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.

Our everyday life depends on network synchronization at various levels. In power grids, power stations must keep a proper synchronization to avoid energy supply disturbances and blackouts1. Sensor networks rely on synchronization among sensors to transmit information2. In the brain, epileptic seizures and Parkinson’s diseases are a strong manifestation of synchronization3. Consensus formation is another example of synchronized activities4567. These complex systems are modeled by networks with diffusive interaction, that is, the interaction between any two coupled elements depends on the difference of their states. So far, research efforts to understand the influence of connectivity on the dynamics have focused on undirected diffusive networks. For instance, it is known that increasing the homogeneity or the number of connections at random enhances synchronization8910111213.

Networks found in nature are often directed and weighted. For example, electrical synapses in neuron networks have asymmetric conductance14, which makes the underlying network directed. Recent work has provided sufficient conditions to guarantee the stability of synchronization in such networks in terms of the network structure and nature of the interaction151617. And although understanding the impact of structural modifications, such as changing weights and adding or deleting links on synchronization is fairly well understood for undirected networks, it remains an open problem for directed networks.

In this letter, using synchronization as a paradigm for network function, we show that improving the network connectivity structure can lead to a functional failure. This phenomenon has dynamical origins. Namely, in directed networks the structural improvement has a suppressing effect in the network spectrum leading to the onset of instabilities associated with the synchronous motion. Furthermore, we identify a class of links for which increasing the weights enhances synchronization. Our results provide a way to understand the dynamical importance of links in directed networks and to develop strategies to avoid functional failures when improving the network structure.

Results

We consider directed networks of identical elements with diffusive interaction. The theory we develop here is general and can include networks of non-identical elements with minor modifications16. The network dynamics is described by

where , is smooth, is the overall coupling strength, and the matrix describes the network structure, i.e., measures the strength of interaction from node to node . We assume that the network solutions are bounded and the local coupling function is smooth satisfying . This last condition guarantees that the synchronous state is a solution of the coupled equations for all values of The overall coupling strength 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 another18. 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 and , connecting the two components.

Figure 1

Improving connectivity leads to desynchronization. The figures show simulation results for the networks on top. All links in black have weight one. In the main plots we show the difference of the first component of x1 and the first component of x5. In a) the node dynamics is given by Hindmarsh-Rose (HR) neurons, and in b) by Rössler dynamics. The global coupling α is chosen such that the nodes synchronize chaotically for the original network. This can be seen in the main plots for times until t = 2000 in blue. The introduction of the new link with weight 0.4 at time t = 2000 leads to a destabilization, displayed in red. The insets show the time series of a single node. For the HR neurons we consider a chaotic bursting mode and for the Rössler dynamics a chaotic state.

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 to node 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 and , all other possible links opposite to the cutset (for instance ) 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).

Figure 2

Desynchronization in a strongly connected network: Strengthening the link in red can lead to synchronization loss. Initially, all the links have weight one, however, increasing the weight on the link decreases the spectral gap. On the other hand, increasing weights on the links , and increases the spectral gap. Furthermore, introducing the new links and with a weak coupling strength also increases the spectral gap.

We illustrate this effect with two different classes of node dynamics, namely, Hindmarsh-Rose neurons in the chaotic bursting regime19 as shown in the inset of Fig. 1a), and chaotic Rössler oscillators20, see inset of Fig. 1b). The state of each node is given by a three-dimensional vector Details on the models can be found in the supplementary material. All nonzero weights in the network on the top left are set to one and we choose the global coupling such that the nodes synchronize, that is, for any nodes and the difference of states vanishes for . This synchronous dynamics can be seen in Fig. 1 for times . At time we add the new link with a weight of , which leads to the strongly connected network on the top right. As can be seen for times this destabilizes the synchronous state.

To understand this phenomenon, we analyze the stability of the synchronization subspace , with . The variational equation of eq. (1) along can be decomposed into blocks of the form

where denotes the Jacobian, . The are the eigenvalues of the Laplacian matrix where is the diagonal matrix with the row sums of on its diagonal1617. We assume that the spectrum of is real, which is the case for many applications such as electrical synapses and resistive coupling in electrical circuits. Now, the decoupled equations of the variational equation given by eq. (2) are -dimensional equations which only differ by the Laplacian eigenvalues . We consider the case where has a simple zero eigenvalue5. In this case, the eigenvalues of 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 dynamics16, the stability condition for synchronization is given by

where . 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 plays a central role for synchronization properties of the network. Structural changes which decrease the real part of can destabilize the synchronous state, see Fig. 3.

Figure 3

Change of the spectral gap. In a schematic representation we illustrate the motion of the spectral gap under structural modifications in case is real. The network on the right has a spectral gap such that . Adding a link as indicated decreases the gap to , which violates the stability condition (3).

Let us consider the generic case where the eigenvalue is simple. Using perturbation analysis21, we obtain the direction of growth of 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 gap of a perturbed Laplacian is given by with

where is the Euclidean inner product, a left and a right eigenvector of the unperturbed Laplacian corresponding to the spectral gap , and we choose the vectors such that .

If the network is undirected then is symmetric and the left and right eigenvectors are dual. As is positive semi-definite we obtain . So increasing weights as well as adding links do not decrease the spectral gap in undirected networks22. 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 represents the cutset pointing from the strong component to , are the respective Laplacians and is again a diagonal matrix with the row sums of on its diagonal. As a consequence of the block structure, eigenvalues of are either eigenvalues of or eigenvalues of . Suppose the eigenvalue with the smallest nonzero real part is located in the second component (this encloses the example from Fig. 1). Using a Perron-Frobenius argument, we can show that the eigenvalue is real, see supplementary material. So, the eigenvectors and are real and by eq. (4) the motion of is along the real axis.

To determine the effects of the network changes on the spectral gap we investigate the left and right eigenvectors and . This decomposition corresponds to the triangular form of the Laplacian. Using the block structure, we can show that and that and are left and right eigenvectors of . Furthermore, again by a Perron-Frobenius argument we can show that both and are positive. By our assumptions, is not an eigenvalue of . Therefore, we can solve the eigenvector equation for in terms of to obtain , where is the identity matrix.

Now we introduce a link in the opposite direction of the cutset, so the Laplacian writes as

where is the matrix describing the new link, and is the associated diagonal matrix, see supplementary material. This yields

where

takes into account the structural changes . Now, such a modification will weaken the stability or even lead to instabilities if . 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, is symmetric and we have . Moreover,

and and are eigenvectors of with eigenvalue . Because is a common eigenvector of both and corresponding to a positive eigenvalue we obtain a decrease of the spectral gap with a rate . This is a main mechanism that generates instabilities: the eigenvectors of lie in the space spanned by the eigenvectors of with positive eigenvalues.

Formally, we can obtain all the structural changes capable for destabilization as a function of the eigenvectors of and . In contrast to undirected networks where there is a well developed theory relating eigenvectors to the underlying graph structure23, for directed graphs the theory is underdeveloped. Therefore, further analytical insights remain a challenge. From a computational point of view though, we can solve this problem for any given network.

For modifications in the direction of the cutset the situation is clear, as eq. (4) reduces to

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 the spectral gap decreases and the synchronous state will become less stable. On the other hand, increasing weights on the links , and respectively increases the spectral gap. Furthermore, introducing the new links and 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 , 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 state1624.

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 attention2526, 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 in127.

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 paradox28. In games, the effect occurs because the players take rational decisions to optimize their strategies. In the case of complex networks of dynamical systems considered here, the effect is dynamical and is a consequence of the motion of eigenvalues of the network Laplacian.

Additional Information

How to cite this article: Pade, J. P. and Pereira, T. Improving Network Structure can lead to Functional Failures. Sci. Rep. 5, 9968; doi: 10.1038/srep09968 (2015).