Nature offers a wealth of systems where basic unities multiply and non-linearly interact with each other to eventually exhibit a synchronous behavior. In particular, biology displays a vast assortment of examples, from myocyte cells coordinating the heartbeats to the synchronous firing pattern of coupled neurons. The importance of self-organized collective behaviors in many biological systems has triggered the interest of scientists since long time, leading to the study of stability and, consequently, robustness of the synchronous state. In this context, a well-know mathematical tool has been developed, the Master Stability Function (MSF) [1], through which it is possible to investigate, among other parameters, the role that different interacting topologies have in the stability of the synchronized behavior [2]. It has been shown, for example, that long-range links enhance the possibility of synchronization [3] or that asymmetry can reduce it [4]. In this work, we studied the phenomenon of synchronization on non-normal networks, i.e., networks having a strong degree of directionality and a hierarchical structure, which appear to be ubiquitous in real scenarios [5]. At variance with the classical MSF approach, here we performed a stability analysis for the Jacobian matrix averaged on a period of the synchronous solution [6], studying in this way the stability as a function of the graph Laplacian eigenvalues (see the figure). In particular, to capture the non-normality effect, we complement the MSF analysis by taking into account the pseudo-spectrum [7]. We found (blue dots) that due to the non-normality of the network, the perturbed eigenvalues become unstable, despite the fact that the spectrum is stable, hence impeding the synchronization of the system. In the small boxes we plot a function d(t) indicating the difference between the behaviors of each oscillator: we observe that when the network is symmetric (green) the system synchronizes (d(t) = 0), while in the non-normal case (blue) it does not. Such an outcome is consistent with our previous results for the pattern formation problem and states that linear analysis could be not sufficient once non-linearity and non-normality are at play [8]. This shows that non-normal networks are more difficult to synchronize if compared with their symmetric analogues. Moreover, by repeating the analysis with a classical MSF approach, we have corroborated our result by showing that non-normality may impede synchronization, even with a negative MSF, as in the case above. We have validated our results by considering several ”realworld networks” of biological interest, e.g., the C. elegans neuronal network.
