# Sinchronization in complex networks

#### Albert Díaz-Guilera

Complexity is usually understood as the emergence of nontrivial collective behaviors from the dynamical evolution of simple units that are interconnected. Synchronization corresponds to a global behavior of a population of individual oscillators. We have basically analyzed the influence of the topological features of complex networks in the dynamical properties of the system.

Relevant references

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

Gemma Rosell-Tarragó and Albert Díaz-Guilera

Physical Review E
(2021)

abstract

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

Gemma Rosell-Tarrago, Albert Diaz-Guilera

Chaos
(2020)

abstract

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.

### Flocking-enhanced social contagion

Demian Levis, Albert Diaz-Guilera, Ignacio Pagonabarraga, and Michele Starnini

Phys Rev Research
(2020)

abstract

Populations of mobile agents—animal groups, robot swarms, or crowds of people—self-organize into a large diversity of states as a result of information exchanges with their surroundings. While in many situations of interest the motion of the agents is driven by the transmission of information from neighboring peers, previous modeling efforts have overlooked the feedback between motion and information spreading. Here we show that such a feedback results in contagion enhanced by flocking. We introduce a reference model in which agents carry an internal state whose dynamics is governed by the susceptible-infected-susceptible (SIS) epidemic process, characterizing the spread of information in the population and affecting the way they move in space. This feedback triggers flocking, which is able to foster social contagion by reducing the epidemic threshold with respect to the limit in which agents interact globally. The velocity of the agents controls both the epidemic threshold and the emergence of complex spatial structures, or swarms. By bridging together soft active matter physics and modeling of social dynamics, we shed light upon a positive feedback mechanism driving the self-organization of mobile agents in complex systems.

### Synchronization invariance under network structural transformations

Lluís Arola-Fernández, Albert Díaz-Guilera, and Alex Arenas

Physical Review E
(2018)

abstract

Synchronization processes are ubiquitous despite the many connectivity patterns that complex systems can show. Usually, the emergence of synchrony is a macroscopic observable; however, the microscopic details of the system, as, e.g., the underlying network of interactions, is many times partially or totally unknown. We already know that different interaction structures can give rise to a common functionality, understood as a common macroscopic observable. Building upon this fact, here we propose network transformations that keep the collective behavior of a large system of Kuramoto oscillators invariant. We derive a method based on information theory principles, that allows us to adjust the weights of the structural interactions to map random homogeneous in-degree networks into random heterogeneous networks and vice versa, keeping synchronization values invariant. The results of the proposed transformations reveal an interesting principle; heterogeneous networks can be mapped to homogeneous ones with local information, but the reverse process needs to exploit higher-order information. The formalism provides analytical insight to tackle real complex scenarios when dealing with uncertainty in the measurements of the underlying connectivity structure.

### Collective motion of active Brownian particles with polar alignment

Aitor Martín-Gómez, Demian Levis, Albert Díaz-Guilera, and Ignacio Pagonabarraga

Soft Matter
(2018)

abstract

We present a comprehensive computational study of the collective behavior emerging from the competition between self-propulsion, excluded volume interactions and velocity-alignment in a two-dimensional model of active particles. We consider an extension of the active brownian particles model where the self-propulsion direction of the particles aligns with the one of their neighbors. We analyze the onset of collective motion (flocking) in a low-density regime (10% surface area) and show that it is mainly controlled by the strength of velocity-alignment interactions: the competition between self-propulsion and crowding effects plays a minor role in the emergence of flocking. However, above the flocking threshold, the system presents a richer pattern formation scenario than analogous models without alignment interactions (active brownian particles) or excluded volume effects (Vicsek-like models). Depending on the parameter regime, the structure of the system is characterized by either a broad distribution of finite-sized polar clusters or the presence of an amorphous, highly fluctuating, large-scale traveling structure which can take a lane-like or band-like form (and usually a hybrid structure which is halfway in between both). We establish a phase diagram that summarizes collective behavior of polar active brownian particles and propose a generic mechanism to describe the complexity of the large-scale structures observed in systems of repulsive self-propelled particles.

### Synchronization in Dynamical Networks of Locally Coupled Self-Propelled Oscillators

Demian Levis, Ignacio Pagonabarraga, and Albert Díaz-Guilera

Phys. Rev. X
(2017)

abstract

he emergent cooperative behavior of mobile physical entities exchanging information with their neighborhood has become an important problem across disciplines, thus requiring a general framework to describe such a variety of situations. We introduce a generic model to tackle this problem by considering the synchronization in time-evolving networks generated by the stochastic motion of self-propelled physical interacting units. This framework generalizes previous approaches and brings a unified picture to understand the role played by the network topology, the motion of the agents, and their mutual interaction. This allows us to identify different dynamic regimes where synchronization can be understood from theoretical considerations. While for noninteracting particles, self-propulsion accelerates synchronization, the presence of excluded volume interactions gives rise to a richer scenario, where self-propulsion has a nonmonotonic impact on synchronization. We show that the synchronization of locally coupled mobile oscillators generically proceeds through coarsening, verifying the dynamic scaling hypothesis, with the same scaling laws as the 2D XY model following a quench. Our results shed light into the generic nature of synchronization in time-dependent networks, providing an efficient way to understand more specific situations involving interacting mobile agents.

### Influence of topology in the mobility enhancement of pulse-coupled oscillator synchronization

A. Beardo, L. Prignano, O. Sagarra, and A. Díaz-Guilera

Phys. Rev. E
(2017)

abstract

In this work we revisit the nonmonotonic behavior (NMB) of synchronization time with velocity reported for systems of mobile pulse-coupled oscillators (PCOs). We devise a control parameter that allows us to predict in which range of velocities NMB may occur, also uncovering the conditions allowing us to establish the emergence of NMB based on specific features of the connectivity rule. Specifically, our results show that if the connectivity rule is such that the interaction patterns are sparse and, more importantly, include a large fraction of nonreciprocal interactions, then the system will display NMB. We furthermore provide a microscopic explanation relating the presence of such features of the connectivity patterns to the existence of local clusters unable to synchronize, termed frustrated clusters, for which we also give a precise definition in terms of simple graph concepts. We conclude that, if the probability of finding a frustrated cluster in a system of moving PCOs is high enough, NMB occurs in a predictable range of velocities.

### Chimera-like states in modular neural networks

Johanne Hizanidis, Nikos E. Kouvaris, Gorka Zamora-López, Albert Díaz-Guilera & Chris G. Antonopoulos

Scientific Reports
(2016)

abstract

Chimera states, namely the coexistence of coherent and incoherent behavior, were previously analyzed in complex networks. However, they have not been extensively studied in modular networks. Here, we consider a neural network inspired by the connectome of the *C. elegans* soil worm, organized into six interconnected communities, where neurons obey chaotic bursting dynamics. Neurons are assumed to be connected with electrical synapses within their communities and with chemical synapses across them. As our numerical simulations reveal, the coaction of these two types of coupling can shape the dynamics in such a way that chimera-like states can happen. They consist of a fraction of synchronized neurons which belong to the larger communities, and a fraction of desynchronized neurons which are part of smaller communities. In addition to the Kuramoto order parameter *ρ*, we also employ other measures of coherence, such as the chimera-like *χ* and metastability *λ* indices, which quantify the degree of synchronization among communities and along time, respectively. We perform the same analysis for networks that share common features with the *C. elegans* neural network. Similar results suggest that under certain assumptions, chimera-like states are prominent phenomena in modular networks, and might provide insight for the behavior of more complex modular networks.

### Synchronization of mobile chaotic oscillator networks

N. Fujiwara, J. Kurths, A. Diaz-Guilera

Chaos
(2016)

abstract

We study synchronization of systems in which agents holding chaotic oscillators move in a two-dimensional plane and interact with nearby ones forming a time dependent network. Due to the uncertainty in observing other agents' states, we assume that the interaction contains a certain amount of noise that turns out to be relevant for chaotic dynamics. We find that a synchronization transition takes place by changing a control parameter. But this transition depends on the relative dynamic scale of motion and interaction. When thetopology change is slow, we observe an intermittent switching between laminar and burst states close to the transition due to small noise. This novel type of synchronization transition and intermittency can happen even when complete synchronization is linearly stable in the absence of noise. We show that the linear stability of the synchronized state is not a sufficient condition for its stability due to strong fluctuations of the transverse Lyapunov exponent associated with a slow network topology change. Since this effect can be observed within the linearized dynamics, we can expect such an effect in the temporal networks with noisy chaotic oscillators, irrespective of the details of the oscillator dynamics. When the topology change is fast, a linearized approximation describes well the dynamics towards synchrony. These results imply that the fluctuations of the finite-time transverse Lyapunov exponent should also be taken into account to estimatesynchronization of the mobile contact networks.

### Self-Organized Stationary Patterns in Networks of Bistable Chemical Reactions

Nikos E. Kouvaris, Michael Sebek, Alexander S. Mikhailov and Istvan Z. Kiss

Angewandte Chemie International Edition
(2016)

abstract

Experiments with networks of discrete reactive bistable electrochemical elements organized in regular and nonregular tree networks are presented to confirm an alternative to the Turing mechanism for the formation of self-organized stationary patterns. The results show that the pattern formation can be described by the identification of domains that can be activated individually or in combinations. The method also enabled the localization of chemical reactions to network substructures and the identification of critical sites whose activation results in complete activation of the system. Although the experiments were performed with a specific nickel electrodissolution system, they reproduced all the salient dynamic behavior of a general network model with a single nonlinearity parameter. Thus, the considered pattern-formation mechanism is very robust, and similar behavior can be expected in other natural or engineered networked systems that exhibit, at least locally, a treelike structure.

### Chimera states in a network-organized public goods game with mutations

Nikos E. Kouvaris, Ruben J. Requejo, Johanne Hizanidis and Albert Diaz-Guilera

Chaos
(2016)

abstract

We found that a network-organized metapopulation of cooperators, defectors, and destructive agents playing the public goods game with mutations can collectively reach global synchronization or chimera states. Global synchronization is accompanied by a collective periodic burst of cooperation, whereas chimera states reflect the tendency of the networked metapopulation to be fragmented in clusters of synchronous and incoherent bursts of cooperation. Numerical simulations have shown that the system's dynamics switches between these two steady states through a first order transition. Depending on the parameters determining the dynamical and topological properties, chimera states with different numbers of coherent and incoherent clusters are observed. Our results present the first systematic study of chimera states and their characterization in the context of evolutionary game theory. This provides a valuable insight into the details of their occurrence, extending the relevance of such states to natural and social systems.

### Remote Synchronization Reveals Network Symmetries and Functional Modules

Vincenzo Nicosia, Miguel Valencia, Mario Chavez, Albert Diaz-Guilera, Vito Latora,

PHYSICAL REVIEW LETTERS
(2013)

abstract

We study a Kuramoto model in which the oscillators are associated with the nodes of a complex network and the interactions include a phase frustration, thus preventing full synchronization. The system organizes into a regime of remote synchronization where pairs of nodes with the same network symmetry are fully synchronized, despite their distance on the graph. We provide analytical arguments to explain this result, and we show how the frustration parameter affects the distribution of phases. An application to brain networks suggests that anatomical symmetry plays a role in neural synchronization by determining correlated functional modules across distant locations. DOI: 10.1103/PhysRevLett.110.174102

### Tuning Synchronization of Integrate-and-Fire Oscillators through Mobility

L. Prignano, O. Sagarra, Albert Diaz-Guilera,

PHYSICAL REVIEW LETTERS
(2013)

abstract

We analyze the emergence of synchronization in a population of moving integrate-and-fire oscillators. Oscillators, while moving on a plane, interact with their nearest neighbor upon firing time. We discover a nonmonotonic dependence of the synchronization time on the velocity of the agents. Moreover, we find that mechanisms that drive synchronization are different for different dynamical regimes. We report the extreme situation where an interplay between the time scales involved in the dynamical processes completely inhibits the achievement of a coherent state. We also provide estimators for the transitions between the different regimes. DOI: 10.1103/PhysRevLett.110.114101

### How Does Network Topology Determine the Synchronization Threshold in a Network of Oscillators?

Lubos Buzna, Sergi Lozano, Albert Díaz-Guilera

Operations Research Proceedings
(2012)

abstract

The reliable functioning of an electrical power grid is dependent on the proper interaction between many of its elements. What is critically important is its ability to keep the frequency across the entire system stable. Considering a simple mathematical model, representing the network of coupled oscillators, we study the stability of frequency synchronization. This model can be interpreted as the dynamical representation of frequency synchronization between the power producing and power consuming units. Assuming a uniform network, we analytically derive the formula estimating the relation between the minimum coupling strength required to ensure the frequency synchronization and the network parameters. This minimum value can be efficiently found by solving a binary optimization problem, using universal solver XPRESS, even for large networks,. We validate the accuracy of the analytical estimation by comparing it with numerical simulations on the realistic network describing the European interconnected high-voltage electricity system, finding good agreement. Moreover, by repeatedly solving the binary optimization problem, we can test the stability of the frequency synchronization with respect to link removals. As the threshold value changes only in few cases, we conclude that the network is resilient in this regard. Since the synchronization threshold depends on the network partition representing the synchronization bottleneck, we also evaluate which network areas become critical for the synchronization when removing single links.

### Chimera states in a network-organized public goods game with mutations

Nikos E. Kouvaris, Ruben J. Requejo, Johanne Hizanidis and Albert Diaz-Guilera

abstract

We show that a network-organized metapopulation of cooperators, defectors and destructors playing the public goods game with mutations, can collectively reach global synchronization or chimera states. Global synchronization is accompanied by a collective periodic burst of cooperation, whereas chimera states reflect the tendency of the networked metapopulation to be fragmented in clusters of synchronous and incoherent bursts of cooperation. Numerical simulations have shown that the system's dynamics alternates between these two steady states through a first order transition. Depending on the parameters determining the dynamical and topological properties, chimera states with different numbers of (in)coherent clusters are observed. Our results present the first example of chimera states in the context of evolutionary game theory. This provides a valuable insight into the details of their occurrence, extending the relevance of such states to natural and social systems.