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 signicance 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 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 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]. Signicant 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 signicant 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 dicult to evaluate. xi = (xi ; :::; xi ) s coordinates of the i-th oscillator, F : s ! s describes In this paper, we report our signicant 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 dene 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- dicult 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 verica- that an interlayer trac 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 trac load by complete synchronization dened by the synchronization a stronger pairwise converging coupling (a good link) manifold M = fx1 = x2 = ::: = xng: 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