Effective centrality and explosive synchronization in complex networks

Synchronization transitions in phase oscillator populations with partial adaptive coupling

The adaptation underlying many realistic processes plays a pivotal role in shaping the collective dynamics of diverse systems. Here, we untangle the generic conditions for synchronization transitions in a system of coupled phase oscillators incorporating the adaptive scheme encoded by the feedback between the coupling and the order parameter via a power-law function with different weights. We mathematically argue that, in the subcritical and supercritical correlation scenarios, there exists no critical adaptive fraction for synchronization transitions converting from the first (second)-order to the second (first)-order. In contrast to the synchronization transitions previously deemed, the explosive and continuous phase transitions take place in the corresponding regions as long as the adaptive fraction is nonzero, respectively. Nevertheless, we uncover that, at the critical correlation, the routes toward synchronization depend crucially on the relative adaptive weights. In particular, we unveil that the emergence of a range of interrelated scaling behaviors of the order parameter near criticality, manifesting the subcritical and supercritical bifurcations, are responsible for various observed phase transitions. Our work, thus, provides profound insights for understanding the dynamical nature of phase transitions, and for better controlling and manipulating synchronization transitions in networked systems with adaptation.

Relaxation dynamics of phase oscillators with generic heterogeneous coupling

The coupled phase oscillator model serves as a paradigm that has been successfully used to shed light on the collective dynamics occurring in large ensembles of interacting units. It was widely known that the system experiences a continuous (second-order) phase transition to synchronization by gradually increasing the homogeneous coupling among the oscillators. As the interest in exploring synchronized dynamics continues to grow, the heterogeneous patterns between phase oscillators have received ample attention during the past years. Here, we consider a variant of the Kuramoto model with quenched disorder in their natural frequencies and coupling. Correlating these two types of heterogeneity via a generic weighted function, we systematically investigate the impacts of the heterogeneous strategies, the correlation function, and the natural frequency distribution on the emergent dynamics. Importantly, we develop an analytical treatment for capturing the essential dynamical properties of the equilibrium states. In particular, we uncover that the critical threshold corresponding to the onset of synchronization is unaffected by the location of the inhomogeneity, which, however, does depend crucially on the value of the correlation function at its center. Furthermore, we reveal that the relaxation dynamics of the incoherent state featuring the responses to external perturbations is significantly shaped by all the considered effects, thereby leading to various decaying mechanisms of the order parameters in the subcritical region. Moreover, we untangle that synchronization is facilitated by the out-coupling strategy in the supercritical region. Our study is a step forward in highlighting the potential importance of the inhomogeneous patterns involved in the complex systems, and could thus provide theoretical insights for profoundly understanding the generic statistical mechanical properties of the steady states toward synchronization.

Dynamical complexity as a proxy for the network degree distribution

We explore the relation between the topological relevance of a node in a complex network and the individual dynamics it exhibits. When the system is weakly coupled, the effect of the coupling strength against the dynamical complexity of the nodes is found to be a function of their topological roles, with nodes of higher degree displaying lower levels of complexity. We provide several examples of theoretical models of chaotic oscillators, pulse-coupled neurons, and experimental networks of nonlinear electronic circuits evidencing such a hierarchical behavior. Importantly, our results imply that it is possible to infer the degree distribution of a network only from individual dynamical measurements.

Emergent explosive synchronization in adaptive complex networks

Adaptation plays a fundamental role in shaping the structure of a complex network and improving its functional fitting. Even when increasing the level of synchronization in a biological system is considered as the main driving force for adaptation, there is evidence of negative effects induced by excessive synchronization. This indicates that coherence alone cannot be enough to explain all the structural features observed in many real-world networks. In this work, we propose an adaptive network model where the dynamical evolution of the node states toward synchronization is coupled with an evolution of the link weights based on an anti-Hebbian adaptive rule, which accounts for the presence of inhibitory effects in the system. We found that the emergent networks spontaneously develop the structural conditions to sustain explosive synchronization. Our results can enlighten the shaping mechanisms at the heart of the structural and dynamical organization of some relevant biological systems, namely, brain networks, for which the emergence of explosive synchronization has been observed.

Functional and Topological Conditions for Explosive Synchronization Develop in Human Brain Networks with the Onset of Anesthetic-Induced Unconsciousness

Sleep, anesthesia, and coma share a number of neural features but the recovery profiles are radically different. To understand the mechanisms of reversibility of unconsciousness at the network level, we studied the conditions for gradual and abrupt transitions in conscious and anesthetized states. We hypothesized that the conditions for explosive synchronization (ES) in human brain networks would be present in the anesthetized brain just over the threshold of unconsciousness. To test this hypothesis, functional brain networks were constructed from multi-channel electroencephalogram (EEG) recordings in seven healthy subjects across conscious, unconscious, and recovery states. We analyzed four variables that are involved in facilitating ES in generic, non-biological networks: (1) correlation between node degree and frequency, (2) disassortativity (i.e., the tendency of highly-connected nodes to link with less-connected nodes, or vice versa), (3) frequency difference of coupled nodes, and (4) an inequality relationship between local and global network properties, which is referred to as the suppressive rule. We observed that the four network conditions for ES were satisfied in the unconscious state. Conditions for ES in the human brain suggest a potential mechanism for rapid recovery from the lightly-anesthetized state. This study demonstrates for the first time that the network conditions for ES, formerly shown in generic networks only, are present in empirically-derived functional brain networks. Further investigations with deep anesthesia, sleep, and coma could provide insight into the underlying causes of variability in recovery profiles of these unconscious states.