From Dynamics to Structure of Complex Networks: Exploiting Heterogeneity in the Sakaguchi-Kuramoto Model

[eng] Most of the real-world complex systems are best described as complex networks and can be mathematically described as oscillatory systems, coupled with the neighbours through the connections of the network. The flashing of fireflies, the neuronal brain signals or the energy flow through the power grid are some examples. Yoshiki Kuramoto came up with a tractable mathematical model that could capture the phenomenology of collective synchronization by suggesting that oscillators were coupled by a sinusoidal function of their phase differences. Later, Yoshiki Kuramoto together with Hidetsugu Sakaguchi presented a generalization of the previous limit-cycle set of oscillators Kuramoto’s model which incorporated a constant phase lag between oscillators. Subsequent studies of the model included the network structure within the model together with the global shift. For a wide range of the phase lag values, the system becomes synchronized to a resulting frequency, i.e., the dynamics reaches a stationary state. In the original work of Kuramoto and Sakaguchi and in most of the consequent later studies, a uniform distribution of phase lag parameters is customarily assumed. However, the intrinsic properties of nodes – that assuredly represent the constituents of real systems – do not need be identical but distributed non-homogeneously among the population. This thesis contributes to the understanding of the Kuramoto-Sakaguchi model with a generalization for nonhomogeneous phase lag parameter distribution. Considering different scenarios concerning the distribution of the frustration parameter among the oscillators represents a major step towards the extension of the original model and provides significant novel insights into the structure and function of the considered network. The first setting that the present thesis considers consists in perturbing the stationary state of the system by introducing a non-zero phase lag shift into the dynamics of a single node. The aim of this work is to sort the nodes by their potential effect on the whole network when a change on their individual dynamics spreads over the entire oscillatory system by disrupting the otherwise synchronized state. In particular, we define functionability, a novel centrality measure that addresses the question of which are the nodes that, when individually perturbed, are best able to move the system away from the fully synchronized state. This issue may be relevant for the identification of critical nodes that are either beneficial – by enabling access to a broader spectrum of states – or harmful – by destroying the overall synchronization. The second scenario that the present thesis addresses considers a more general configuration in which the phase lag parameter is an intrinsic property of each node, not necessarily zero, and hence exploring the potential heterogeneity of the frustration among oscillators. We obtain the analytical solution of frustration parameters so as to achieve any phase configuration, by linearizing the most general model. We also address the fact that the question ’among all the possible solutions, which is the one that makes the system achieve a particular phase configuration with the minimum required cost?’ is of particular relevance when we consider the plausible real nature of the system. Finally, the homogenous distribution of phase lag parameters is revisited in the last scenario. As studied in the literature, a certain degree of symmetry is an attribute of real-world networks. Nevertheless, beyond structural or topological symmetry, one should consider the fact that real- world networks are exposed to fluctuations or errors, as well as mistaken insertions or removals. In the present thesis, we provide an alternative notion to approximate symmetries, which we call ‘Quasi-Symmetries’ and are defined such that they remain free to impose any invariance of a particular network property and are obtained from the stationary state of the Kuramoto-Sakaguchi model with a homogeneous phase lag distribution. A first contribution is exploring the distributions of structural similarity among all pairs of nodes. Secondly, we define the ‘dual network’, a weighted network –and its corresponding binarized counterpart– that effectively encloses all the information of quasi-symmetries in the original one.[cat] La major part dels sistemes complexos presents en la natura i la societat es poden descriure com a xarxes complexes. Molts d’aquests sistemes es poden modelitzar matemàticament com un sistema oscil·latori, on les unitats queden acoblades amb els components veïns a través de les connexions de la xarxa. Yoshiki Kuramoto i Hidetsugu Sakaguchi van presentar la generalització del ben conegut model d’oscil·ladors de Kuramoto, on s’incorporava un terme de desfasament entre parelles d’oscil·ladors. Aquesta tesi contribueix en la comprensió d’aquest model, tot considerant una distribució no homogènia d’aquest paràmetre de desfasament o frustració. S’han considerat tres escenaris diferents, tots ells donant lloc a resultats que permeten una millor descripció de l’estructura i funció de la xarxa que s’està considerant. Una primera configuració consisteix en pertorbar l’estat estacionari tot introduint un desfasament en la dinàmica d’un node de manera aïllada. Seguidament, definim la funcionabilitat, una mesura de centralitat única que respon a la pregunta de, quins nodes, quan són pertorbats individualment, són més capaços d’allunyar el sistema de l’estat sincronitzat. Aquest fet podria suposar un comportament beneficiós o perjudicial per sistemes reals. El segon escenari considera la configuració més flexible, explorant la potencial heterogeneïtat dels paràmetres de frustració dels diferents nodes. Obtenim la solució analítica d’aquesta distribució per tal d’assolir qualsevol configuració de les fases dels oscil·ladors, a través de la linearització del model. També contestem a la pregunta: “de totes les possibles solucions, quina és la que implica un menor cost per tal d’assolir una configuració en particular?”. Finalment, en l’últim escenari, proporcionem una definició alternativa al concepte de simetria aproximada d’una xarxa, i que anomenem “Quasi simetries”. Aquestes són definides sense imposar invariàncies en les propietats del sistema, sinó que s’obtenen de l’estat estacionari del model de Kuramoto-Sakaguchi model, tot considerant una distribució homogènia dels paràmetres de frustració

Optimal cost tuning of frustration: Achieving desired states in the Kuramoto-Sakaguchi model

There are numerous examples of studied real-world systems that can be described as dynamical systems characterized by individual phases and coupled in a networklike structure. Within the framework of oscillatory models, much attention has been devoted to the Kuramoto model, which considers a collection of oscillators interacting through a sinus function of the phase differences. In this paper, we draw on an extension of the Kuramoto model, called the Kuramoto-Sakaguchi model, which adds a phase lag parameter to each node. We construct a general formalism that allows us to compute the set of lag parameters that may lead to any phase configuration within a linear approximation. In particular, we devote special attention to the cases of full synchronization and symmetric configurations. We show that the set of natural frequencies, phase lag parameters, and phases at the steady state is coupled by an equation and a continuous spectra of solutions is feasible. In order to quantify the system's strain to achieve that particular configuration, we define a cost function and compute the optimal set of parameters that minimizes it. Despite considering a linear approximation of the model, we show that the obtained tuned parameters for the case of full synchronization enhance frequency synchronization in the nonlinear model as well

Functionability in complex networks: Leading nodes for the transition from structural to functional networks through remote asynchronization

Complex networks are essentially heterogeneous not only in the basic properties of the constituent nodes, such as their degree, but also in the effects that these have on the global dynamical properties of the network. Networks of coupled identical phase oscillators are good examples for analyzing these effects, since an overall synchronized state can be considered a reference state. A small variation of intrinsic node parameters may cause the system to move away from synchronization, and a new phase-locked stationary state can be achieved. We propose a measure of phase dispersion that quantifies the functional response of the system to a given local perturbation. As a particular implementation, we propose a variation of the standard Kuramoto model in which the nodes of a complex network interact with their neighboring nodes, by including a node-dependent frustration parameter. The final stationary phase-locked state now depends on the particular frustration parameter at each node and also on the network topology. We exploit this scenario by introducing individual frustration parameters and measuring what their effect on the whole network is, measured in terms of the phase dispersion, which depends only on the topology of the network and on the choice of the particular node that is perturbed. This enables us to define a characteristic of the node, its functionability, that can be computed analytically in terms of the network topology. Finally, we provide a thorough comparison with other centrality measures

A complex network framework to model cognition: unveiling correlation structures from connectivity

Several approaches to cognition and intelligence research rely on statistics-based model testing, namely, factor analysis. In the present work, we exploit the emerging dynamical system perspective putting the focus on the role of the network topology underlying the relationships between cognitive processes. We go through a couple of models of distinct cognitive phenomena and yet find the conditions for them to be mathematically equivalent. We find a nontrivial attractor of the system that corresponds to the exact definition of a well-known network centrality and hence stresses the interplay between the dynamics and the underlying network connectivity, showing that both of the two are relevant. Correlation matrices evince there must be a meaningful structure underlying real data. Nevertheless, the true architecture regarding the connectivity between cognitive processes is still a burning issue of research. Regardless of the network considered, it is always possible to recover a positive manifold of correlations. Furthermore, we show that different network topologies lead to different plausible statistical models concerning the correlation structure, ranging from one to multiple factor models and richer correlation structures

Quasi-symmetries in complex networks: a dynamical model approach

The existence of symmetries in complex networks has a significant effect on network dynamic behaviour. Nevertheless, beyond topological symmetry, one should consider the fact that real-world networks are exposed to fluctuations or errors, as well as mistaken insertions or removals. Therefore, the resulting approximate symmetries remain hidden to standard symmetry analysis¿fully accomplished by discrete algebra software. There have been a number of attempts to deal with approximate symmetries. In the present work we provide an alternative notion of these weaker symmetries, which we call 'quasi-symmetries'. Differently from other definitions, quasi-symmetries remain free to impose any invariance of a particular network property and they are obtained from the phase differences at the steady-state configuration of an oscillatory dynamical model: the Kuramoto-Sakaguchi model. The analysis of quasi-symmetries unveils otherwise hidden real-world networks attributes. On the one hand, we provide a benchmark to determine whether a network has a more complex pattern than that of a random network with regard to quasi-symmetries, namely, if it is structured into separate quasi-symmetric groups of nodes. On the other hand, we define the 'dual-network', a weighted network (and its corresponding binnarized counterpart) that effectively encodes all the information of quasi-symmetries in the original network. The latter is a powerful instrument for obtaining worthwhile insights about node centrality (obtaining the nodes that are unique from that act as imitators with respect to the others) and community detection (quasi-symmetric groups of nodes)

A complex network framework to model cognition: unveiling correlation structures from connectivity

Several approaches to cognition and intelligence research rely on statistics-based model testing, namely, factor analysis. In the present work, we exploit the emerging dynamical system perspective putting the focus on the role of the network topology underlying the relationships between cognitive processes. We go through a couple of models of distinct cognitive phenomena and yet find the conditions for them to be mathematically equivalent. We find a nontrivial attractor of the system that corresponds to the exact definition of a well-known network centrality and hence stresses the interplay between the dynamics and the underlying network connectivity, showing that both of the two are relevant. Correlation matrices evince there must be a meaningful structure underlying real data. Nevertheless, the true architecture regarding the connectivity between cognitive processes is still a burning issue of research. Regardless of the network considered, it is always possible to recover a positive manifold of correlations. Furthermore, we show that different network topologies lead to different plausible statistical models concerning the correlation structure, ranging from one to multiple factor models and richer correlation structures

Pla d’actuació per prevenir els efectes de les onades de calor sobre la salut (POCS 2021): informe de les actuacions realitzades i dels resultats obtinguts 

Calor elevada; Mesures de prevenció; ResultatsCalor elevada; Medidas de prevención; ResultadosHigh heat; Preventive measures; ResultsEl Pla d'actuació per prevenir els efectes de les onades de calor sobre la salut (POCS) a Catalunya és un pla estacional que s'activa anualment de l'1 de juny fins al 30 de setembre. Està coordinat per l'Agència de Salut Pública de Catalunya (ASPCAT) i el Servei Català de la Salut i compta amb la participació de més d'una quinzena d'entitats. L’informe presenta les actuacions realitzades i dels resultats obtinguts en relació al pla d’actuació per prevenir els efectes de les onades de calor sobre la salut (POCS) durant l'estiu de l'any 2021