Does the brain behave like a (complex) network? I. Dynamics.

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Consistency and identifiability of football teams: a network science perspective.

We investigated the ability of football teams to develop a particular playing style by looking at their passing patterns. Using the information contained in the pass sequences during matches, we constructed the pitch passing networks of teams, whose nodes are the divisions of the pitch for a given spatial scale and links account for the number of passes from region to region. We translated football passings networks into their corresponding adjacency matrices. We calculated the correlations between matrices of the same team to quantify how consistent the passing patterns of a given team are. Next, we quantified the differences with other teams' matrices and obtained an identifiability parameter that indicates how unique are the passing patterns of a given team. Consistency and identifiability rankings were calculated during a whole season, allowing to detect those teams of a league whose passing patterns are different from the rest. Furthermore, we found differences between teams playing at home or away. Finally, we used the identifiability parameter to investigate what teams imposed their passing patterns over the rivals during a given match.

Identifiability of structural networks of nonlinear electronic oscillators.

The interplay between structure and function is critical in the understanding of complex systems, their dynamics and their behavior. We investigated the interplay between structural and functional networks by means of the differential identifiability framework, which here quantifies the ability of identifying a particular network structure based on (1) the observation of its functional network and (2) the comparison with a prior observation under different initial conditions. We carried out an experiment consisting of the construction of [Formula: see text] different structural networks composed of [Formula: see text] nonlinear electronic circuits and studied the regions where network structures are identifiable. Specifically, we analyzed how differential identifiability is related to the coupling strength between dynamical units (modifying the level of synchronization) and what are the consequences of increasing the amount of noise existing in the functional networks. We observed that differential identifiability reaches its highest value for low to intermediate coupling strengths. Furthermore, it is possible to increase the identifiability parameter by including a principal component analysis in the comparison of functional networks, being especially beneficial for scenarios where noise reaches intermediate levels. Finally, we showed that the regime of the parameter space where differential identifiability is the highest is highly overlapped with the region where structural and functional networks correlate the most.

Defining a historic football team: Using Network Science to analyze Guardiola's F.C. Barcelona.

The application of Network Science to social systems has introduced new methodologies to analyze classical problems such as the emergence of epidemics, the arousal of cooperation between individuals or the propagation of information along social networks. More recently, the organization of football teams and their performance have been unveiled using metrics coming from Network Science, where a team is considered as a complex network whose nodes (i.e., players) interact with the aim of overcoming the opponent network. Here, we combine the use of different network metrics to extract the particular signature of the F.C. Barcelona coached by Guardiola, which has been considered one of the best teams along football history. We have first compared the network organization of Guardiola's team with their opponents along one season of the Spanish national league, identifying those metrics with statistically significant differences and relating them with the Guardiola's game. Next, we have focused on the temporal nature of football passing networks and calculated the evolution of all network properties along a match, instead of considering their average. In this way, we are able to identify those network metrics that enhance the probability of scoring/receiving a goal, showing that not all teams behave in the same way and how the organization Guardiola's F.C. Barcelona is different from the rest, including its clustering coefficient, shortest-path length, largest eigenvalue of the adjacency matrix, algebraic connectivity and centrality distribution.

Brain synchronizability, a false friend.

Synchronization plays a fundamental role in healthy cognitive and motor function. However, how synchronization depends on the interplay between local dynamics, coupling and topology and how prone to synchronization a network is, given its topological organization, are still poorly understood issues. To investigate the synchronizability of both anatomical and functional brain networks various studies resorted to the Master Stability Function (MSF) formalism, an elegant tool which allows analysing the stability of synchronous states in a dynamical system consisting of many coupled oscillators. Here, we argue that brain dynamics does not fulfil the formal criteria under which synchronizability is usually quantified and, perhaps more importantly, this measure refers to a global dynamical condition that never holds in the brain (not even in the most pathological conditions), and therefore no neurophysiological conclusions should be drawn based on it. We discuss the meaning of synchronizability and its applicability to neuroscience and propose alternative ways to quantify brain networks synchronization.

Role of inter-hemispheric connections in functional brain networks.

Today the human brain can be modeled as a graph where nodes represent different regions and links stand for statistical interactions between their activities as recorded by different neuroimaging techniques. Empirical studies have lead to the hypothesis that brain functions rely on the coordination of a scattered mosaic of functionally specialized brain regions (modules or sub-networks), forming a web-like structure of coordinated assemblies (a network of networks. NoN). The study of brain dynamics would therefore benefit from an inspection of how functional sub-networks interact between them. In this paper, we model the brain as an interconnected system composed of two specific sub-networks, the left (L) and right (R) hemispheres, which compete with each other for centrality, a topological measure of importance in a networked system. Specifically, we considered functional scalp EEG networks (SEN) derived from high-density electroencephalographic (EEG) recordings and investigated how node centrality is shaped by interhemispheric connections. Our results show that the distribution of centrality strongly depends on the number of functional connections between hemispheres and the way these connections are distributed. Additionally, we investigated the consequences of node failure on hemispherical centrality, and showed how the abundance of inter-hemispheric links favors the functional balance of centrality distribution between the hemispheres.

Inter-layer synchronization in non-identical multi-layer networks.

Inter-layer synchronization is a dynamical process occurring in multi-layer networks composed of identical nodes. This process emerges when all layers are synchronized, while nodes in each layer do not necessarily evolve in unison. So far, the study of such inter-layer synchronization has been restricted to the case in which all layers have an identical connectivity structure. When layers are not identical, the inter-layer synchronous state is no longer a stable solution of the system. Nevertheless, when layers differ in just a few links, an approximate treatment is still feasible, and allows one to gather information on whether and how the system may wander around an inter-layer synchronous configuration. We report the details of an approximate analytical treatment for a two-layer multiplex, which results in the introduction of an extra inertial term accounting for structural differences. Numerical validation of the predictions highlights the usefulness of our approach, especially for small or moderate topological differences in the intra-layer coupling. Moreover, we identify a non-trivial relationship connecting the betweenness centrality of the missing links and the intra-layer coupling strength. Finally, by the use of multiplexed layers of electronic circuits, we study the inter-layer synchronization as a function of the removed links.

Optimal phase synchronization in networks of phase-coherent chaotic oscillators.

We investigate the existence of an optimal interplay between the natural frequencies of a group of chaotic oscillators and the topological properties of the network they are embedded in. We identify the conditions for achieving phase synchronization in the most effective way, i.e., with the lowest possible coupling strength. Specifically, we show by means of numerical and experimental results that it is possible to define a synchrony alignment function J(ω,L) linking the natural frequencies ωi of a set of non-identical phase-coherent chaotic oscillators with the topology of the Laplacian matrix L, the latter accounting for the specific organization of the network of interactions between oscillators. We use the classical Rössler system to show that the synchrony alignment function obtained for phase oscillators can be extended to phase-coherent chaotic systems. Finally, we carry out a series of experiments with nonlinear electronic circuits to show the robustness of the theoretical predictions despite the intrinsic noise and parameter mismatch of the electronic components.

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.

The cost of attack in competing networks.

Real-world attacks can be interpreted as the result of competitive interactions between networks, ranging from predator-prey networks to networks of countries under economic sanctions. Although the purpose of an attack is to damage a target network, it also curtails the ability of the attacker, which must choose the duration and magnitude of an attack to avoid negative impacts on its own functioning. Nevertheless, despite the large number of studies on interconnected networks, the consequences of initiating an attack have never been studied. Here, we address this issue by introducing a model of network competition where a resilient network is willing to partially weaken its own resilience in order to more severely damage a less resilient competitor. The attacking network can take over the competitor's nodes after their long inactivity. However, owing to a feedback mechanism the takeovers weaken the resilience of the attacking network. We define a conservation law that relates the feedback mechanism to the resilience dynamics for two competing networks. Within this formalism, we determine the cost and optimal duration of an attack, allowing a network to evaluate the risk of initiating hostilities.

Enhancing the stability of the synchronization of multivariable coupled oscillators.

Synchronization processes in populations of identical networked oscillators are the focus of intense studies in physical, biological, technological, and social systems. Here we analyze the stability of the synchronization of a network of oscillators coupled through different variables. Under the assumption of an equal topology of connections for all variables, the master stability function formalism allows assessing and quantifying the stability properties of the synchronization manifold when the coupling is transferred from one variable to another. We report on the existence of an optimal coupling transference that maximizes the stability of the synchronous state in a network of Rössler-like oscillators. Finally, we design an experimental implementation (using nonlinear electronic circuits) which grounds the robustness of the theoretical predictions against parameter mismatches, as well as against intrinsic noise of the system.

Synchronization of intermittent behavior in ensembles of multistable dynamical systems.

We propose a methodology to analyze synchronization in an ensemble of diffusively coupled multistable systems. First, we study how two bidirectionally coupled multistable oscillators synchronize and demonstrate the high complexity of the basins of attraction of coexisting synchronous states. Then, we propose the use of the master stability function (MSF) for multistable systems to describe synchronizability, even during intermittent behavior, of a network of multistable oscillators, regardless of both the number of coupled oscillators and the interaction structure. In particular, we show that a network of multistable elements is synchronizable for a given range of topology spectra and coupling strengths, irrespective of specific attractor dynamics to which different oscillators are locked, and even in the presence of intermittency. Finally, we experimentally demonstrate the feasibility and robustness of the MSF approach with a network of multistable electronic circuits.

Synchronization of interconnected networks: the role of connector nodes.

In this Letter we identify the general rules that determine the synchronization properties of interconnected networks. We study analytically, numerically, and experimentally how the degree of the nodes through which two networks are connected influences the ability of the whole system to synchronize. We show that connecting the high-degree (low-degree) nodes of each network turns out to be the most (least) effective strategy to achieve synchronization. We find the functional relation between synchronizability and size for a given network of networks, and report the existence of the optimal connector link weights for the different interconnection strategies. Finally, we perform an electronic experiment with two coupled star networks and conclude that the analytical results are indeed valid in the presence of noise and parameter mismatches.

Generalized synchronization in relay systems with instantaneous coupling.

We demonstrate the existence of generalized synchronization in systems that act as mediators between two dynamical units that, in turn, show complete synchronization with each other. These are the so-called relay systems. Specifically, we analyze the Lyapunov spectrum of the full system to elucidate when complete and generalized synchronization appear. We show that once a critical coupling strength is achieved, complete synchronization emerges between the systems to be synchronized, and at the same point, generalized synchronization with the relay system also arises. Next, we use two nonlinear measures based on the distance between phase-space neighbors to quantify the generalized synchronization in discretized time series. Finally, we experimentally show the robustness of the phenomenon and of the theoretical tools here proposed to characterize it.

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.

Topological measure locating the effective crossover between segregation and integration in a modular network.

We introduce an easily computable topological measure which locates the effective crossover between segregation and integration in a modular network. Segregation corresponds to the degree of network modularity, while integration is expressed in terms of the algebraic connectivity of an associated hypergraph. The rigorous treatment of the simplified case of cliques of equal size that are gradually rewired until they become completely merged, allows us to show that this topological crossover can be made to coincide with a dynamical crossover from cluster to global synchronization of a system of coupled phase oscillators. The dynamical crossover is signaled by a peak in the product of the measures of intracluster and global synchronization, which we propose as a dynamical measure of complexity. This quantity is much easier to compute than the entropy (of the average frequencies of the oscillators), and displays a behavior which closely mimics that of the dynamical complexity index based on the latter. The proposed topological measure simultaneously provides information on the dynamical behavior, sheds light on the interplay between modularity and total integration, and shows how this affects the capability of the network to perform both local and distributed dynamical tasks.

Explosive first-order transition to synchrony in networked chaotic oscillators.

Critical phenomena in complex networks, and the emergence of dynamical abrupt transitions in the macroscopic state of the system are currently a subject of the outmost interest. We report evidence of an explosive phase synchronization in networks of chaotic units. Namely, by means of both extensive simulations of networks made up of chaotic units, and validation with an experiment of electronic circuits in a star configuration, we demonstrate the existence of a first-order transition towards synchronization of the phases of the networked units. Our findings constitute the first prove of this kind of synchronization in practice, thus opening the path to its use in real-world applications.

Synchronization waves in geometric networks.

We report synchronization of networked excitable nodes embedded in a metric space, where the connectivity properties are mostly determined by the distance between units. Such a high clustered structure, combined with the lack of long-range connections, prevents full synchronization and yields instead the emergence of synchronization waves. We show that this regime is optimal for information transmission through the system, as it enhances the options of reconstructing the topology from the dynamics. Measurements of topological and functional centralities reveal that the wave-synchronization state allows detection of the most structurally relevant nodes from a single observation of the dynamics, without any a priori information on the model equations ruling the evolution of the ensemble.

Entraining the topology and the dynamics of a network of phase oscillators.

We show that the topology and dynamics of a network of unsynchronized Kuramoto oscillators can be simultaneously controlled by means of a forcing mechanism which yields a phase locking of the oscillators to that of an external pacemaker in connection with the reshaping of the network's degree distribution. The entrainment mechanism is based on the addition, at regular time intervals, of unidirectional links from oscillators that follow the dynamics of a pacemaker to oscillators in the pristine graph whose phases hold a prescribed phase relationship. Such a dynamically based rule in the attachment process leads to the emergence of a power-law shape in the final degree distribution of the graph whenever the network is entrained to the dynamics of the pacemaker. We show that the arousal of a scale-free distribution in connection with the success of the entrainment process is a robust feature, characterizing different networks' initial configurations and parameters.

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.