Complex dynamical networks: effects of heterogeneity, adaptivity, and topology

The project focussed upon the collective dynamics of complex adaptive networks. The results established important relations between the structural and functional properties of such networks. This contributed towards the understanding of the working principles of realistic dynamical networks, such as neuronal networks with plasticity and heterogeneity. The project developed new methods for the study of adaptive networks. These methods include the Master Stability Function for adaptive networks. While this is a standard tool for studying synchronization in dynamical networks with a fixed topology, here we have extended the master stability function to a class of adaptive networks, hence, it allows for reducing the stability problem of synchronization for adaptive networks to a low-dimensional system, by decoupling topological and dynamical properties. Furthermore, a mean-field approach was developed to study synchronization, clustering, and transitions between these states. A novel analytical framework for synchronization in multilayer-networks was also developed. New phenomena have been discovered in adaptive dynamical networks. In particular the project demonstrated the emergence of two distinct first-order nonequilibrium phase transitions in a finite-size adaptive network of heterogeneous phase oscillators. Depending on the nature of the defects in the internal frequency distribution, either an abrupt single-step transition to full synchronization or a more gradual multi-step transition is observed. Another novel phenomenon which we reported was the recurrent synchronization, which is governed by asymmetric adaptation rules. Furthermore, a theory of symmetry-induced generalized splay states was developed, extending the previously known splay states in coupled neural systems. Contributions have been made to several applications: general neural networks, neural networks modelling epileptic seizures, power grids, disease modelling for cancer and sepsis. In particular, we proved that large classes of power grid network models can be viewed as a special class of adaptive networks, and thus the toolbox from adaptive network theory can be applied to power grid models. We developed a two-layer physiological network model of adaptively coupled phase oscillators describing either a healthy state (full synchronization) or a pathological state (two-frequency cluster) that corresponds to cancer or sepsis. We studied the effect of blinking coupling functions, coupling delay in neuronal networks, external forcing in adaptive networks, and noise in discrete-time networks upon synchronization.