I. Leyva

Reorganization of functional networks in mild cognitive impairment.

Whether the balance between integration and segregation of information in the brain is damaged in Mild Cognitive Impairment (MCI) subjects is still a matter of debate. Here we characterize the functional network architecture of MCI subjects by means of complex networks analysis. Magnetoencephalograms (MEG) time series obtained during a memory task were evaluated by synchronization likelihood (SL), to quantify the statistical dependence between MEG signals and to obtain the functional networks. Graphs from MCI subjects show an enhancement of the strength of connections, together with an increase in the outreach parameter, suggesting that memory processing in MCI subjects is associated with higher energy expenditure and a tendency toward random structure, which breaks the balance between integration and segregation. All features are reproduced by an evolutionary network model that simulates the degenerative process of a healthy functional network to that associated with MCI. Due to the high rate of conversion from MCI to Alzheimer Disease (AD), these results show that the analysis of functional networks could be an appropriate tool for the early detection of both MCI and AD.

Synchronization interfaces and overlapping communities in complex networks.

We show that a complex network of phase oscillators may display interfaces between domains (clusters) of synchronized oscillations. The emergence and dynamics of these interfaces are studied for graphs composed of either dynamical domains (influenced by different forcing processes), or structural domains (modular networks). The obtained results allow us to give a functional definition of overlapping structures in modular networks, and suggest a practical method able to give information on overlapping clusters in both artificially constructed and real world modular networks.

Principles of recovery from traumatic brain injury: Reorganization of functional networks

Recovery after brain injury is an excellent platform to study the mechanism underlying brain plasticity, the reorganization of networks. Do complex network measures capture the physiological and cognitive alterations that occurred after a traumatic brain injury and its recovery? Patients as well as control subjects underwent resting-state MEG recording following injury and after neurorehabilitation. Next, network measures such as network strength, path length, efficiency, clustering and energetic cost were calculated. We show that these parameters restore, in many cases, to control ones after recovery, specifically in delta and alpha bands, and we design a model that gives some hints about how the functional networks modify their weights in the recovery process. Positive correlations between complex network measures and some of the general index of the WAIS-III test were found: changes in delta-based path-length and those in Performance IQ score, and alpha-based normalized global efficiency and Perceptual Organization Index. These results indicate that: 1) the principle of recovery depends on the spectral band, 2) the structure of the functional networks evolves in parallel to brain recovery with correlations with neuropsychological scales, and 3) energetic cost reveals an optimal principle of recovery.

Explosive transitions to synchronization in networks of phase oscillators.

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject of the utmost interest. The occurrence of a first-order phase transition to synchronization of an ensemble of networked phase oscillators was reported, so far, for very particular network architectures. Here, we show how a sharp, discontinuous transition can occur, instead, as a generic feature of networks of phase oscillators. Precisely, we set conditions for the transition from unsynchronized to synchronized states to be first-order, and demonstrate how these conditions can be attained in a very wide spectrum of situations. We then show how the occurrence of such transitions is always accompanied by the spontaneous setting of frequency-degree correlation features. Third, we show that the conditions for abrupt transitions can be even softened in several cases. Finally, we discuss, as a possible application, the use of this phenomenon to express magnetic-like states of synchronization.

Sparse repulsive coupling enhances synchronization in complex networks.

Through the last years, different strategies to enhance synchronization in complex networks have been proposed. In this work, we show that synchronization of nonidentical dynamical units that are attractively coupled in a small-world network is strongly improved by just making phase-repulsive a tiny fraction of the couplings. By a purely topological analysis that does not depend on the dynamical model, we link the emerging dynamical behavior with the structural properties of the sparsely coupled repulsive network.

Explosive synchronization in weighted complex networks.

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject of the utmost interest. Given a set of phase oscillators networking with a generic wiring of connections and displaying a generic frequency distribution, we show how combining dynamical local information on frequency mismatches and global information on the graph topology suggests a judicious and yet practical weighting procedure which is able to induce and enhance explosive, irreversible, transitions to synchronization. We report extensive numerical and analytical evidence of the validity and scalability of such a procedure for different initial frequency distributions, for both homogeneous and heterogeneous networks, as well as for both linear and nonlinear weighting functions. We furthermore report on the possibility of parametrically controlling the width and extent of the hysteretic region of coexistence of the unsynchronized and synchronized states.

Introduction to Focus Issue: Mesoscales in Complex Networks

Although the functioning of real complex networks is greatly determined by modularity, the majority of articles have focused, until recently, on either their local scale structure or their macroscopical properties. However, neither of these descriptions can adequately describe the important features that complex networks exhibit due to their organization in modules. This Focus Issue precisely presents the state of the art on the study of complex networks at that intermediate level. The reader will find out why this mesoscale level has become an important topic of research through the latest advances carried out to improve our understanding of the dynamical behavior of modular networks. The contributions presented here have been chosen to cover, from different viewpoints, the many open questions in the field as different aspects of community definition and detection algorithms, moduli overlapping, dynamics on modular networks, interplay between scales, and applications to biological, social, and technological fields.

Phase Locking Induces Scale-Free Topologies in Networks of Coupled Oscillators

An initial unsynchronized ensemble of networking phase oscillators is further subjected to a growing process where a set of forcing oscillators, each one of them following the dynamics of a frequency pacemaker, are added to the pristine graph. Linking rules based on dynamical criteria are followed in the attachment process to force phase locking of the network with the external pacemaker. We show that the eventual locking occurs in correspondence to the arousal of a scale-free degree distribution in the original graph.

Inter-layer synchronization in multiplex networks of identical layers

Inter-layer synchronization is a distinctive process of multiplex networks whereby each node in a given layer evolves synchronously with all its replicas in other layers, irrespective of whether or not it is synchronized with the other units of the same layer. We analytically derive the necessary conditions for the existence and stability of such a state, and verify numerically the analytical predictions in several cases where such a state emerges. We further inspect its robustness against a progressive de-multiplexing of the network, and provide experimental evidence by means of multiplexes of nonlinear electronic circuits affected by intrinsic noise and parameter mismatch.

Unveiling Protein Functions through the Dynamics of the Interaction Network

Protein interaction networks have become a tool to study biological processes, either for predicting molecular functions or for designing proper new drugs to regulate the main biological interactions. Furthermore, such networks are known to be organized in sub-networks of proteins contributing to the same cellular function. However, the protein function prediction is not accurate and each protein has traditionally been assigned to only one function by the network formalism. By considering the network of the physical interactions between proteins of the yeast together with a manual and single functional classification scheme, we introduce a method able to reveal important information on protein function, at both micro- and macro-scale. In particular, the inspection of the properties of oscillatory dynamics on top of the protein interaction network leads to the identification of misclassification problems in protein function assignments, as well as to unveil correct identification of protein functions. We also demonstrate that our approach can give a network representation of the meta-organization of biological processes by unraveling the interactions between different functional classes.