Improving Network Structure can lead to Functional Failures

In many real-world networks the ability to synchronize is a key property for their performance. Recent work on undirected networks with diffusive interaction revealed that improvements in the network connectivity such as making the network more connected and homogeneous enhances synchronization. However, real-world networks have directed and weighted connections. In such directed networks, understanding the impact of structural changes on the network performance remains a major challenge. Here, we show that improving the structure of a directed network can lead to a failure in the network function. For instance, introducing new links to reduce the minimum distance between nodes can lead to instabilities in the synchronized motion. This effect only occurs in directed networks. Our results allow to identify the dynamical importance of a link and thereby have a major impact on the design and control of directed networks.

 is smooth, α ≥ 0 is the overall coupling strength, and the matrix W describes the network structure, i.e., ≥ W 0 ij measures the strength of interaction from node j to node i. We assume that the network solutions are bounded and the local coupling function H is smooth satisfying ( ) = H 0 0. This last condition guarantees that the synchronous state is a solution of the coupled equations for all values of α. The overall coupling strength α represents the fixed energy cost per connection.
Here, we consider weakly connected networks, that is, when ignoring the link's directions the network is connected. A directed network is strongly connected if every node can be reached by every other node through a directed path. Of course, a strongly connected network is also weakly connected. If a directed network is not strongly connected the links can be partitioned into two different classes: links belonging to some strongly connected subnetwork, called strongly connected component, and links belonging to some cutset. A cutset is a set of links which point from one strongly connected component to another 18 . In physical terms, this situation corresponds to a master-slave configuration. In the top left inset of Fig. 1 we show a network composed of two strongly connected subnetworks indicated by the gray dotted ellipses. The cutset consists of the links → 1 4 and → 1 5, connecting the two components. The smaller strongly connected component does not influence the larger component as there are no links from the smaller to the larger component. Nonetheless, the network still supports stable synchronous dynamics. The particular effects of directed networks come into play once we try to improve the network structure. Introducing a new link pointing from node 4 to node 1 improves the connection structure significantly in the sense that the whole network is now strongly connected: any two nodes in the network are connected by a directed path. However, this structural improvement has fundamental consequences for the dynamics: the synchronous state becomes unstable.
An important remark is that the effect of suppressing synchronization is not restricted to the master-slave configuration. The effect has nontrivial origins. For instance, except for the links → 4 1 and → 5 1, all other possible links opposite to the cutset (for instance → 4 2) will make the synchronous state more stable. Moreover, synchronization loss can also be observed in strongly connected networks, if links are added or reinforced (see Fig. 2 and explanation below).
We illustrate this effect with two different classes of node dynamics, namely, Hindmarsh-Rose neurons in the chaotic bursting regime 19 as shown in the inset of Fig. 1a), and chaotic Rössler oscillators 20 , see inset of Fig. 1b). The state of each node is given by a three-dimensional vector = ( , , ). x Details on the models can be found in the supplementary material. All nonzero weights W ij in the network on the top left are set to one and we choose the global coupling α such that the nodes synchronize, that is, for any nodes i and j the difference of states ( ) − ( ) This synchronous dynamics can be seen in Fig. 1  , which leads to the strongly connected network on the top right. As can be seen for times > t 2000 this destabilizes the synchronous state. To understand this phenomenon, we analyze the stability of the synchronization subspace The variational equation of eq. (1) along γ ( ) t can be decomposed into n blocks of the form where Df denotes the Jacobian, Γ = ( ) H D 0 . The λ i are the eigenvalues of the Laplacian matrix = − L D W where D is the diagonal matrix with the row sums of W on its diagonal 16,17 . We assume that the spectrum of Γ is real, which is the case for many applications such as electrical synapses and resistive coupling in electrical circuits. Now, the n decoupled equations of the variational equation given by eq.
(2) are m-dimensional equations which only differ by the Laplacian eigenvalues λ i . We consider the case where L has a simple zero eigenvalue 5 . In this case, the eigenvalues of L have nonnegative real parts, so we can order them increasingly according to their real parts The eigenvalues with non-zero real part correspond to dynamics transverse to the synchronization subspace. Therefore, if the corresponding equations (2) have stable trivial solutions, synchronization in eq. (1) is stable. For a large class of coupling functions and local dynamics 16 , the stability condition for synchronization is given by More involved stability conditions can be tackled, but the analysis becomes more technical without providing new insight into the phenomenon. Condition (3) shows that the spectral gap . λ 2 plays a central role for synchronization properties of the network. Structural changes which decrease the real part of λ 2 can destabilize the synchronous state, see Fig. 3.
Let us consider the generic case where the eigenvalue λ 2 is simple. Using perturbation analysis 21 , we obtain the direction of growth of λ 2 as a function of the structural modifications in the network. This analysis leads to a characterization of links which are capable of destabilizing the network. The spectral where ⋅ , ⋅ is the Euclidean inner product, u a left and v a right eigenvector of the unperturbed Laplacian L corresponding to the spectral gap λ 2 , and we choose the vectors such that , > u v 0. If the network is undirected then L is symmetric and the left and right eigenvectors are dual. As  L is positive semi-definite we obtain λ ′ ≥ 0 2 . So increasing weights as well as adding links do not decrease the spectral gap in undirected networks 22 . This is an essential difference to directed networks.
Let us now consider a directed network composed of two strongly connected components as in the example in Fig. 1. Our approach is general and this choice is for the sake of simplicity. In this situation, the Laplacian can be represented in block form where C represents the cutset pointing from the strong component W 1 to W 2 , , L 1 2 are the respective Laplacians and D c is again a diagonal matrix with the row sums of C on its diagonal. As a consequence of the block structure, eigenvalues of L are either eigenvalues of L 1 or eigenvalues of . Suppose the eigenvalue with the smallest nonzero real part is located in the second component W 2 (this encloses the example from Fig. 1). Using a Perron-Frobenius argument, we can show that the eigenvalue λ 2 is real, see supplementary material. So, the eigenvectors u and v are real and by eq. (4) the motion of λ 2 is along the real axis.
To determine the effects of the network changes on the spectral gap we investigate the left and right eigenvectors , = (  takes into account the structural changes ∆. Now, such a modification will weaken the stability or even lead to instabilities if λ ′<0 2 . Determining the modifications that yield a decrease of the spectral gap is an involved problem, and we shall tackle it elsewhere. Here, we will focus on the example of Fig. 1, as it contains all central concepts without technical intricacies.
In the example from Fig. 1, Because all the involved quantities are positive, the spectral gap does not decrease when reinforcing the cutset: it represents a stabilising class of links. By increasing strengths in the cutset we are guaranteed to enhance synchronization.
As we mentioned earlier, our findings are not restricted to the case of strong components connected by cutsets but can also be observed in strongly connected networks themselves as shown in Fig. 2. Here, all the links have weight one initially. When we increase the weight on link → 2 3 the spectral gap decreases and the synchronous state will become less stable. On the other hand, increasing weights on the links → 3 2, → 2 1 and → 2 4 respectively increases the spectral gap. Furthermore, introducing the new links → 1 2 and → 4 2 with a weak coupling strength also increases the spectral gap. As the network is strongly connected, the observed effects cannot be explained by simple master-slave behaviour. See supplementary material for simulations and more details on this network.

Discussion
Our results revealed that directed and undirected networks behave essentially distinct under structural changes. Namely, assuming the stability condition eq. (3), synchronization loss caused by structural improvements is a property inherent to directed networks exclusively. If α λ α ( )  R c 2 , the network modification may not destroy synchronization. However, it worsens the quality in the sense that the transient towards synchronization becomes larger. The network improvement can also affect the basin of attraction of the synchronization state 16,24 .
An important problem is to characterize the links capable of suppressing synchronization a priori in terms of their topological role in the network. This problem remains undisclosed. We provided a partial answer for a class of stabilizing links: the cutsets. However, our approach allows one to study such links with a low computational cost in terms of eq. (7).
Recently, interconnected networks have attracted much attention 25,26 , as they can exhibit catastrophic cascades of failures when connections are undirected. Our results suggest that interconnected networks in which interconnections are represented by (directed) cutsets behave qualitatively different. Furthermore, our results shed light on how to plan and design network modifications without destroying the network performance, as discussed for power-grids in 1,27 .
The catastrophic effects of structural improvements on the network function have a long history in game theory. In the realm of games such effects are known as Braess's paradox 28 . In games, the effect occurs because the players take rational decisions to optimize their strategies. In the case of complex