The Effects of Structural Perturbations on the Synchronizability of Diffusive Networks
Abstract
We investigate the effects of structural perturbations on the networks ability to synchronize. We establish a classification of directed links according to their impact on synchronizability. We focus on adding directed links in weakly connected networks having a strongly connected component acting as driver. When the connectivity of the driver is not stronger than the connectivity of the slave component, we can always make the network strongly connected while hindering synchronization. On the other hand, we prove the existence of a perturbation which makes the network strongly connected while increasing the synchronizability. Under additional conditions, there is a node in the driving component such that adding a single link starting at an arbitrary node of the driven component and ending at this node increases the synchronizability.
Keywords
Ordinary differential equations Synchronization Stability theory Phase transitions Perturbations Graphs and linear algebra Network models deterministicMathematics Subject Classification
05C82 34D06 82B26 93C73 05C50 90B101 Introduction
Synchronization is an important phenomenon in realworld networks. For instance, in power grids, power stations must work in 50 Hz synchrony in order to avoid blackouts (Dörfler and Bullo 2012; Motter et al. 2013). In sensor networks, synchronization among the sensors is vital for the transmission of information (Papadopoulos et al. 2005; Yadav et al. 2017). On the other hand, synchronization of subcortical brain areas such as in the Thalamus is strongly believed to be the origin of motor diseases such as Dystonia and Parkinson (Hammond et al. 2007; Milton and Jung 2003; Starr et al. 2005). In all of the mentioned examples, the stability of synchronous states is crucial for the network’s function or dysfunction, respectively. Motivated by these observations, stability properties of synchronous states in systems of coupled elements have been investigated intensively (Barahona and Pecora 2002; Pecora and Carroll 1998; Pikovsky et al. 2001; Field 2017; Li et al. Oct 2007).
An important class, mimicking the above examples, is given by networks of identical elements which are coupled in a diffusive manner. That is, networks for which the dynamics of a node depend on the difference between its own state and its input. A special focus has been on unravelling the connection between such a network’s coupling topology and its overall dynamics (Pereira et al. 2017; Jalili 2013; Nishikawa et al. 2003, 2017; Agarwal and Field 2010; Bick and Field 2017).
While certain correlations have been observed, there are few rigorous results determining the relation between a network’s structure and its dynamical properties (Wu and Chua 1996; Pogromsky and Nijmeijer 2001; Wang and Chen 2002; Ujjwal et al. 2016). There is even less known about the impact of structural perturbations of a network on its dynamical properties such as the stability of synchrony (see for instance Milanese et al. Apr 2010). A particularly interesting and important question in this category is the following: assume that a link’s weight in a network is perturbed or a new link with a small weight is added to the network. What is the impact on the dynamics? For instance, in interaction graphs of gene networks, it has been shown that adding links between two stable systems can lead to dynamics with positive topological entropy (Poignard 2013). In diffusive systems such as laser networks, it was shown that the addition of a link can lead to synchronization loss (Pade and Pereira 2015; Hart et al. 2015). In this article, we focus on the question whether these structural perturbations lead to higher or lower synchronizability. In the main body, we give rigorous answers to this question for directed networks. In undirected networks, under our assumptions, undirected perturbations will never decrease the synchronizability. Here, the only nontrivial situation appears when introducing directed perturbations to undirected networks. We deal with this case in the “Appendix”. Let us first introduce the model and motivate the main questions with some examples.
1.1 Model and Examples
1.2 Structural Perturbations in Directed Networks—About Masters and Slaves
Directed networks always consist of one or several strongly connected subnetworks in which every node is reachable from any other node through a directed path. If there is more than one strongly connected subnetwork, two such subnetworks can be connected through unidirectional links pointing from one subnetwork to another. In the top right of Fig. 1, we show a network composed of two strongly connected subnetworks (without the red link), which is weakly connected; starting from the smallest connected subnetwork, it is not possible to reach the larger connected subnetwork through a directed path. In the physics literature, this configuration is called master–slave coupling as the subnetwork consisting of nodes 1, 2 and 3 drives the subnetwork consisting of nodes 4 and 5.
An example for this is found in Fig. 1a. Introducing the new link (in red) makes the whole network strongly connected: there is a directed path connecting any two vertices in the network. Therefore, the addition of the link significantly improves the connectivity properties of the network. However, this structural improvement has a surprising consequence for the dynamics: the network synchronization is lost, as can be seen in the simulation in Fig. 1a.
Hinderance of synchronization is not about breaking a master–slave configuration One may think that this synchronization loss appears because we are breaking the master–slave configuration. This rationale is justified as master–slave configurations are known to synchronize well (Nishikawa and Motter 2006). However, the synchronization loss is not related to the master–slave breaking. Indeed, adding a different connection which also makes the network strongly connected stabilizes the synchronous state (see Fig. 1b).
Hinderance of synchronization is not about reinforcing the hub Synchronization loss in the example of Fig. 1a appeared as an additional link was added to the hub of the largest subnetwork (the most connected node in the network). However, running experiments on random graphs with hubs, we found several counterexamples in which linking to the hub improves synchronization.
To sum up, while in some settings, master–slave configurations and the presence of hubs play an important role for the behaviour of a network under structural perturbations (Pereira et al. 2017; Pereira 2010; Belykh et al. 2005), for networks with diffusive dynamics near synchronization adding extra links generates nonlinearities which can either enhance or hinder synchronization. Our main result (Theorem 1) gives an almost complete explanation of the complex behaviour of such weakly connected directed networks when a master–slave configuration is reinforced or destroyed, respectively.
1.3 Informal Statement of the Main Result
Using the master stability approach to tackle the transverse stability of the synchronization manifold M (Pereira et al. 2014), we can in fact reduce the stability problem to the spectral analysis of graph Laplacians. The rather mild assumptions needed for this approach are specified in Sect. 2.2. We emphasize that under these assumptions, the master stability function is unbounded. Furthermore, the left bound exclusively depends on the spectral gap \(\lambda _2\), i.e. the other Laplacian eigenvalues do not play a role for linear stability considerations. Let us now give an informal statement of our main result.
In Theorem 1, we consider networks consisting of two strongly connected components. The general case of a higher number of strongly connected components is a straightforward generalization.

Strengthening the driving facilitates synchronization, leading to shorter transients towards synchronization and augmenting the basin of attraction.

Master–Slave configurations are nonoptimal. It is always possible to break the master–slave configuration in a way that favours synchronization (e.g. Fig. 1). Provided the overall connectivity of the network is poor, it is even possible to find one or several nodes in the master component such that the addition of an arbitrary single link ending at this node and breaking the master–slave configuration increases the synchronizability. In fact, if additionally, the Laplacian of the master component has zero column sums, then any perturbation in opposite direction of the cutset enhances the synchronizability.

Breaking Master–Slave configurations can hinder Synchronization. If the connectivity of the master component is not much stronger than the connectivity of the slave component (a precise condition of this will be given in Theorem 1), we can always find a cutset such that there is a perturbation in opposite direction of this cutset for which synchronization is hindered. Our result reveals the role the eigenvectors of the master network play in the destabilization of the synchronous motion. For example, if \(\alpha _k\) is an eigenvalue of the Laplacian of the master and close to \(\lambda _2\) the spectral gap in the slave component (this is the case in our illustration), then the eigenvector \(\varvec{X}_k\) associated with \(\alpha _k\) encodes the important information about the possible destabilization. For instance, assume that the ith entry of \(\varvec{X}_k\) is the maximal (or minimal) one. If the slave network is driven by a link coming from the ith node, then it is possible to destabilize the synchronization.
2 Notations and Definitions
2.1 Weighted Graphs and Laplacian Matrices
We consider networks of identical elements with diffusive interaction. It will be useful to interpret the coupling structure of the network as a graph. We recall some basic facts on graph theory.
Definition 1
 (i)
We say that the graph is undirected if \((i,j)\in \mathcal {E}\iff (j,i)\in \mathcal {E}\) and \(w(i,j)=w(j,i)\) for all \((i,j)\in \mathcal {E}\). Otherwise, the graph is directed and edges are assigned orientations. A directed graph is also called digraph.
 (ii)
\(\mathcal {G}=(\mathcal {N},\mathcal {E},w)\) is a subgraph of \(\mathcal {G}^{\prime }=(\mathcal {N}^{\prime },\mathcal {E}^{\prime },w^{\prime })\) if \(\mathcal {N}\subseteq \mathcal {N}^{\prime }\), and \(\mathcal {E}\subseteq \mathcal {E}^{\prime }\). In this case, we write \(\mathcal {G}\subseteq \mathcal {G}^{\prime }\).
 (iii)The adjacency matrix \(\varvec{W}\in \mathbb {R}^{N\times N}\) of the graph \(\mathcal {G}\) is defined through$$\begin{aligned} W_{ij}=\left\{ \begin{array}{cc} w(i,j) &{} \text {if } (i,j)\in \mathcal {E}\\ 0 &{} else \end{array}\right. \end{aligned}$$
To deal with synchronization of networks, we will focus on graphs exhibiting some sort of connectedness.
Definition 2
 (i)
A digraph \(\mathcal {G}\) is strongly connected if every node is reachable from every other node through a directed path.
 (ii)
The digraph is weakly connected if it is not strongly connected and the underlying graph which is obtained by ignoring the links’ directions is connected. A maximal strongly connected subgraph of a weakly connected digraph is called strongly connected component, or strong component. The maximal set of links which connects two strong components is called cutset.
 (iii)
A spanning diverging tree of a digraph is a weakly connected subgraph such that one node, the root node, has no incoming edges and all other nodes have one incoming edge.
2.2 Synchronizability of Networks: Assumptions
Although equations of the form (1) are heavily used in the context of network synchronization, it was only very recently that a stability result has been established for the general case of timedependent solutions (Pereira et al. 2014). In order to guarantee for the stability of synchronous motion, we make the following assumptions:
A1 (Structural assumption)\(\mathcal {G}\) has a spanning diverging tree.
Assumption B1 guarantees that the nodes’ dynamics admit an invariant compact set, for instance an equilibrium, a periodic orbit or a chaotic orbit (such as is the case for the motivating example from Fig. 1).
The second dynamical condition B2 guarantees that the synchronous state \({x}_{1}\left( t\right) ={x}_{2}\left( t\right) =\cdots ={x}_{N}\left( t\right) \) is a solution of the coupled equations for all values of the overall coupling strength \(\alpha \): when starting with identical initial conditions, the coupling term vanishes and all the nodes behave in the same manner.
The last condition B3 remarks that for undirected graphs the zero eigenvalue of the graph Laplacian is nonsimple iff the underlying graph is disconnected (Brouwer and Haemers 2011). In this case, the stability condition would be violated. Indeed, in order to observe synchronization, it is clear that one should consider networks which are connected in some sense. We remark that the assumption that the \(\beta _j\) are real is true for many applications. The general case of complex eigenvalues \(\beta _j\) can be tackled in a similar way, but the analysis becomes more technical without providing new insight into the phenomena (Pereira et al. 2014).
2.3 Critical Threshold for Synchronization
2.3.1 Measures of Synchronization
We can use the critical coupling \(\alpha _c\) in order to define a measure of synchronizability.
Definition 3
Indeed, the range of coupling strengths which yield stable synchronization is larger for \(( \mathcal {G}_1, \varvec{f}_1, \varvec{H}_1)\). Fixing the dynamics \(\varvec{f}\) and the coupling function \(\varvec{H}\), we can now measure whether structural changes in the graph will favour or hinder synchronization. Assume we have a network \((\mathcal {G}, \varvec{f}, \varvec{H})\) and a perturbed network \((\tilde{\mathcal {G}}, \varvec{f}, \varvec{H})\) with corresponding spectral gaps \(\lambda _2\) and \(\tilde{\lambda }_2\).
A direct consequence of the definition of synchronizability is that if \(\mathfrak {R}(\lambda _2)<\mathfrak {R}(\tilde{\lambda }_2)\), the perturbed network \((\tilde{\mathcal {G}}, \varvec{f}, \varvec{H})\) is more synchronizable than \((\mathcal {G}, \varvec{f}, \varvec{H})\).
We also say that the modification favours synchronization. Otherwise, if \(\mathfrak {R}(\lambda _2)>\mathfrak {R}(\tilde{\lambda }_2)\), we say the structural perturbation hinders synchronization. This enables us to reduce the stability problem to an algebraic problem, i.e. the behaviour of the spectral gap under structural perturbations. We will use this approach throughout the whole article.
3 Main Result
In this section, we state our main result, Theorem 1, on perturbations of directed networks. We emphasize that, given assumption A1, the result is structurally generic, a term which we introduced in an earlier paper (Poignard et al. 2018). In order to explain the notion of structural genericity, consider the set of Laplacians corresponding to networks with identical coupling topologies but potentially different weights. In this set, the subset of Laplacians for which our results are valid is dense and its complement is of zero Lebesgue measure. In other words, given any network topology satisfying A1, our result is valid up to a small perturbation of the weights of the existing links of this network. This structural genericity is stronger than the classical one for which it is usually necessary to perturb drastically the structure of the original network itself. For more details on structural genericity, see Theorem 6.6 for the directed case and Theorem 3.1 for the undirected case in Poignard et al. (2018).
3.1 Structural Perturbations in Directed Networks
Remark 1
In the rest of the paper, to avoid cumbersome formulations, we will employ the formulation “a structural perturbation \(\varvec{\Delta }\) in direction of the cutset” to refer to a structural perturbation in direction of the cutset induced by a nonnegative matrix \(\varvec{\Delta } \in \mathbb {R}^{m\times n}\)” and similarly for structural perturbations in opposite direction of the cutset.
Notation
Observe, as in Definition 4, that the spectral gap map \(\varepsilon \mapsto \lambda _2\left( \varvec{L}_p(\varepsilon \varvec{\Delta })\right) \) is regular because of the simplicity of \(\lambda _2\left( \varvec{L_W}\right) \) (Horn and Johnson 1985). In Poignard et al. (2018), we proved that having simple eigenvalues is a structurally generic property for graph Laplacians of weakly connected digraphs that satisfy Assumption A1.
Notice that we can possibly have \(s\left( \varvec{\Delta }\right) \in \mathbb {C}\) since the matrices involved in this notation are no more symmetric. However, we will prove in Sect. 4 (Lemma 2) that in the case where \(\lambda _2\left( \varvec{L_W}\right) \) is an eigenvalue of \(\varvec{L_{2}}+\varvec{D_{C}}\), then \(\lambda _2\left( \varvec{L_W}\right) \) is real positive and therefore \(s\left( \varvec{\Delta }\right) \in \mathbb {R}\). We can now state our second main result in the directed case:
Theorem 1
 (i)
Invariance of synchronizability If the spectral gap \(\lambda _2\) of \(\varvec{L_W}\) is an eigenvalue of \(\varvec{L_1}\), then the network’s synchronizability is invariant under arbitrary structural perturbations \(\varvec{\Delta }\) in direction of the cutset.
 (ii)
Improving synchronizability by reinforcing the cutset If \(\lambda _{2}\) is an eigenvalue of \(\varvec{L_{2}}+\varvec{D_{C}}\), then the network’s synchronizability increases for arbitrary structural perturbations \(\varvec{\Delta }\) in direction of the cutset.
 (iii)Nonoptimality of master–slave configurations Assume \(\lambda _{2}\) is an eigenvalue of \(\varvec{L_{2}}+\varvec{D_{C}}\). Then, we have the following statements:
 (a)
There exists a structural perturbation \(\varvec{\Delta }\) in opposite direction of the cutset such that \(s(\varvec{\Delta })> 0\).
 (b)
There exists a constant \(\delta (\varvec{L}_1)>0\) and at least one node \(1\le k_0\le n\) (in the driving component) such that if we have \(0<\lambda _2<\delta (\varvec{L}_1)\), then \(s(\varvec{\Delta })> 0\) for any structural perturbation \(\varvec{\Delta }\) consisting of only one link in opposite direction of the cutset and ending at node \(k_0\).
 (c)
If, moreover, \(\varvec{L}_1\) has zero column sums, then there exists a constant \(\delta (\varvec{L}_1)>0\) such that if \(0<\lambda _2<\delta (\varvec{L}_1)\), we have \(s(\varvec{\Delta })> 0\) for any structural perturbation \(\varvec{\Delta }\) in opposite direction of the cutset.
 (a)
 (iv)
Hindering synchronizability by breaking the master–slave configuration There exists a cutset \(\varvec{C}\) for which \(\lambda _{2}\) is an eigenvalue of \(\varvec{L_{2}}+\varvec{D_{C}}\) and a perturbation \(\varvec{\Delta }\) in opposite direction of \(\varvec{C}\) such that: if \(\varvec{L}_1\) admits a positive eigenvalue sufficiently small, then we have \(s(\varvec{\Delta })\le 0\).
Let us make a few remarks. In the proof of items (i) and (ii), we repeatedly apply a perturbation result in order to handle nonsmall perturbations. This is not possible in items (iii)(a)–(c) and item (iv) because every perturbation in opposite direction of the cutset makes the graph strongly connected and thus qualitatively changes the network’s structure. In item (iii) a, the perturbation can be realised by turning an arbitrary node in the slave component into a hub having directed connections to all the nodes in the master component.
As we have shown numerically in the example in Fig. 1 and also in Pade and Pereira (2015), not all perturbations in opposite direction of the cutset are increasing the synchronizability when \(\varvec{L}_1\) does not have zero column sums. This is stated in item (iv): an example where the situation described in this item occurs is when the master component is an undirected subnetwork, i.e when \(L_1\) is symmetric, in which case all the eigenvalues are nonnegative.
In items (ii)–(iv), we assume that the spectral gap is an eigenvalue of \(\varvec{L_{2}}+\varvec{D_{C}}\). This happens for instance when the entries in the cutset \(\varvec{C}\) are very small (see Lemma 2). Topologically, this means that the master component is very well connected in comparison with the intensity and/or density of the driving force. It is worth remarking that the connection density of the second component does not play a role in this scenario. The rest of the paper is devoted to the proofs of our two main results and of other results completing our study.
4 Proof of the Main Result
The following standard result from matrix theory is the technical starting point for the rest of this article (Horn and Johnson 1985). It allows us to determine the dynamical effect of structural perturbations up to first order in the strength of the perturbation.
Lemma 1
In order to track the motion of the spectral gap through this representation, we first investigate the structure of the eigenvectors of \(\varvec{L_W}\) in the following two auxilliary lemmata. First, observe that the matrix \(\varvec{L}_{2}+\varvec{D_{C}}\) is nonnegative diagonally dominant (Berman and Plemmons 1994; Horn and Johnson 1985). This property enables us to find a Perron–Frobenius like result.
Lemma 2
Let \(\varvec{L_W}\) be as in Theorem 1. Then, \(\varvec{L}_{2}+\varvec{D_{C}}\) has a minimal simple, real and positive eigenvalue with corresponding positive left and right eigenvectors.
Proof
This Lemma shows that the spectral gap and the corresponding eigenvectors are real in this case. So, when changing the coupling structure, the motion of \(\lambda _{2}\) will be along the real axis by Lemma 1. Next, we investigate the structure of the eigenvectors of \(\varvec{L_W}\).
Lemma 3
Proof
Proof of Theorem 1
We will use throughout the proof that the smallest eigenvalue of \(\varvec{L}_2+\varvec{D_C}\) is simple, real and positive by Lemma 2.
Ad (iii)(b). Here, we first use a result proved in Poignard et al. (2018) on the structure of Laplacian spectra in the case of strongly connected digraphs. Such graphs admit a spanning diverging tree, and therefore, by Theorem 6.6 in Poignard et al. (2018), we have that for a generic choice of the nonzero weights of \(\varvec{L}_1\), the spectrum of this matrix is simple. Under this genericity assumption, we can thus suppose that \(\varvec{L}_1\) is diagonalizable.
In item (iv) of this theorem, the choice of the cutset \(\varvec{C}\) for which we hinder synchronization is not that sharp, indeed suppose only one entry in \(\varvec{X}_k\) is negative. Then, we can apply the same reasoning to the vector \(\varvec{X}_k\), for which \(n1\) entries will be non negative. In this case, the suitable cutsets \(\varvec{C}\) will be more numerous.
5 Discussion
In this paper, we have investigated the effect of structural perturbations on the transverse stability of the synchronous manifold in diffusively coupled networks. Establishing a connection between topological properties of a network and its synchronizability has been a challenge for the last few decades. So far, most of the existing literature focuses on establishing correlations supported by numerical simulations. Here, we present a first step in proving rigorous results for both, undirected and directed networks.
For directed networks, we have investigated the behaviour of a network when its cutset is perturbed. There is only scenario we did not investigate here: when the spectral gap is an eigenvalue of \(L_1\), determining the effect of a perturbation in opposite direction of the cutset cannot be solved in the framework presented above. It is of course possible to write down a similar term as in Eq. (19). However, in this case it involves left and right eigenvectors of \(L_1\). One would thus need to investigate eigenvectors of Laplacians of strongly connected digraphs, and more precisely the signs of their entries. To our knowledge, there have been no attempts to do so, yet.
Even more involved is the question whether there exists a classification of links according to their dynamical impact in strongly connected networks. To our knowledge, no results have been obtained for the general case so far either. This is also due to the fact that there are few attempts to extend the approaches on undirected graphs due to Fiedler (see Fiedler 1973) to directed graphs. As shown here, related results would essentially improve our understanding of the dynamical impact of a link in directed networks.
Notes
Acknowledgements
We are in debt with Mike Field, Chris Bick, Anna Dias, Jeroen Lamb, Matteo Tanzi, and Serhyi Yanchuk for useful discussions.
Funding
CP was supported by FAPESP Cemeai grant 2013/073750 and by EPSRC Centre for Predictive Modelling in Healthcare grant EP/N014391/1. TP was supported by FAPESP Cemeai grant 2013/073750, Russian Science Foundation grant 144100044 at the Lobayevsky University of Nizhny Novgorod, by the European Research Council [ERC AdG grant number 339523 RGDD] and by Serrapilheira Institute. JPP was supported by the DFG Research Center Matheon in Berlin. This study did not generate any new data.
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