Synchronization dynamics in non-normal networks: the trade-off for optimality

TL;DR

The paper addresses how to design networks that maintain synchronization robustness in the presence of strong directionality and non-normality. It uses the Master Stability Function (MSF) to relate synchronous stability to the Laplacian spectrum $\mathcal{L}$, and employs a first-order averaging (Magnus) to derive a time-independent dispersion relation that captures stability across modes. Validation is performed with a Brusselator-based reaction-diffusion model on toy networks, including a normal bidirectional circulant and a non-normal chain, with pseudo-spectrum analysis revealing non-normal transient amplification. The study concludes that there is no single optimal topology; robustness comes from a trade-off between directedness and non-normality, with implications for designing real-world networks with robust synchrony.

Abstract

Synchronization is an important behavior that characterizes many natural and human made systems composed by several interacting units. It can be found in a broad spectrum of applications, ranging from neuroscience to power-grids, to mention a few. Such systems synchronize because of the complex set of coupling they exhibit, the latter being modeled by complex networks. The dynamical behavior of the system and the topology of the underlying network are strongly intertwined, raising the question of the optimal architecture that makes synchronization robust. The Master Stability Function (MSF) has been proposed and extensively studied as a generic framework to tackle synchronization problems. Using this method, it has been shown that for a class of models, synchronization in strongly directed networks is robust to external perturbations. In this paper, our approach is to transform the non-autonomous system of coupled oscillators into an autonomous one, showing that previous results are model-independent. Recent findings indicate that many real-world networks are strongly directed, being potential candidates for optimal synchronization. Inspired by the fact that highly directed networks are also strongly non-normal, in this work, we address the matter of non-normality by pointing out that standard techniques, such as the MSF, may fail in predicting the stability of synchronized behavior. These results lead to a trade-off between non-normality and directedness that should be properly considered when designing an optimal network, enhancing the robustness of synchronization.

Figures (5)

• Figure 1: The network toy models for the case of a normal bidirectional circulant network, panel $a)$, and a non-normal bidirectional chain, panel $b)$. $c)$ Normalized Henrici's departure from non-normality as a function of tunning parameter $\epsilon$ for the non-normal model. We observe that starting from $0$, the network is symmetric, and the non-normality increases as the weight of the reciprocal edges decreases, taking the maximal value of non-normality in the limit when $\varepsilon=0$. In this case, the Laplacian spectrum is degenerate.
• Figure 2: $a)$ MSF for the Brusselator model with $b=2.5$, $c=1$ (limit cycle regime), $D_{\varphi}=0.7$, $D_{\psi}=5$ on a circulant network of $20$ node; $\Lambda_\alpha$ indicates the the Laplacian's eigenvalues, of which we plot only the real part. In this setting the system should remain stable after a perturbation: in fact, when the network is symmetric ($\varepsilon=0$), the discrete MSF (black dots) lies on the continuous one (magenta line); however, when we introduce an asymmetry in the topology as $\varepsilon$ decreases (red and blue dots), the MSF reaches the instability region, and the system loses synchronization. $b)$ The equivalent rapresentation in the complex domain where the instability region is shaded magenta and the discrete Laplacian spectrum is denoted by the symbols. For the network topology with at least one eigenvalue that lies in the instability region, the synchronized state is lost.
• Figure 3: $a)$ The comparison of the MSF and dispersion relation for the Brusselator with model parameters $b=3$, $c=1.8$, $D_{\varphi}=0.7$, $D_{\psi}=5$. We depict in magenta the MSF of the system in a limit cycle regime and cyan the dispersion relation of the averaged autonomous system. Inset: Similar comparison for a set of parameters where the instability occurs. Notice also the lack of continuity of the stability interval of eigenvalues. $b)$ The same representation in the complex domain. We see that for the chosen values of the parameters, the two approaches give an excellent agreement in predicting the instability interval.
• Figure 4: Desynchronization in a non-normal network. The parameters for the Brusselator model are as follows $b=2.5$, $c=1$, $D_{\varphi}=0.7$, $D_{\psi}=5$ on the (directed chain) non-normal network of $20$ nodes with $\varepsilon=0.1$ of Fig. \ref{['fig:net_model']}$b)$. As it can be observed from panels $a,)$ and $b)$, respectively, for the MSF and the stability region, the set of parameters is such that the MSF is neatly stable. Nevertheless, the instability occurs as shown by the pattern evolution in panel $c)$ at odd with the outcome that would have been expected from the symmetrized version. Such a result is strong evidence of the role of the network non-normality in the nonlinear dynamics of the system under investigation. The mechanism that drives the instability in the non-normal linearized regime manifests in the transition growth of the perturbations vector $\mathbf{x}(t)$ eq. \ref{['eq:compact']}, the blue curve in panel $d)$, before the system relaxes to the oscillatory state of the equilibrium. Such growth might transform in a permanent instability for the nonlinear system $\textbf{u}(t)=\left[\boldsymbol{\varphi}(t),\boldsymbol{\psi}(t)\right]$, red curve.
• Figure 5: $a)$ The pseudo-spectral description of the stability of the directed chain of $20$ nodes for the Brusselator model with $b=2.5$, $c=1.12$, $D_{\varphi}=0.7$, $D_{\psi}=5$, and an initial condition perturbation of the average magnitude $\delta=0.1$. We show the pseudo-spectra for three different values of the control parameter $\varepsilon$ for the chain network, emphasizing the considerably large difference between the pseudo-spectra regions and the spectrum of the Laplacian matrix. Inset: the pseudo-spectra for many other values of the perturbation magnitude $\delta$ for the chain with $\varepsilon=0.1$. Notice that although the eigenvalues do not lie inside the instability region due to the lack of an imaginary part, the pseudo-spectra might do. $b)$ The comparison between the expected outcome as predicted from the MSF and the actual outcome as measures by the standard deviation of the desynchronized pattern. The stability basin (shaded grey) projected onto the limit cycle plane for the non-normal case, panel $c_1)$ and the symmetrized (normal) one, panel $c_2)$, calculated over $300$ different initial conditions (of the same averaged magnitude) and a perturbation whose maximum magnitude varies from $10^{-3}$ to $1$. Inset: In the $y$-axis we plot the points of limit cycle we perturb and in the $x$-axis the magnitude of the perturbation; the colormap gives the fraction of orbits that conserve the synchronized regime. It can be clearly noticed that the attraction basin for the non-normal network is strongly reduced, though not at the same amount compared to where the perturbation occurs.