Abstract
Synchronization of non-identical oscillators coupled through complex networks is an important example of collective behavior, and it is interesting to ask how the structural organization of network interactions influences this process. Several studies have explored and uncovered optimal topologies for synchronization by making purposeful alterations to a network. On the other hand, the connectivity patterns of many natural systems are often not static, but are rather modulated over time according to their dynamics. However, this co-evolution and the extent to which the dynamics of the individual units can shape the organization of the network itself are less well understood. Here, we study initially randomly connected but locally adaptive networks of Kuramoto oscillators. In particular, the system employs a co-evolutionary rewiring strategy that depends only on the instantaneous, pairwise phase differences of neighboring oscillators, and that conserves the total number of edges, allowing the effects of local reorganization to be isolated. We find that a simple rule—which preserves connections between more out-of-phase oscillators while rewiring connections between more in-phase oscillators—can cause initially disordered networks to organize into more structured topologies that support enhanced synchronization dynamics. We examine how this process unfolds over time, finding a dependence on the intrinsic frequencies of the oscillators, the global coupling, and the network density, in terms of how the adaptive mechanism reorganizes the network and influences the dynamics. Importantly, for large enough coupling and after sufficient adaptation, the resulting networks exhibit interesting characteristics, including degree–frequency and frequency–neighbor frequency correlations. These properties have previously been associated with optimal synchronization or explosive transitions in which the networks were constructed using global information. On the contrary, by considering a time-dependent interplay between structure and dynamics, this work offers a mechanism through which emergent phenomena and organization can arise in complex systems utilizing local rules.
Highlights
How dynamical processes unfold on networks with non-trivial coupling between individual units remains an1054-1500/2017/27(7)/073115/26VC Author(s) 2017.073115-2 Papadopoulos et al.important question in complex systems science.1–4 Examples of such dynamical systems on networks include the timedependent patterns of electrical activity in populations of neurons,5–11 the spread of information or disease across social networks,12–15 or regulatory mechanisms in biological networks.16–19 In each case, the way the system evolves over time is dependent on the specific form of the dynamics, intrinsic properties of each element, and the architecture of connectivity
Motivated by these types of systems, here we address the question of how both structured network topology and global synchronization can develop through a co-evolution of the underlying network connectivity and the dynamics
We find that co-evolution of the network and dynamics can promote the degree of synchronization in the system, which occurs in tandem with the development of specific correlations between the topology and the natural frequencies of the oscillators
Summary
This model has been extended to study systems with heterogeneous network topologies, in order to investigate how the architecture of complex connectivity affects the onset of synchronization in diverse oscillator populations.24,46 Such efforts have provided important insights into the nature of the synchronization transition in different graph models including those that display a scale-free degree distribution, those with small-world architecture, and those with community structure.. We find that co-evolution of the network and dynamics can promote the degree of synchronization in the system, which occurs in tandem with the development of specific correlations between the topology and the natural frequencies of the oscillators In previous work, these features have been imposed by purposeful selection, or have been shown to arise in studies on optimizing synchronization. V, we discuss the implications of our findings and conclude
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