Layered complex networks as fluctuation amplifiers

Abstract

In complex networked systems theory, an important question is how to evaluate the system robustness to external perturbations. With this task in mind, I investigate the propagation of noise in multi-layer networked systems. I find that, for a two layer network, noise originally injected in one layer can be strongly amplified in the other layer, depending on how well-connected are the complex networks in each layer and on how much the eigenmodes of their Laplacian matrices overlap. These results allow to predict potentially harmful conditions for the system and its sub-networks, where the level of fluctuations is important, and how to avoid them. The analytical results are illustrated numerically on various synthetic networks.

Faster network disruption from layered oscillatory dynamics

“…(4), spatially uncorrelated and white in time noise acting on the first layer x i 's might lead to amplified fluctuations in the second layer y i 's. 1 Indeed, the variance of the degrees of freedom in the first layer is generated by η i 's which are i.i.d. and white in time, and yields 1…”

“…It was recently shown that noise acting on layer might be strongly amplified in the other layers depending on the network connectivities. 1 Besides being amplified, the noise structure seems to align with the lowest-lying eigenmodes of the network Laplacian. Building on these results, I investigate the first escape time from the initial basin FIG.…”

“…13,14 In such layered dynamics, noise acting on one sub-network propagates to other layers with system-specific correlations. 1 In other words, when going through a layer, the noise acquire some specific statistical properties, which is then injected into other layers. Including such system specific correlation in the noise might increase the level of fluctuations and thus the risk of transitions between equilibria.…”

“…(c) Time-evolution of the winding numbers q 1 and q 2 in each layer (c) of the two-layers system shown in panel (a), corresponding to Eqs. (1) with parameters such that injected noise amplitudes are the same in both layers. The two layers have the same cyclic coupling network of size n = 40 nodes.…”

“…The latters might have serious consequences on the desired functioning of some layers, one of them being the amplification of fluctuations. 1 Together with the inherent multistability of nonlinearly coupled dynamical systems, noise acting on one layer may be more prompt to drive other layers outside their initial basin of attraction, through a transition to another equilibrium, if such a state exists. Here, I investigate how first escape times are affected by noise propagating through a layered system and compare it to uncorrelated noise.…”