Trade-Off Between Multiplicity and Specificity in the Inter-layer Connectivity of non-identical Multilayer Networks

TL;DR

The paper investigates how inter-layer connectivity topology—specifically one-to-one symmetry versus random one-to-many coupling—controls dynamics in non-identical multilayer networks. Using diffusive coupling of Rössler oscillators, it quantifies global and intra-layer synchronization, amplitude death, and multi-stability across connectivity schemes, densities, and mismatches. The main findings show that one-to-one connectivity preserves layer symmetry and enhances intra-layer synchronization while inducing amplitude death at lower coupling and mismatch; random connectivity shifts bifurcation thresholds, fosters cluster synchronization guided by homomorphisms, and produces remanent, multi-stable states in faster layers. These results suggest design principles for real-world networks, indicating that symmetry supports memory and ILS, whereas asymmetry can mitigate amplitude death at low coupling.

Abstract

We study the coupled dynamics of multilayer networks with symmetric (MLs) and asymmetric (MLas) inter-layer connections. The symmetric inter-layer connections arise from a one-to-one correspondence between the nodes of different layers. In contrast, asymmetry results from the multiplicity of inter-layer connections, achieved by randomizing the links while preserving their overall density, thereby allowing one-to-many inter-layer connections. We investigate how different types of inter-layer coupling impact the dynamics of non-identical multilayer networks. We find that the specificity of one-to-one inter-layer connections facilitates intra-layer synchronization (ILS). In contrast, for networks with random inter-layer connectivity, ILS depends on how randomness affects intra-layer homomorphism (the set of permutations that preserve the network structure). Furthermore, amplitude death (AD) in MLs is observed at lower connectivity strength and frequency mismatch than the MLas. Moreover, AD in MLs depends on the density and topology, but does not depend on the size of the networks. On the other hand, AD in MLas is influenced by network size in addition to density, topology, and inter-layer mismatches. Moreover, both the MLs and MLas exhibit multi-stability, with the faster layer exhibiting a remanent periodic phase-locked oscillation, irrespective of the topology and inter-layer connectivity. In addition, remnant synchrony between nodes with homomorphic relationships is observed in the slower layer. Overall, we propose that symmetric inter-layer connections should be preferable for achieving intra-layer synchronization-regardless of global synchronization-and for sustaining permanent memory in multilayer networks with mismatched nodes across layers. However, to mitigate AD at low coupling values and layer mismatch, asymmetric inter-layer connectivity is more advantageous.

Figures (11)

• Figure 1: Bi-layer small network with one-to-one and one-to-many inter-layer connections. Displaying the simplest bi-layer network with different inter-layer configurations. Subplot (a) illustrates one-to-one inter-layer connectivity, while diagrams (b) and (c) depict the two possible one-to-many inter-layer connection configurations. The dotted (solid) lines represent inter-(intra-) layer connections. In our simulations, we use a similar strength of inter-layer and intra-layer connectivity to solely see the impact of the inter-layer connection topology.
• Figure 2: Hopf bifurcation for the one-to-one and one-to-many inter-layer connectivity w.r.t. coupling strength ($\varepsilon$). The plots (a-c) in the top panel illustrate how the largest eigenvalue of the Jacobian varies for the equilibrium state at the origin. The plots (d-f) in the bottom panel show the bifurcation plots for the three different inter-layer connectivity schemes displayed in the top panel. Here, the solid (dashed) lines display the stable (unstable) stationary states, and the closed (open) circles represent the stable (unstable) periodic states. Additionally, the various acronyms in the figure have the following meanings: HB: Hopf Bifurcation, PD: Period Doubling Bifurcation. In all the cases we have considered, $\omega_1=1$, $\omega_2=1.4$.
• Figure 3: Synchronization for the one-to-one and one-to-many inter-layer connectivity The top panel shows the variation of the synchronization order parameters ($\sigma_1, \sigma_2$, and $\sigma$ for three different inter-layer configurations of $6$ nodes bi-layer networks. The bottom panels display the bifurcation plots for all nodes in these configurations, further illustrating the range of coupling for AD and synchronization. Subplots (a) and (d) show synchronization of all the nodes of the same layers in the non-AD regime. Subplots (b) and (e) show no intra-layer synchronization, whereas subplot (e) shows the cluster synchronization between the two nodes from the second layer, with a homomorphic relation unaffected by the inter-layer connectivity. Finally, subplots (c) and (f) exhibit intra-layer synchronization in the second layer, where all nodes maintain a homomorphic relationship with each other. In contrast, in the first layer, the two nodes synchronize, as they do not have inter-layer connections and so share a homomorphic relationship. For all these plots $N_{l_1} = N_{l_2} = 3$, and $\omega_1=1$, $\omega_2=1.4$ for all above cases.
• Figure 4: Synchronization for the one-to-one and one-to-many inter-layer connectivity between 3-layers Sub-figures (c) and (d) show the variation of the synchronization order parameters $\sigma_1, \sigma_2$, $\sigma_3$, and $\sigma$ for two different inter-layer configurations of $9$ nodes three-layer networks. The schematics (a) and (b) show the connectivity chosen to plot the order parameters. For all these plots $N_{1} = N_{2} = N_{3} = 3$, $\omega_1=1.4$, $\omega_2=1.6$, and $\omega_3=0.6$. All the graphs are plotted for the average of $20$ random initial conditions.
• Figure 5: Parameter space for One-to-one and one-to-many inter layer connectivity. The sub figs show that variation of the $R$, $r_1$, and $r_2$ versus the $\epsilon$ and intrinsic frequency $\omega_2$ of the second layer in a Bi-layer network of size $N_{1} = N_{2} = 50$. For this plot, we consider both layers to have a ring topology with an average degree $\langle k \rangle = 2$ and $\omega_1 = 1$. For the top (bottom) panel, the inter-layer connectivity is one-to-one (random). All the graphs are plotted for the average of the various different random initial conditions.