Synchronization, Collective Oscillations, and Information Flow in Duplex Networks

Abstract

In many real-world systems, partial synchronization is the dominant dynamical regime and, in systems such as the brain, is often accompanied by collective oscillations in which multiple overlapping modes interact to produce complex rhythmic activity. Here, we investigate duplex networks with reactive interlayer links, where full synchronization cannot be achieved. We show that when interlayer frequency differences between mirror nodes are uniformly distributed with sufficient width, the network self-organizes into collective macroscopic oscillations composed of multiple interacting modes. By linking macroscopic phase transitions to microscopic directed information transfer between nodes, we uncover the mechanisms underlying the emergence of these multimodal dynamics.

Figures (12)

• Figure 1: A schematic illustration of a duplex network with two fully connected layers. Solid lines show the intralayer connections, while dashed lines denote the interlayer couplings that connect each node to its mirror node in the other layer.
• Figure 1: Comparison of intralayer transfer entropy matrices for layer I (top row, red) and layer II (bottom row, blue) along the backward path of the phase transition at different coupling strengths. From left to right, $\sigma = 3.5$, $2.5$, $2.3$, and $0.5$, corresponding to the progression from coherent to incoherent states.
• Figure 1: Snapshot from a video comparing the dynamics of layer II in a duplex network with $\alpha = \frac{\pi}{2}$ under two different interlayer frequency mismatch distributions: Gaussian (first row) and uniform (second row). Panels ( a) and ( d) show the synchronization phase transition as a function of the intralayer coupling constant $\sigma$ for the forward (light curves) and backward (dark curves) paths. Red curves correspond to layer I, and blue curves to layer II. The vertical black dashed line indicates the coupling value at which the dynamical evolution is examined in the video ($\sigma = 2.34$). Panels ( b) and ( e) show the temporal evolution of the synchronization order parameter of layer II at $\sigma = 2.34$, along the backward path. The vertical red dashed lines indicate the time instants at which the similarity matrices are extracted and displayed in panels ( c) and ( f). The network structures and frequency arrangements are identical to those in Fig. 2 of the main text, and the nodes are ordered according to the intrinsic frequency differences with their corresponding mirror nodes, $\delta\omega_i$, in ascending order.
• Figure 2: Phase transition behavior in two globally similar frequency configurations with distinct interlayer frequency-mismatch distributions. Panels (a--c) correspond to a uniform (step-function) distribution of mirror-node frequency differences, while panels (d--f) correspond to a Gaussian distribution. In both cases, layers I and II (each with $N = 1000$ nodes) share identical natural frequencies across the two configurations, although the frequencies in each layer are drawn from different Gaussian distributions. The interlayer interaction includes a frustration parameter $\alpha = \frac{\pi}{2}$, and both configurations have the same average frequency mismatch, $\Delta\omega = 0.56$; thus, any differences in behavior arise solely from the specific pairing of mirror nodes. The first column shows the natural frequencies of layer II nodes versus those of layer I, with marginal plots indicating their respective distributions (red for layer I and blue for layer II). The second column displays the mirror-node frequency differences (Freq-diff, $\delta \omega_i$) as a function of node index, ordered from low to high values, with the corresponding distributions shown alongside each panel (uniform in the top row and Gaussian in the bottom row). The third column presents the phase transition along the forward (light lines) and backward (dark lines) paths, with red and blue curves representing layers I and II, respectively; shaded regions denote the temporal standard deviation.
• Figure 2: Procedure used to extract the transfer entropy matrices reported in Fig. 7 of the main text for $\sigma = 2.30$ (first row) and $\sigma = 0.5$ (second row). Matrices in each row are shown from left to right: raw transfer entropy, surrogate analysis, and surrogate-filtered transfer entropy.