Adaptive Complex Network

Abstract

Synchrony is a phenomenon that pervades all fields of science. The Kuramoto model is a prominent model that describes synchronization in systems (networks). It models each element of the network as an oscillator with an individual phase. Due to global coupling, a phase transition is realized, such that some oscillators of the network synchronize. An example of this model is the synchronization of chemical oscillators. However, this model is sometimes insufficient. It is found that in some parts of the brain, synchronization is required, but that excessive synchronization may lead to epilepsy. This reveals that negative feedback is required in order to avoid excessive synchronization. In this study, the Kuramoto model is extended to an adaptive network by introducing two opposing adaptation rules, such that the strength of coupling can differ per pair of oscillators. The anti-Hebbian rule promotes links between oscillators that are in anti-phase. It is found that networks with this adaptation rule organize themselves in a way that links occur between oscillators whose frequencies are most distant, and that other links are weakened or pruned completely. This suggests that networks with this adaptation rule are able to avoid excessive synchronization. However, the network is still able to sustain explosive synchronization. The second rule, the Hebbian rule, promotes links between oscillators that are in phase. Networks with this rule do not prune links. Again explosive synchronization is revealed in this network. A stability analysis is performed to obtain more fundamental insights in the dynamics of the network. The results of this thesis can help obtaining deeper understanding the dynamics and principles of link pruning and explosive synchronization in complex networks: phenomena that are observed in, among others, the field of neuroscience.