Trade-off between multiplicity and specificity in the interlayer connectivity of nonidentical multilayer networks

We study the coupled dynamics of multilayer networks with symmetric (MLs) and asymmetric (MLa) interlayer connections. The symmetric interlayer connections arise from a one-to-one correspondence between the nodes of different layers. In contrast, asymmetry results from the multiplicity of interlayer connections, achieved by randomizing the links while preserving their overall density, thereby allowing one-to-many interlayer connections. We investigate how different types of interlayer coupling impact the dynamics of nonidentical multilayer networks. We find that the specificity of one-to-one interlayer connections facilitates intralayer synchronization (ILS). In contrast, for networks with random interlayer connectivity, ILS depends on how randomness affects intralayer 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 MLa. Moreover, AD in MLs for regular intralayer connectivity depends on the density and topology but does not depend on the size of the networks. On the other hand, AD in MLa is influenced by network size in addition to density, topology, and interlayer mismatches. Moreover, both the MLs and MLa exhibit multistability, with the faster layer exhibiting a remanent periodic phase-locked oscillation, irrespective of the topology and interlayer connectivity. In addition, remnant synchrony between nodes with homomorphic relationships is observed in the slower layer. Overall, we propose that symmetric interlayer connections should be preferable for achieving intralayer 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 interlayer connectivity is more advantageous.