Synchronization and phase transitions on multiplex networks

Abstract

Synchronization means rhythmic activity by connected dynamical units. Since 1665 when C. Huygens first noticed the synchronization of hanging pendulums, the question of how a collective behavior can emerge among the interacting elements of a population has always attracted significant attention from the science community. Further, daily life is full of phase transitions whether it is boiling of water to vapor or ferromagnetic transitions; therefore, it is no surprise that phase transitions occupy a prominent place in our longing to understand the natural phenomena around us. In network science also, coupled dynamical units usually exhibit a smooth and reversible transition from an incoherent state to a coherent or synchronized state; however, in rare cases, it can be dis continuous and irreversible as well. The latter is also called explosive synchronization (ES), and it brings various surprising results. Most of the studies on synchronization or explosive synchronization so far are confined to the classical concept of single or single-layer networks that treats all links among the nodes at the same foot. However, as realized recently, a group of nodes having different types of interactions should be treated as a multilayer or multiplex network. The multiplex networks are a special form of multilayer networks in which the same set of nodes are replicated in different layers; also, each node is connected to itself (also called mirror nodes) across the layers. This thesis can be summarized into two broad questions. The first question one would probably ask while constructing a multiplex network is how to put interlayer connections? If there is any correlation among the mirror nodes, what is its impact on the synchronization ability of connected oscillators? Secondly, what are the different techniques using which one can change a continuous phase transition on multiplex net works into a discontinuous one or vice versa? What are the underlying causes behind the emergence of ES, and what factors govern the width of hysteresis associated with ES? We investigate identical and non-identical phase oscillators. For the first case, we choose diffusive coupling, while for the latter, Kuramoto type coupling or Kuramoto Os cillators are considered. Due to their analytical solvability and relevance to real systems, the two models occupy a prominent place in the field of collective dynamics. For identi x cal oscillators, we find that if nodes in individual layers have moderate (strong) degree degree correlations, strong negative (moderate) degree-degree correlations among the mirror nodes are beneficial for global synchronizability (GS). Furthermore, as deter mined by degree-degree correlations, increasing connections in the multiplex networks is not always beneficial for the GS. For non-identical oscillators, we focus on the type of phase transition exhibited by the ensemble. We show that using three techniques, i.e., interlayer adaptation, interlayer phase-shifted interactions, and natural frequency dis placement between the layers, one can induce ES in a two-layer multiplex network of Kuramoto oscillators. In all three cases, suppression of synchronization is accountable for the onset of ES. Since the ES is accompanied by a hysteresis, different parame ters affecting the hysteresis size are discussed in detail. Our observations suggest that interlayer coupling and natural frequency mismatch between the mirror nodes play an important role in the emergence of ES. The robustness of ES against changes in net work parameters (for instance, network topology, natural frequency distribution, etc.) is tested. A mean-field analysis is performed to justify the numerical simulations. Extensive work on synchronization so far reveals that it does not require any proof of its relevance to real-world systems. It has applications in almost all branches of science—from physics, chemistry, and biology to ecology, sociology, and technology. Similarly, the discontinuous phase transitions are undesired in many real-world situa tions; therefore, it is worthy to investigate them thoroughly. We hope our findings on synchronization and explosive synchronization will strengthen our current understand ing of collective dynamics existing in real-world systems that can be modeled in the multiplex framework.