). and Systems I: Fundamental Theory and Applications [43] Igor Belykh, Reimbay Reimbayev, and Kun Zhao, Syn- 43, 161165 (1996). ergistic eect of repulsive inhibition in synchronization [26] Stefano Boccaletti, Jürgen Kurths, Grigory Osipov, of excitatory networks, Physical Review E 91, 062919 DL Valladares, and CS Zhou, The synchronization of (2015). chaotic systems, Physics Reports 366, 1101 (2002). [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). 438, 4344 (2005). [28] Chai Wah Wu, Synchronization in Coupled Chaotic Cir- [45] Bruno Eckhardt, Edward Ott, Steven H Strogatz, cuits & Systems, Vol. 41 (World Scienti c, 2002). Daniel M Abrams, and Allan McRobie, Modeling [29] Xiao Fan Wang and Guanrong Chen, Synchronization in walker synchronization on the Millennium Bridge, Phys- scale-free dynamical networks: robustness and fragility, ical Review E 75, 021110 (2007). IEEE Transactions on Circuits and Systems I: Funda- [46] Igor V Belykh, Russell Jeter, and Vladimir N Be- mental Theory and Applications 49, 5462 (2002). lykh, Bistable gaits and wobbling induced by pedestrian- [30] Vladimir N Belykh, Igor V Belykh, and Martin Hasler, bridge interactions, Chaos: An Interdisciplinary Journal Connection graph stability method for synchronized of Nonlinear Science 26, 116314 (2016). coupled chaotic systems, Physica D: Nonlinear Phenom- [47] Igor Belykh, Russell Jeter, and Vladimir Belykh, Foot ena 195, 159187 (2004). 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 lin- (2010). ear dynamics in random directed networks, IEEE Trans- [49] Mikko Kivelä, Alex Arenas, Marc Barthelemy, James P actions on Automatic Control 51, 12071210 (2006). Gleeson, Yamir Moreno, and Mason A Porter, Multi- [33] Zhi Li and Guanrong Chen, Global synchronization and layer networks, Journal of Complex Networks 2, 203271 asymptotic stability of complex dynamical networks, (2014). IEEE Transactions on Circuits and Systems II: Express [50] Stefano Boccaletti, Ginestra Bianconi, Regino Criado, Briefs 53, 2833 (2006). Charo I Del Genio, Jesús Gómez-Gardenes, Miguel Ro- [34] Igor Belykh, Martin Hasler, Menno Lauret, and Henk mance, Irene Sendina-Nadal, Zhen Wang, and Massim- Nijmeijer, Synchronization and graph topology, Inter- iliano Zanin, The structure and dynamics of multilayer national Journal of Bifurcation and Chaos 15, 34233433 networks, Physics Reports 544, 1122 (2014). (2005). [51] Manlio De Domenico, Clara Granell, Mason A Porter, [35] Igor Belykh and Andrey Shilnikov, When weak inhibi- and Alex Arenas, The physics of spreading processes in tion synchronizes strongly desynchronizing networks of multilayer networks, Nature Physics 12, 901 (2016). bursting neurons, Physical Review Letters 101, 078102 [52] Filippo Radicchi and Ginestra Bianconi, Redundant in- 17
terdependencies boost the robustness of multiplex net- 038701 (2015). works, Physical Review X 7, 011013 (2017). [57] Edward N Lorenz, Deterministic nonperiodic ow, [53] Francesco Sorrentino, Synchronization of hypernetworks Journal of the Atmospheric Sciences 20, 130141 (1963). of coupled dynamical systems, New Journal of Physics [58] Rabinder N Madan, Chua's Circuit: a Paradigm for 14, 033035 (2012). Chaos (World Scienti c Publishing, 1993). [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 ma- (1959). trices, Physical Review E 86, 056102 (2012). [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: An Interdisciplinary Journal of works with multiple interaction layers, Science Advances Nonlinear Science 16, 015102 (2006). 2, e1601679 (2016). [61] Reimbay Reimbayev, Kevin Daley, and Igor Belykh, [56] Xiyun Zhang, Stefano Boccaletti, Shuguang Guan, 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).