Entropy of Badminton Strike Positions.

The aim of the current study was twofold: (i) to investigate the distribution of the strike positions of badminton players while quantifying the corresponding standard entropy and using an alternative metric (spatial entropy) related to winning and losing points and random positions; and (ii) to evaluate the standard entropy of the receiving positions. With the datasets of 259 badminton matches, we focused on the positions of players' strokes and the outcome of each point. First, we identified those regions of the court from which hits were most likely to be struck. Second, we computed the standard entropy of stroke positions, and then the spatial entropy, which also considers the order and clustering of the hitting locations in a two-dimensional Euclidean space. Both entropy quantifiers revealed high uncertainty in the striking position; however, specific court locations (i.e., the four corners) are preferred over the rest. When the outcome of each point was taken into account, we observed that the hitting patterns with lower entropy were associated with higher probabilities of winning points. On the contrary, players striking from more random positions were more prone to losing the points.

Biological conservation law as an emerging functionality in dynamical neuronal networks.

Scientists strive to understand how functionalities, such as conservation laws, emerge in complex systems. Living complex systems in particular create high-ordered functionalities by pairing up low-ordered complementary processes, e.g., one process to build and the other to correct. We propose a network mechanism that demonstrates how collective statistical laws can emerge at a macro (i.e., whole-network) level even when they do not exist at a unit (i.e., network-node) level. Drawing inspiration from neuroscience, we model a highly stylized dynamical neuronal network in which neurons fire either randomly or in response to the firing of neighboring neurons. A synapse connecting two neighboring neurons strengthens when both of these neurons are excited and weakens otherwise. We demonstrate that during this interplay between the synaptic and neuronal dynamics, when the network is near a critical point, both recurrent spontaneous and stimulated phase transitions enable the phase-dependent processes to replace each other and spontaneously generate a statistical conservation law-the conservation of synaptic strength. This conservation law is an emerging functionality selected by evolution and is thus a form of biological self-organized criticality in which the key dynamical modes are collective.

The resumption of sports competitions after COVID-19 lockdown: The case of the Spanish football league.

In this work, we present a stochastic discrete-time SEIR Susceptible-Exposed-Infectious-Recoveredmodel adapted to describe the propagation of COVID-19 during a football tournament. Specifically, we are concerned about the re-start of the Spanish national football league, La Liga, which is currently -May 2020- stopped with 11 fixtures remaining. Our model includes two additional states of an individual, confined and quarantined, which are reached when an individual presents COVID-19 symptoms or has undergone a virus test with a positive result. The model also accounts for the interaction dynamics of players, considering three different sources of infection: the player social circle, the contact with his/her team colleagues during training sessions, and the interaction with rivals during a match. Our results highlight the influence of the days between matches, the frequency of virus tests and their sensitivity on the number of players infected at the end of the season. Following our findings, we finally propose a variety of strategies to minimise the probability that COVID-19 propagates in case the season of La Liga was re-started after the current lockdown.

Permutation Entropy and Statistical Complexity in Mild Cognitive Impairment and Alzheimer's Disease: An Analysis Based on Frequency Bands.

We present one of the first applications of Permutation Entropy (PE) and Statistical Complexity (SC) (measured as the product of PE and Jensen-Shanon Divergence) on Magnetoencephalography (MEG) recordings of 46 subjects suffering from Mild Cognitive Impairment (MCI), 17 individuals diagnosed with Alzheimer's Disease (AD) and 48 healthy controls. We studied the differences in PE and SC in broadband signals and their decomposition into frequency bands ( δ , θ , α and ß ), considering two modalities: (i) raw time series obtained from the magnetometers and (ii) a reconstruction into cortical sources or regions of interest (ROIs). We conducted our analyses at three levels: (i) at the group level we compared SC in each frequency band and modality between groups; (ii) at the individual level we compared how the [PE, SC] plane differs in each modality; and (iii) at the local level we explored differences in scalp and cortical space. We recovered classical results that considered only broadband signals and found a nontrivial pattern of alterations in each frequency band, showing that SC does not necessarily decrease in AD or MCI.

Spatial and Temporal Entropies in the Spanish Football League: A Network Science Perspective.

We quantified the spatial and temporal entropy related to football teams and their players by means of a pass-based interaction. First, we calculated the spatial entropy associated to the positions of all passes made by a football team during a match, obtaining a spatial entropy ranking of Spanish teams during the 2017/2018 season. Second, we investigated how the player's average location in the field is related to the amount of entropy of his passes. Next, we constructed the temporal passing networks of each team and computed the deviation of their network parameters along the match. For each network parameter, we obtained the permutation entropy and the statistical complexity of its temporal fluctuations. Finally, we investigated how the permutation entropy (and statistical complexity) of the network parameters was related to the total number of passes made by a football team. Our results show that (i) spatial entropy changes according to the position of players in the field, and (ii) the organization of passing networks change during a match and its evolution can be captured measuring the permutation entropy and statistical complexity of the network parameters, allowing to identify what parameters evolve more randomly.

Analysis of player speed and angle toward the ball in soccer.

The study analyzes how the magnitude and angle of the speed of soccer players change according to the distance to the ball and the phases of the game, namely the defensive and attacking phases. We observed how the role played in the team (goalkeeper, defender, midfielder, or forward) strongly determines the speed pattern of players. As a general trend, the speed's modulus is incremented as their position is closer to the ball, however, it is slightly decreased when arriving at it. Next, we studied how the angle of the speed with the direction to the ball is related to the distance to the ball and the game phases. We observed that, during the defensive phase, goalkeepers are the players that run more parallel to the ball, while forwards are the ones running more directly to the ball position. Importantly, this behavior changes dramatically during the attacking phase. Finally, we show how the proposed methodology can be used to analyze the speed-angle patterns of specific players to understand better how they move on the pitch according to the distance to the ball.

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.

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.

Antiphase synchronization in multiplex networks with attractive and repulsive interactions.

A series of recent publications, within the framework of network science, have focused on the coexistence of mixed attractive and repulsive (excitatory and inhibitory) interactions among the units within the same system, motivated by the analogies with spin glasses as well as to neural networks, or ecological systems. However, most of these investigations have been restricted to single layer networks, requiring further analysis of the complex dynamics and particular equilibrium states that emerge in multilayer configurations. This article investigates the synchronization properties of dynamical systems connected through multiplex architectures in the presence of attractive intralayer and repulsive interlayer connections. This setting enables the emergence of antisynchronization, i.e., intralayer synchronization coexisting with antiphase dynamics between coupled systems of different layers. We demonstrate the existence of a transition from interlayer antisynchronization to antiphase synchrony in any connected bipartite multiplex architecture when the repulsive coupling is introduced through any spanning tree of a single layer. We identify, analytically, the required graph topologies for interlayer antisynchronization and its interplay with intralayer and antiphase synchronization. Next, we analytically derive the invariance of intralayer synchronization manifold and calculate the attractor size of each oscillator exhibiting interlayer antisynchronization together with intralayer synchronization. The necessary conditions for the existence of interlayer antisynchronization along with intralayer synchronization are given and numerically validated by considering Stuart-Landau oscillators. Finally, we also analytically derive the local stability condition of the interlayer antisynchronization state using the master stability function approach.

Distance Between Players During a Soccer Match: The Influence of Player Position.

In this study, we analyse the proximity between professional players during a soccer match. Specifically, we are concerned about the time a player remains at a distance to a rival that is closer than 2 m, which has a series of consequences, from the risk of contagion during a soccer match to the understanding of the tactical performance of players during the attacking/defensive phases. Departing from a dataset containing the Euclidean positions of all players during 60 matches of the Spanish national league (30 from LaLiga Santander and 30 from LaLiga Smartbank, respectively, the first and second divisions), we analysed 1,670 participations of elite soccer players. Our results show a high heterogeneity of both the player-player interaction time (from 0 to 14 min) and the aggregated time with all opponents (from <1 to 44 min). Furthermore, when the player position is taken into account, we observe that goalkeepers are the players with the lowest exposure (lower than 1 min), while forwards are the players with the highest values of the accumulated time (~21 min). In this regard, defender-forward interactions are the most frequent. To the best of our knowledge, this is the largest dataset describing the proximity between soccer players. Therefore, we believe these results may be crucial to the development of epidemiological models aiming the predict the risk of contagion between players and, furthermore, to understand better the statistics of all actions that involve proximity between players.

Using symbolic networks to analyse dynamical properties of disease outbreaks.

We introduce a new methodology, which is based on the construction of epidemic networks, to analyse the evolution of epidemic time series. First, we translate the time series into ordinal patterns containing information about local fluctuations in disease prevalence. Each pattern is associated with a node of a network, whose (directed) connections arise from consecutive appearances in the series. The analysis of the network structure and the role of each pattern allows them to be classified according to the enhancement of entropy/complexity along the series, giving a different point of view about the evolution of a given disease.

Influence of Players' Maximum Running Speed on the Team's Ranking Position at the End of the Spanish LaLiga.

The maximum running speed that a football player can attain during match play has become one of the most popular variables to assess a player's physical talent. However, the influence of a player's maximum running speed on football performance has not yet been properly investigated. The aim of this study was to determine the influence of a player's peak/maximum running speed on the team's ranking position at the end of a national league. A second aim was to investigate differences in maximum running speed among playing positions. To fulfil this aim, the peak/maximum running speeds of 475 male professional football players were recorded for 38 fixtures of the Spanish first-division league (LaLiga) from the 2017-2018 season (7838 data points). Players' peak running speeds in each match were assessed with a validated multicamera tracking system and associated software (Mediacoach®). Players' maximum running speed was established as the fastest running speed they attained during the entire season. Most players (53.5% of the total) had a maximum running speed in the range of 32.0-33.9 km/h, with only three players (0.6%) with a maximum running speed of over 35.0 km/h. Overall, forwards were faster than defenders and both types of players were faster than midfielders (33.03 ± 1.35 > 32.72 ± 1.32 > 32.08 ± 1.63 km/h; p < 0.001). There was no association between teams' maximum running speed and ranking position at the end of the league (r = -0.356, p = 0.135). The correlations between teams' maximum speeds and ranking position were low for defenders (r = -0.334, p = 0.163), midfielders (r = 0.125, p = 0.610), and forwards (r = -0.065, p = 0.791). As a result, the variance in the ranking position at the end of the season explained by a team's maximum speed was of only 7.5%. Finally, as an average for all teams, players' peak running speeds remained stable at ~30.7 ± 0.6 km/h throughout the whole season. These results suggest that successful and less successful football teams have squads with players able to obtain similar maximum running speeds during match play throughout the season. Hence, players' maximum running speeds have a poor association with the team's ranking position at the end of the Spanish professional national league.

Experimental datasets of networks of nonlinear oscillators: Structure and dynamics during the path to synchronization.

The analysis of the interplay between structural and functional networks require experiments where both the specific structure of the connections between nodes and the time series of the underlying dynamical units are known at the same time. However, real datasets typically contain only one of the two ways (structural or functional) a network can be observed. Here, we provide experimental recordings of the dynamics of 28 nonlinear electronic circuits coupled in 20 different network configurations. For each network, we modify the coupling strength between circuits, going from an incoherent state of the system to a complete synchronization scenario. Time series containing 30000 points are recorded using a data-acquisition card capturing the analogic output of each circuit. The experiment is repeated three times for each network structure allowing to track the path to the synchronized state both at the level of the nodes (with its direct neighbours) and at the whole network. These datasets can be useful to test new metrics to evaluate the coordination between dynamical systems and to investigate to what extent the coupling strength is related to the correlation between functional and structural networks.

Entraining synthetic genetic oscillators.

We propose a new approach for synchronizing a population of synthetic genetic oscillators, which consists in the entrainment of a colony of repressilators by external modulation. We present a model where the repressilator dynamics is affected by periodic changes in temperature. We introduce an additional plasmid in the bacteria in order to correlate the temperature variations with the enhancement of the transcription rate of a certain gene. This can be done by introducing a promoter that is related to the heat shock response. This way, the expression of that gene results in a protein that enhances the overall oscillations. Numerical results show coherent oscillations of the population for a certain range of the external frequency, which is in turn related to the natural oscillation frequency of the modified repressilator. Finally we study the transient times related with the loss of synchronization and we discuss possible applications in biotechnology of large-scale production coupled to synchronization events induced by heat shock.

Taming out-of-equilibrium dynamics on interconnected networks.

A wide variety of social, biological or technological systems can be described as processes taking place on networked structures in continuous interaction with other networks. We propose here a new methodology to describe, anticipate and manage, in real time, the out-of-equilibrium dynamics of processes that evolve on interconnected networks. This goal is achieved through the full analytical treatment of the phenomenology and its reduction to a two-dimensional flux diagram, allowing us to predict at every time step the dynamical consequences of modifying the links between the different ensembles. Our results are consistent with real data and the methodology can be translated to clustered networks and/or interconnected networks of any size, topology or origin, from the struggle for knowledge on innovation structures to international economic relations or disease spreading on social groups.

Community structures and role detection in music networks.

We analyze the existence of community structures in two different social networks using data obtained from similarity and collaborative features between musical artists. Our analysis reveals some characteristic organizational patterns and provides information about the driving forces behind the growth of the networks. In the similarity network, we find a strong correlation between clusters of artists and musical genres. On the other hand, the collaboration network shows two different kinds of communities: rather small structures related to music bands and geographic zones, and much bigger communities built upon collaborative clusters with a high number of participants related through the period the artists were active. Finally, we detect the leading artists inside their corresponding communities and analyze their roles in the network by looking at a few topological properties of the nodes.

Disorder and decision cost in spatial networks.

We introduce the concept of decision cost of a spatial graph, which measures the disorder of a given network taking into account not only the connections between nodes but their position in a two-dimensional map. The influence of the network size is evaluated and we show that normalization of the decision cost allows us to compare the degree of disorder of networks of different sizes. Under this framework, we measure the disorder of the connections between airports of two different countries and obtain some conclusions about which of them is more disordered. The introduced concepts (decision cost and disorder of spatial networks) can easily be extended to Euclidean networks of higher dimensions, and also to networks whose nodes have a certain fitness property (i.e., one-dimensional).

Frequency-based brain networks: From a multiplex framework to a full multilayer description.

We explore how to study dynamical interactions between brain regions by using functional multilayer networks whose layers represent different frequency bands at which a brain operates. Specifically, we investigate the consequences of considering the brain as (i) a multilayer network, in which all brain regions can interact with each other at different frequency bands; and as (ii) a multiplex network, in which interactions between different frequency bands are allowed only within each brain region and not between them. We study the second-smallest eigenvalue λ 2 of the combinatorial supra-Laplacian matrix of both the multiplex and multilayer networks, as λ 2 has been used previously as an indicator of network synchronizability and as a biomarker for several brain diseases. We show that the heterogeneity of interlayer edge weights and, especially, the fraction of missing edges crucially modify the value of λ 2, and we illustrate our results with both synthetic network models and real data obtained from resting-state magnetoencephalography. Our work highlights the differences between using a multiplex approach and a full multilayer approach when studying frequency-based multilayer brain networks.